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An Introduction to Grids Graphs and Networks

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This is a mathematics book about graphs, networks and grids. Graphs are studied through grids.

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Page 1: An Introduction to Grids Graphs and Networks
Page 2: An Introduction to Grids Graphs and Networks

AN INTRODUCTION TO GRIDS,GRAPHS, AND NETWORKS

Page 3: An Introduction to Grids Graphs and Networks
Page 4: An Introduction to Grids Graphs and Networks

AN INTRODUCTIONTO GRIDS, GRAPHS,AND NETWORKS

C. Pozrikidis

3

Page 5: An Introduction to Grids Graphs and Networks

3

Oxford University Press is a department of the University of Oxford. It furthers the University’sobjective of excellence in research, scholarship, and education by publishing worldwide.

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You must not circulate this work in any other formand you must impose this same condition on any acquirer.

Library of Congress Cataloging-in-Publication DataPozrikidis, C. (Constantine), 1958– author.

An introduction to grids, graphs, and networks / C. Pozrikidis.p. cm.

Includes bibliographical references and index.ISBN 978–0–19–999672–8 (alk. paper)

1. Graph theory. 2. Differential equations, Partial—Numerical solutions. 3. Finite differences. I. Title.QA166.P69 2014

511’.5—dc232013048508

1 3 5 7 9 8 6 4 2

Printed in the United States of Americaon acid-free paper

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CONTENTS

Preface xi

1. One-Dimensional Grids 1

1.1. Poisson Equation in One Dimension 1

1.2. Dirichlet Boundary Condition at Both Ends 3

1.3. Neumann–Dirichlet Boundary Conditions 6

1.4. Dirichlet–Neumann Boundary Conditions 8

1.5. Neumann Boundary Conditions 10

1.6. Periodic Boundary Conditions 13

1.7. One-Dimensional Graphs 161.7.1. Graph Laplacian 171.7.2. Adjacency Matrix 181.7.3. Connectivity Lists and Oriented Incidence Matrix 19

1.8. Periodic One-Dimensional Graphs 201.8.1. Periodic Adjacency Matrix 211.8.2. Periodic Oriented Incidence Matrix 221.8.3. Fourier Expansions 221.8.4. Cosine Fourier Expansion 241.8.5. Sine Fourier Expansion 24

2. Graphs and Networks 26

2.1. Elements of Graph Theory 262.1.1. Adjacency Matrix 262.1.2. Node Degrees 282.1.3. The Complete Graph 292.1.4. Complement of a Graph 292.1.5. Connectivity Lists and the Oriented Incidence Matrix 30

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vi / / CONTENTS

2.1.6. Connected and Unconnected Graphs 302.1.7. Pairwise Distance and Diameter 302.1.8. Trees 312.1.9. Random and Real-Life Networks 31

2.2. Laplacian Matrix 322.2.1. Properties of the Laplacian Matrix 332.2.2. Complete Graph 342.2.3. Estimates of Eigenvalues 352.2.4. Spanning Trees 362.2.5. Spectral Expansion 362.2.6. Spectral Partitioning 362.2.7. Complement of a Graph 382.2.8. Normalized Laplacian 382.2.9. Graph Breakup 39

2.3. Cubic Network 39

2.4. Fabricated Networks 412.4.1. Finite-Element Network on a Disk 422.4.2. Finite-Element Network on a Square 432.4.3. Delaunay Triangulation of an Arbitrary Set of Nodes 432.4.4. Delaunay Triangulation of a Perturbed Cartesian Grid 432.4.5. Finite Element Network Descending from an Octahedron 442.4.6. Finite Element Network Descending from an Icosahedron 45

2.5. Link Removal and Addition 462.5.1. Single and Multiple Link 472.5.2. Link Addition 49

2.6. Infinite Lattices 502.6.1. Bravais Lattices 502.6.2. Archimedean Lattices 532.6.3. Laves Lattices 562.6.4. Other Two-Dimensional Lattices 572.6.5. Cubic Lattices 58

2.7. Percolation Thresholds 592.7.1. Link (Bond) Percolation Threshold 592.7.2. Node Percolation Threshold 612.7.3. Computation of Percolation Thresholds 62

3. Spectra of Lattices 67

3.1. Square Lattice 673.1.1. Isolated Network 683.1.2. Periodic Strip 69

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CONTENTS // vii

3.1.3. Doubly Periodic Network 733.1.4. Doubly Periodic Sheared Network 77

3.2. Möbius Strips 793.2.1. Horizontal Strip 803.2.2. Vertical Strip 833.2.3. Klein Bottle 84

3.3. Hexagonal Lattice 863.3.1. Isolated Network 873.3.2. Doubly Periodic Network 893.3.3. Alternative Node Indexing 92

3.4. Modified Union Jack Lattice 933.4.1. Isolated Network 943.4.2. Doubly Periodic Network 95

3.5. Honeycomb Lattice 983.5.1. Isolated Network 993.5.2. Brick Representation 1013.5.3. Doubly Periodic Network 1023.5.4. Alternative Node Indexing 110

3.6. Kagomé Lattice 1113.6.1. Isolated Network 1123.6.2. Doubly Periodic Network 115

3.7. Simple Cubic Lattice 122

3.8. Body-Centered Cubic (bcc) Lattice 124

3.9. Face-Centered Cubic (fcc) Lattice 126

4. Network Transport 130

4.1. Transport Laws and Conventions 1304.1.1. Isolated and Embedded Networks 1304.1.2. Nodal Sources 1314.1.3. Linear Transport 1324.1.4. Nonlinear Transport 133

4.2. Uniform Conductances 1334.2.1. Isolated Networks 1344.2.2. Embedded Networks 134

4.3. Arbitrary Conductances 1354.3.1. Scaled Conductance Matrix 1364.3.2. Weighed Adjacency Matrix 136

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viii / / CONTENTS

4.3.3. Weighed Node Degrees 1374.3.4. Kirchhoff Matrix 1384.3.5. Weighed Oriented Incidence Matrix 1394.3.6. Properties of the Kirchhoff Matrix 1394.3.7. Normalized Kirchhoff Matrix 1404.3.8. Summary of Notation 141

4.4. Nodal Balances in Arbitrary Networks 1424.4.1. Isolated Networks 1424.4.2. Embedded Networks and the Modified Kirchhoff Matrix 1424.4.3. Properties of the Modified Kirchhoff Matrix 143

4.5. Lattices 1454.5.1. Square Lattice 1454.5.2. Möbius Strip 1494.5.3. Hexagonal Lattice 1504.5.4. Modified Union Jack Lattice 1504.5.5. Simple Cubic Lattice 151

4.6. Finite Difference Grids 153

4.7. Finite Element Grids 1564.7.1. One-Dimensional Grid 1564.7.2. Two-Dimensional Grid 157

5. Green’s Functions 161

5.1. Embedded Networks 1615.1.1. Green’s Function Matrix 1625.1.2. Normalized Green’s Function 163

5.2. Isolated Networks 1645.2.1. Moore–Penrose Green’s Function 1645.2.2. Spectral Expansion 1665.2.3. Normalized Moore–Penrose Green’s Function 1675.2.4. One-Dimensional Network 1685.2.5. Periodic One-Dimensional Network 1695.2.6. Free-Space Green’s Function in One Dimension 1715.2.7. Complete Network 1715.2.8. Discontiguous Networks 172

5.3. Lattice Green’s Functions 1735.3.1. Periodic Green’s Functions 1735.3.2. Free-Space Green’s Functions 175

5.4. Square Lattice 1775.4.1. Periodic Green’s Function 1775.4.2. Free-Space Green’s Function 179

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CONTENTS // ix

5.4.3. Helmholtz Equation Green’s Function 1905.4.4. Kirchhoff Green’s Function 190

5.5. Hexagonal Lattice 1915.5.1. Periodic Green’s Function 1915.5.2. Free-Space Green’s Function 192

5.6. Modified Union Jack Lattice 1965.6.1. Periodic Green’s Function 1975.6.2. Free-Space Green’s Function 198

5.7. Honeycomb Lattice 2005.7.1. Periodic Green’s Function 2015.7.2. Free-Space Green’s Function 203

5.8. Simple Cubic Lattice 2065.8.1. Periodic Green’s Function 2065.8.2. Free-Space Green’s Function 207

5.9. Body-Centered Cubic (bcc) Lattice 209

5.10. Face-Centered Cubic (fcc) Lattice 211

5.11. Free-Space Lattice Green’s Functions 2125.11.1. Probability Lattice Green’s Function 213

5.12. Finite Difference Solution in Terms of Green’s Functions 216

6. Network Performance 220

6.1. Pairwise Resistance 2206.1.1. Embedded Networks 2216.1.2. Isolated Networks 2236.1.3. One-Dimensional Network 2256.1.4. One-Dimensional Periodic Network 2266.1.5. Infinite Lattices 2266.1.6. Triangle Inequality 2276.1.7. Random Walks 227

6.2. Mean Pairwise Resistance 2286.2.1. Spectral Representation 2286.2.2. Complete Network 2296.2.3. One-Dimensional Isolated Network 2296.2.4. One-Dimensional Periodic Network 2306.2.5. Periodic Lattice Patches 231

6.3. Damaged Networks 2346.3.1. Damaged Kirchhoff Matrix 2356.3.2. Embedded Networks 236

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6.3.3. One Damaged Link 2386.3.4. Clipped Links 2406.3.5. Isolated Networks 240

6.4. Reinforced Networks 240

6.5. Damaged Lattices 2426.5.1. One Damaged Link 2426.5.2. Effective-Medium Theory 2456.5.3. Percolation Threshold 246

6.6. Damaged Square Lattice 247

6.7. Damaged Honeycomb Lattice 251

6.8. Damaged Hexagonal Lattice 2556.8.1. Longitudinal Transport 2556.8.2. Lateral Transport 257

Appendices

A.Eigenvalues of Matrices 259

A.1. Eigenvalues and Eigenvectors 259

A.2. The Characteristic Polynomial 260A.2.1. Eigenvalues, Trace, and the Determinant 261A.2.2. Powers, Inverse, and Functions of a Matrix 262A.2.3. Hermitian Matrices 262A.2.4. Diagonal Matrix of Eigenvalues 263

A.3. Eigenvectors and Principal Vectors 263A.3.1. Properties of Eigenvectors 264A.3.2. Left Eigenvectors 264A.3.3. Matrix of Eigenvectors 265A.3.4. Eigenvalues and Eigenvectors of the Adjoint 266A.3.5. Eigenvalues of Positive Definite Hermitian Matrices 266

A.4. Circulant Matrices 267

A.5. Block Circulant Matrices 268

B. The Sherman–Morrison and Woodbury Formulas 269

B.1. The Woodbury Formula 269

B.2. The Sherman–Morrison Formula 273

References 278

Index 281

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PREFACE

Cartesian, curvilinear, and other unstructured grids are used for the numerical so-lution of ordinary and partial differential equations using finite difference, finiteelement, finite volume, and related methods. Graphs are broadly defined as finiteor infinite sets of vertices connected by edges in structured or unstructured config-urations. Infinite lattices and tiled surfaces are described by highly ordered graphsparametrized by an appropriate number of indices. Networks consist of nodes con-nected by physical or abstract links with an assigned conductance in spontaneous orengineered configurations. In physical and engineering applications, networks arevenues for conducting or convecting a transported entity, such as heat, mass, ordigitized information according to a prevailing transport law. The performance ofnetworks is an important topic in the study of complex systems with applications inenergy, material, and information transport.

The analysis of grids, graphs, and networks involves overlapping and comple-mentary topics that benefit from a unified discussion. For example, finite differenceand finite element grids can be regarded as networks whose link conductance isdetermined by the differential equation whose solution is sought as well as by thechosen finite difference or finite element approximation. Particular topics of inter-est include the properties of the node adjacency, Laplacian, and Kirchhoff matrices;the evaluation of percolation thresholds for infinite, periodic, and finite systems; thecomputation of the regular and generalized lattice Green’s function describing theresponse to a nodal source; the pairwise resistance of any two nodes; the overall char-acterization of the network robustness; and the performance of damaged networkswith reference to operational and percolation thresholds.

My goal in this text is to provide a concise and unified introduction to grids,graphs, and networks to a broad audience in the engineering, physical, biological,and social sciences. The approach is practical, in that only the necessary theoreticaland mathematical concepts are introduced. Theory and computation are discussedalongside, and formulas amenable to computer programming are provided. The pre-requisite is familiarity with college-level linear algebra, calculus, and elementarynumerical methods.

One important new concept is the distinction between isolated and embeddednetworks. The former stand in isolation as though they were suspended in vacuum,

xi

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xii / / PREFACE

whereas the latter are connected to exterior nodes where a nodal potential, such astemperature, pressure, or electrical voltage, is specified. Regular Green’s functionsdescribing the discrete field due to a nodal impulse are available in the case of em-bedded or infinite networks, whereas generalized Green’s functions describing thediscrete field due to a nodal impulse in the presence of distributed sinks are availablein the case of isolated networks. Discrete Green’s functions can be used as buildingblocks for computing general solutions subject to given constraints.

This book is suitable for self-study and as a text in an upper-level undergraduateor entry-level graduate course in sciences, engineering, and applied mathematics.The material serves as a reference of terms and concepts and as a resource of topicsfor further study.

C. PozrikidisSeptember, 2013

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AN INTRODUCTION TO GRIDS,GRAPHS, AND NETWORKS

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/ / / 1 / / / ONE-DIMENSIONAL GRIDS

Afinite difference grid for solving ordinary or partial differentialequations consists of rectilinear or curvilinear grid lines that can be regarded as con-veying links intersecting at nodes. This interpretation provides us with a point ofdeparture for making an analogy between numerical grids, mathematical graphs,and physical or abstract networks. We begin in this chapter by developing finitedifference equations for an elementary ordinary equation with the objective of iden-tifying similarities between grids and graphs, and then we generalize the frameworkto higher dimensions.

1.1 POISSON EQUATION IN ONE DIMENSION

Consider the Poisson equation in one dimension for an unknown function of onevariable, f (x),

(1.1.1)d2f

dx2+ g(x) = 0,

to be solved in a finite domain, [a, b], where g(x) is a given source function. Wheng(x) = 0, the Poisson equation reduces to Laplace’s equation. When g(x) = αf (x),the Poisson equation reduces to Helmholtz’s equation, where α is a real or complexconstant.

A numerical solution can be found on a uniform finite difference grid with Kdivisions defined by K + 1 nodes, as shown in Figure 1.1.1. Nodes numbered 0 and

2 i i + 1i − 1

x0 1

b

K + 1K K + 2

a

FIGURE 1.1.1 A finite difference with K uniform divisions along the x axis.Dirichlet or Neumann boundary conditions are specified at the two ends

of the solution domain.

1

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2 / / AN INTRODUCTION TO GRIDS, GRAPHS, AND NETWORKS

K+2 are phantom nodes, lying outside the solution domain, introduced to implementthe Neumann boundary condition, when specified, as discussed later in this chapter.

Applying the Poisson equation at the ith node, approximating the second deriva-tive with a central difference by setting

(1.1.2) f ′′(xi) � fi–1 – 2fi + fi+1�x2

+ O(�x2)

with an error of order �x2, and rearranging, we obtain the difference equation

(1.1.3) –fi–1 + 2fi – fi+1 = �x2gi

to be applied at an appropriate number of nodes. To simplify the notation, we havedenoted

(1.1.4) fi ≡ f (xi), gi ≡ g(xi).

The signs on the left- and right-hand sides of (1.1.3) were chosen intentionally toconform with standard notation in graph theory regarding the Laplacian, as discussedin Section 1.7.

Collecting all available difference equations and implementing the boundary con-ditions provides us with a system of linear algebraic equations for a suitable numberof unknown nodal values contained in a solution vector, ψ ,

(1.1.5) L · ψ = b,

where the centered dot denotes the matrix–vector product. The size and specific formof the coefficient matrix, L, solution vector, ψ , and vector on the right-hand side, b,depend on the choice of boundary conditions. Several possibilities are discussed inthis chapter.

FactorizationWe will see that, for any type of boundary conditions—Neumann, Dirichlet, orperiodic—the coefficient matrix of the linear system admits the factorization

(1.1.6) L = R · RT ,

where R is a square or rectangular matrix, the superscript T denotes the matrixtranspose, and the centered dot denotes the usual matrix product (e.g., [35]). This fac-torization can be regarded as the discrete counterpart of the definition of the secondderivative as the sequential application of the first derivative,

(1.1.7)d2

dx2=d

dx

d

dx.

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One-Dimensional Gr ids / / 3

It is important to note that the commutative property R · RT = RT · R is not alwayssatisfied.

The counterpart of the factorization (1.1.6) in n dimensions is

(1.1.8) ∇2 = ∇ · ∇,

where

(1.1.9) ∇2 =∂2

∂x21+∂2

∂x21+ · · · + ∂

2

∂x2n

is the scalar Laplacian operator,

(1.1.10) ∇ =

(∂

∂x1+∂

∂x2+ · · · + ∂

∂xn

)is the vectorial gradient operator, and the centered dot denotes the inner vectorproduct. In two dimensions n = 2, and in three dimensions n = 3.

Exercise

1.1.1 Helmholtz equation

Write the counterpart of the difference equation (1.1.3) for the Helmholtz equationin one dimension,

(1.1.11)d2f

dx2+ αf = 0,

where α is a real or complex constant.

1.2 DIRICHLET BOUNDARY CONDITION AT BOTH ENDS

When the Dirichlet boundary condition is specified at both ends of the solutiondomain, the first and last values, f1 and fK+1, are known. Collecting the differenceequations (1.1.3) for the interior nodes, i = 2, . . . ,K, we obtain a system of linearequations,

(1.2.1) LDD · ψDD = bDD,

where

(1.2.2) ψDD ≡

⎡⎢⎢⎢⎢⎢⎣f2f3,...fK–1fK

⎤⎥⎥⎥⎥⎥⎦, bDD =

⎡⎢⎢⎢⎢⎢⎣�x2g2 + f1�x2g3...�x2gK–1�x2gK + fK+1

⎤⎥⎥⎥⎥⎥⎦,

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4 / / AN INTRODUCTION TO GRIDS, GRAPHS, AND NETWORKS

are (K – 1)-dimensional vectors and

(1.2.3) LDD =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

2 –1 0 · · · 0 0 0–1 2 –1 · · · 0 0 00 –1 2 · · · 0 0 0...

......

. . ....

......

0 0 0 · · · 2 –1 00 0 0 · · · –1 2 –10 0 0 · · · 0 –1 2

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦is (K – 1) × (K – 1) symmetric tridiagonal Toeplitz matrix. By definition, a Toeplitzmatrix consists of constant diagonal lines. The superscript DD emphasizes that theDirichlet condition is specified at both ends.

DecompositionWe can decompose

(1.2.4) LDD = 2 I –�DD,

where I is the (K – 1) × (K – 1) identity matrix and

(1.2.5) �DD =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 1 0 · · · 0 0 01 0 1 · · · 0 0 00 1 0 · · · 0 0 0...

......

. . ....

......

0 0 0 · · · 0 1 00 0 0 · · · 1 0 10 0 0 · · · 0 1 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦is a (K – 1) × (K – 1) symmetric bidiagonal Toeplitz matrix with zeros along thediagonal.

Eigenvalues and EigenvectorsThe eigenvalues of the matrices �DD and LDD are

(1.2.6) λ�m = 2 cosαm

and

(1.2.7) λLm = 2 – 2 cosαm = 4 sin2(12αm

),

where

(1.2.8) αm =m

for m = 1, . . . ,K – 1.

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One-Dimensional Gr ids / / 5

The corresponding shared eigenvectors, u(m), normalized so that their length isequal to unity, u(m) · u(m) = 1, are

(1.2.9) u(m)j =

(2

K

)1/2sin(jαm)

for m, j = 1, . . . ,K –1. It is interesting that all eigenvectors are pure harmonic waves,with higher-order eigenvalues corresponding to shorter wavelengths.

FactorizationWe can factorize

(1.2.10) LDD = RDD · RDDT ,

where

(1.2.11) RDD =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

–1 1 0 · · · 0 0 0 00 –1 1 0 · · · 0 0 00 0 –1 1 · · · 0 0 0...

......

. . ....

......

...0 0 0 · · · –1 1 00 0 0 · · · 0 –1 1 00 0 0 · · · 0 0 –1 1

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦is a rectangular (K – 1)×K matrix implementing forward difference approximationsto the first derivative. The transpose of RDD,

(1.2.12) RDDT =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

–1 0 0 · · · 0 0 01 –1 0 · · · 0 0 00 1 –1 · · · 0 0 0...

......

. . ....

......

0 0 0 · · · –1 0 00 0 0 · · · 1 –1 00 0 0 · · · 0 1 –10 0 0 · · · 0 0 1

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦,

is a rectangular K×(K–1) matrix implementing backward difference approximationsto the first derivative.

Exercises

1.2.1 Sinusoidal Source

Solve the linear system (1.2.1) for a = 0 and g(x) = γ sin2(2πx/b), where γ is aconstant. The boundary conditions specify that f (0) = 0 and f (b) = fb, where fb is a

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6 / / AN INTRODUCTION TO GRIDS, GRAPHS, AND NETWORKS

given constant. Carry out computations for K = 2, 4, 8, 16, and 32, and discuss theaccuracy of the numerical results with reference to the exact solution.

1.2.2 Factorization

Confirm the factorization (1.2.10).

1.3 NEUMANN–DIRICHLET BOUNDARY CONDITIONS

Now assume that the Neumann boundary condition is prescribed at the right end ofthe solution domain, x = a, specifying that

(1.3.1) f ′(x1) = –q1,

where q1 is a given constant, while the Dirichlet boundary condition is prescribedat the left end of the solution domain, x = b. specifying the value of fK+1. Fol-lowing standard practice, we introduce a phantom node labeled zero, as shown inFigure 1.1.1, approximate the first derivative with second-order accuracy using acentral difference as

(1.3.2) f ′(x1) � f2 – f02�x

+ O(�x2),

and write

(1.3.3) f0 = f2 + 2�x q1.

The difference equations for i = 1, . . . ,K provide us with a system of linearequations,

(1.3.4) LND · ψND = bND,

where

(1.3.5) ψND ≡

⎡⎢⎢⎢⎢⎢⎣f1f2,...fK–1fK

⎤⎥⎥⎥⎥⎥⎦, bND =

⎡⎢⎢⎢⎢⎢⎣12 �x

2 g1 +�x q1�x2g2

...�x2gK–1�x2gK + fK+1

⎤⎥⎥⎥⎥⎥⎦,

are K-dimensional vectors and

(1.3.6) LND =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

1 –1 0 · · · 0 0 0–1 2 –1 · · · 0 0 00 –1 2 · · · 0 0 0...

......

. . ....

......

0 0 0 · · · 2 –1 00 0 0 · · · –1 2 –10 0 0 · · · 0 –1 2

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

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One-Dimensional Gr ids / / 7

is a K×K symmetric, tridiagonal, nearly Toeplitz matrix. If the first diagonal elementwere equal to 2, this matrix would have been a perfect Toeplitz matrix.

DecompositionWe can decompose

(1.3.7) LND = 2 I –�ND,

where I is the K × K identity matrix and

(1.3.8) �ND =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

1 1 0 · · · 0 0 01 0 1 · · · 0 0 00 1 0 · · · 0 0 0...

......

. . ....

......

0 0 0 · · · 0 1 00 0 0 · · · 1 0 10 0 0 · · · 0 1 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦is a K×K square, symmetric, tridiagonal, nearly Toeplitz matrix. If the first diagonalelement were equal to 0, this would have been a perfect Toeplitz matrix.

Eigenvalues and EigenvectorsThe eigenvalues of �ND and LND are

(1.3.9) λ�m = 2 cosαm

and

(1.3.10) λLm = 2 – 2 cosαm = 4 sin2(12 αm

),

where

(1.3.11) αm =m – 1/2

K + 1/2π

for m = 1, . . . ,K.The corresponding shared eigenvectors, u(m), normalized so that their length is

equal to unity, u(m) · u(m) = 1, are

(1.3.12) u(m)j =( 4

2K + 1

)1/2cos[(j – 1

2

)αm

]for m, j = 1, . . . ,K. All eigenvectors are pure harmonic waves.

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8 / / AN INTRODUCTION TO GRIDS, GRAPHS, AND NETWORKS

FactorizationWe can factorize

(1.3.13) LND = RND · RNDT ,

where

(1.3.14) RND =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

–1 0 0 · · · 0 0 01 –1 0 · · · 0 0 00 1 –1 · · · 0 0 0...

......

. . ....

......

0 0 0 · · · –1 0 00 0 0 · · · 1 –1 00 0 0 · · · 0 1 –1

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦is a K × K square lower bidiagonal K × K Toeplitz matrix implementing backwarddifference approximations to the first derivative. Its transpose,

(1.3.15) RNDT =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

–1 1 0 · · · 0 0 00 –1 0 · · · 0 0 00 0 –1 · · · 0 0 0...

......

. . ....

......

0 0 0 · · · –1 1 00 0 0 · · · 0 –1 10 0 0 · · · 0 0 –1

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦,

is a K × K square upper bidiagonal K × K Toeplitz matrix implementing forwarddifference approximations to the first derivative.

Exercise

1.3.1 Factorization

Confirm by direct multiplication the factorization (1.3.13).

1.4 DIRICHLET–NEUMANN BOUNDARY CONDITIONS

In the third case study, we assume that a Dirichlet boundary condition specifyingthe value of f1 is prescribed at the left end of the solution domain, and a Neumannboundary condition specifying that

(1.4.1) f ′(xK+1) = qK+1

is prescribed at the right end of the solution domain, where qK+1 is a given con-stant. We proceed by introducing a phantom node numbered K + 2, as shown inFigure 1.1.1, approximate the first derivative with second-order accuracy as

(1.4.2) f ′(xK+1) � fK+2 – fK2�x

+ O(�x2),

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One-Dimensional Gr ids / / 9

and obtain

(1.4.3) fK+2 = fK + 2�x qK+1.

The difference equations for i = 2,K + 1 provide us with a linear system,

(1.4.4) LDN · ψDN = bDN,

where

(1.4.5) ψDN ≡

⎡⎢⎢⎢⎢⎢⎣f2f3,...fKfK+1

⎤⎥⎥⎥⎥⎥⎦ , bDN =

⎡⎢⎢⎢⎢⎢⎣�x2g2 + f1�x2g3

...�x2gK

12 �x

2 gK+1 +�x qK+1

⎤⎥⎥⎥⎥⎥⎦are K-dimensional vectors and

(1.4.6) LDN =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

2 –1 0 · · · 0 0 0–1 2 –1 · · · 0 0 00 –1 2 · · · 0 0 0...

......

. . ....

......

0 0 0 · · · 2 –1 00 0 0 · · · –1 2 –10 0 0 · · · 0 –1 1

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦is a K × K symmetric, tridiagonal, nearly Toeplitz matrix.

DecompositionWe can decompose

(1.4.7) LDN = 2 I –�DN,

where I is the N × N identity matrix and

(1.4.8) �DN =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 1 0 · · · 0 0 01 0 1 · · · 0 0 00 1 0 · · · 0 0 0...

......

. . ....

......

0 0 0 · · · 0 1 00 0 0 · · · 1 0 10 0 0 · · · 0 1 1

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦is a K × K symmetric, tridiagonal, nearly Toeplitz matrix.

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10 / / AN INTRODUCTION TO GRIDS, GRAPHS, AND NETWORKS

Eigenvalues and EigenvectorsThe eigenvalues of �DN and LDN are

(1.4.9) λ�m = 2 cosαm

and

(1.4.10) λLm = 2 – 2 cosαm = 4 sin2(12αm

),

where

(1.4.11) αm =m – 1

2

K + 12

π

for m = 1, . . . ,K.The corresponding shared eigenvectors, u(m), normalized so that their length is

equal to unity, u(m) · u(m) = 1, are

(1.4.12) u(m)j =

(4

2K + 1

)1/2cos[(K – j + 1

2

)αm

]for m, j = 1, . . . ,K. All eigenvectors are pure harmonic waves.

FactorizationWe can factorize

(1.4.13) LDN = RDN · RDNT = RNDT · RND,

where RDN = RNDT and the matrix RND is given in (1.3.15).

Exercise

1.4.1 Eigenvalues and eigenvectors

Confirm by direct substitution the eigenvalues and eigenvectors given in (1.4.10) and(1.4.12).

1.5 NEUMANN BOUNDARY CONDITIONS

In the fourth and most important case, the Neumann boundary condition is prescribedat both ends of the solution domain,

(1.5.1) f ′(x1) = –q1, f ′(xK+1) = qK+1.

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One-Dimensional Gr ids / / 11

where q1 and qK+1 are two given constants. Working in the familiar way, we collectthe difference equations for i = 1, . . . ,K + 1 into a linear system,

(1.5.2) LNN · ψNN = bNN,

where

(1.5.3) ψNN ≡

⎡⎢⎢⎢⎢⎢⎣f1f2...fKfK+1

⎤⎥⎥⎥⎥⎥⎦, bNN =

⎡⎢⎢⎢⎢⎢⎣12 �x

2g1 +�x q1�x2g2

...�x2gK

12 �x

2gK+1 +�x qK+1

⎤⎥⎥⎥⎥⎥⎦are (K + 1)-dimensional vectors and

(1.5.4) LNN =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

1 –1 0 · · · 0 0 0–1 2 –1 · · · 0 0 00 –1 2 · · · 0 0 0...

......

. . ....

......

0 0 0 · · · 2 –1 00 0 0 · · · –1 2 –10 0 0 · · · 0 –1 1

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦is a (K + 1) × (K + 1) symmetric tridiagonal matrix. If the first and last diagonalelements were equal to 2, this would have been a perfect Toeplitz matrix.

DecompositionWe can decompose

(1.5.5) L = 2 I –�NN,

where I is the (K + 1) × (K + 1) identity matrix and

(1.5.6) �NN =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

1 1 0 · · · 0 0 01 0 1 · · · 0 0 00 1 0 · · · 0 0 0...

......

. . ....

......

0 0 0 · · · 0 1 00 0 0 · · · 1 0 10 0 0 · · · 0 1 1

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦is a nearly upper and lower bidiagonal matrix. Note the presence of two nonzero topand bottom diagonal elements.

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12 / / AN INTRODUCTION TO GRIDS, GRAPHS, AND NETWORKS

Eigenvalues and EigenvectorsThe eigenvalues of �NN and LNN are

(1.5.7) λ�m = 2 cosαm

and

(1.5.8) λLm = 2 – 2 cosαm = 4 sin2(

12αm

),

where

(1.5.9) αm =m – 1

K + 1π

for m = 1, . . . ,K + 1.The corresponding shared eigenvectors, u(m), normalized so that their length is

equal to unity, u(m) · u(m) = 1, are

(1.5.10) u(m)j = Am( 2

K + 1

)1/2cos[(j – 1

2

)αm

]for m, j = 1, . . . ,K + 1, where Am = 1, except that A1 = 1/

√2. The presence of a

zero eigenvalue of the Laplacian, λL1 = 0, corresponding to a constant eigenvector,confirms that the Laplacian matrix is singular. The rest of the eigenvectors are pureharmonic waves.

Cursory inspection reveals the interesting identity

(1.5.11) f · LNN · f =K∑i=1

( fi – fi+1)2 ≥ 0

for any arbitrary nodal field, f, which demonstrates that the matrix LNN is positivesemidefinite. If u is an eigenvector of LNN with corresponding eigenvector λ, then

(1.5.12) u · LNN · u = λu · u ≥ 0.

This inequality confirms that the eigenvalues of LNN are zero or positive.It is worth remarking that the eigenvalues of the Laplacian matrix are approx-

imations of those of the Laplace equation, , in the interval [a, b], satisfying theequation

(1.5.13)d2u

dx2+

�x2u = 0

with homogeneous Neumann boundary conditions at both ends, u′(a) = 0 andu′(b) = 0, where u(x) is an eigenfunction, L = K�x, and L = b – a. We find that

(1.5.14) m =(m – 1

K

)2π2, um(x) = cos

(m – 1

K

πx

�x

),

for m ≥ 1. The eigenvalues of the Laplacian matrix, λLm, agree with the eigenvaluesm for small m and large K.

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One-Dimensional Gr ids / / 13

FactorizationWe can factorize

(1.5.15) LNN = RNN · RNNT ,

where

(1.5.16) RNN =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

–1 0 0 · · · 0 0 01 –1 0 · · · 0 0 00 1 –1 · · · 0 0 0...

......

. . ....

......

0 0 0 · · · –1 0 00 0 0 · · · 1 –1 00 0 0 · · · 0 1 –10 0 0 · · · 0 0 1

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦is a (K+1)×K rectangular matrix implementing backward difference approximationsto the first derivative. Its transpose,

(1.5.17) RNNT =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

–1 1 0 · · · 0 0 0 00 –1 1 · · · 0 0 0 00 0 –1 · · · 0 0 0 0...

......

. . ....

......

...0 0 0 · · · –1 1 0 00 0 0 · · · 0 –1 1 00 0 0 · · · 0 0 –1 1

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦,

is a K × (K + 1) matrix implementing forward difference approximations to the firstderivative.

Exercise

1.5.1 Eigenvalues of the Laplacian

(a) Derive the eigenvalues and eigenvectors shown in (1.5.14). (b) Prepare and dis-cuss a plot of the eigenvalues given in (1.5.14) and those of the Laplacian matrix forK = 2, 4, 8, 16, and 32.

1.6 PERIODIC BOUNDARY CONDITIONS

When the solution of the differential equation (1.1.1) is required to be periodic, wespecify that f1 = fK+1 and compile the difference equations for i = 1, . . . ,K to obtaina linear system,

(1.6.1) LP · ψP = bP,

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14 / / AN INTRODUCTION TO GRIDS, GRAPHS, AND NETWORKS

where

(1.6.2) ψP ≡

⎡⎢⎢⎢⎢⎢⎣f1f2...fK–1fK

⎤⎥⎥⎥⎥⎥⎦ , bP = �x2

⎡⎢⎢⎢⎢⎢⎣g1g2...gK–1gK

⎤⎥⎥⎥⎥⎥⎦ ,

are K-dimensional vectors and

(1.6.3) LP =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

2 –1 0 · · · 0 0 –1–1 2 –1 · · · 0 0 00 –1 1 · · · 0 0 0...

......

. . ....

......

0 0 0 · · · 2 –1 00 0 0 · · · –1 2 –1

–1 0 0 · · · 0 –1 2

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦,

is a K × K symmetric and nearly tridiagonal matrix. Note the presence of a north-eastern and a southwestern element, both equal to –1, implementing the periodicitycondition.

DecompositionWe can decompose

(1.6.4) LP = 2 I –�P,

where I is the K × K identity matrix and

(1.6.5) �P =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 1 0 · · · 0 0 11 0 1 · · · 0 0 00 1 0 · · · 0 0 0...

......

. . ....

......

0 0 0 · · · 0 1 00 0 0 · · · 1 0 11 0 0 · · · 0 1 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦is a K × K symmetric, nearly bidiagonal Toeplitz matrix. Note the presence of twounit corner elements, equal to 1, implementing the periodicity condition.

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One-Dimensional Gr ids / / 15

Eigenvalues and EigenvectorsThe eigenvalues of �P and LP are

(1.6.6) λ�m = 2 cosαm

and

(1.6.7) λLm = 2 – 2 cosαm = 4 sin2(12 αm

)for m = 1, . . . ,K, where

(1.6.8) αm =m – 1

K2π .

The corresponding shared eigenvectors, u(m), normalized so that their norm isequal to unity, u(m) · u(m)∗ = 1, are

(1.6.9) u(m)j =1√Kexp (–i jαm)

for m, j = 1, . . . ,K, where i is the imaginary unit and an asterisk denotes the complexconjugate. The presence of a zero eigenvalue of the Laplacian, λL1 = 0, correspondingto a constant eigenvector, confirms that the matrix LP is singular. The rest of theeigenvectors are pure harmonic waves.

Complex eigenvectors appear because two eigenvalues, λm1 and λm2 , are identicalwhen

(1.6.10) m1 + m2 = K + 2.

The real part of the complex exponential in (1.6.9) can be retained for one eigenvalue,yielding a cosine, and the imaginary part can be retained for the other eigenvalue,yielding a sine.

Cursory inspection reveals the interesting identity

(1.6.11) f · LP · f =K∑i=1

(fi – fi+1)2 ≥ 0,

where f is an arbitrary nodal field satisfying the mandatory periodicity condi-tion fK+1 = f1, which demonstrates that the matrix LNN is positive semidefinite.Consequently, the eigenvalues of LP are zero or positive.

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16 / / AN INTRODUCTION TO GRIDS, GRAPHS, AND NETWORKS

FactorizationWe can factorize

(1.6.12) LP = RPT · RP = RP · RPT ,

where

(1.6.13) RP =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

–1 0 0 · · · 0 0 11 –1 0 · · · 0 0 00 1 –1 · · · 0 0 0...

......

. . ....

......

0 0 0 · · · –1 0 00 0 0 · · · 1 –1 00 0 0 · · · 0 1 –1

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦is a K×K square nearly lower bidiagonal matrix implementing backward differenceapproximations. Its transpose,

(1.6.14) RPT =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

–1 1 0 · · · 0 0 00 –1 1 · · · 0 0 00 0 –1 · · · 0 0 0...

......

. . ....

......

0 0 0 · · · –1 1 00 0 0 · · · 0 –1 11 0 0 · · · 0 0 –1

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦,

is a K × K square nearly upper bidiagonal matrix implementing forward differenceapproximations. Note the presence of one nonzero corner element implementing theperiodicity condition.

Exercise

1.6.1 Eigenvalues and eigenvectors

Confirm by direct substitution the eigenvalues and eigenvectors given in (1.6.7) and(1.6.9).

1.7 ONE-DIMENSIONAL GRAPHS

The finite difference grid discussed previously in this chapter is now regarded asa graph consisting of N nodes, also called vertices, connected by L = N – 1 links(edges), as illustrated in Figure 1.7.1. In an alternative interpretation, the finite dif-ference grid is a network consisting of conducting or conveying links. For example,the links can be regarded as segments of a fluid-carrying pipe.

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One-Dimensional Gr ids / / 17

1

2 N1 i + 1i − 1 i

i LLinks:

Nodes:

FIGURE 1.7.1 Illustration of a one-dimensional graph consisting of Nnodes connected by L = N – 1 links.

1.7.1 Graph Laplacian

The N × N matrix L ≡ LNN, corresponding to two Neumann boundary conditionsdiscussed in Section 1.5, is the Laplacian of the one-dimensional network, given by

(1.7.1) L =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

1 –1 0 · · · 0 0 0–1 2 –1 · · · 0 0 00 –1 2 · · · 0 0 0...

......

. . ....

......

0 0 0 · · · 2 –1 00 0 0 · · · –1 2 –10 0 0 · · · 0 –1 1

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦.

Note that the sum of the elements in each row or column is zero. Sometimes, thegraph Laplacian is also called the combinatorial Laplacian.

The eigenvalues of L are given by

(1.7.2) λm = 2 – 2 cosαn = 4 sin2(12αn

)

for n = 1, . . . ,N, where

(1.7.3) αn =n – 1

Nπ .

The corresponding eigenvectors, u(n), normalized so that u(n) · u(n) = 1, are given by

(1.7.4) u(n)i = An( 2N

)1/2cos[(2j – 1) αn

]for i, n = 1, . . . ,N, where An = 1, except that A1 = 1/

√2. The presence of a zero

eigenvalue, λ1 = 0, corresponding to a uniform eigenvector with equal elements,confirms that the Laplacian matrix is singular. The rest of the eigenvectors are pureharmonic waves.

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18 / / AN INTRODUCTION TO GRIDS, GRAPHS, AND NETWORKS

1.7.2 Adjacency Matrix

In graph theory, an N×N adjacency matrix is introduced,A, defined such that Aij = 1if nodes i and j are connected by a grid line or link, and Aij = 0 otherwise, with theconvention that Aii = 0. Thus, by convention, the diagonal line of the adjacencymatrix is zero.

In the case of the one-dimensional grid presently considered, the adjacencymatrix is

(1.7.5) A =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 1 0 · · · 0 0 01 0 1 · · · 0 0 00 1 0 · · · 0 0 0...

......

. . ....

......

0 0 0 · · · 0 1 00 0 0 · · · 1 0 10 0 0 · · · 0 1 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦.

The eigenvalues of this matrix are

(1.7.6) μn = 2 cos( n

N + 1π)

for n = 1, . . . ,N.The eigenvalues of the adjacency matrix provide us with measure of the network

properties, independent of node and link labeling. In particular, the number of pathsthat return to an arbitrary node after s steps have been made, summed over all startingnodes, is

(1.7.7) ns =N∑n = 1

μsn,

where s is an integer. We observe that n0 = N, in agreement with physical intuition.In our one-dimensional network, ns = 0 if s is an odd integer and ns = 0 if s is aneven integer. Physically, an even number of steps are necessary for an equal numberof forward and backward steps. For a one-dimensional network with N = 13 nodes,we find that

(1.7.8) n2 = 26, n4 = 74, n6 = 236, n8 = 794, n10 = 2756.

Node DegreesThe degree of the ith node, denoted by di, is defined as the number of links attachedto the node, which is equal to the sum of the elements in the corresponding row or

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One-Dimensional Gr ids / / 19

column of the adjacency matrix, A. In the case of the one-dimensional grid presentlyconsidered, we have

(1.7.9) d1 = 1, di = 2, dN = 1,

for i = 2, . . .N – 1.

Laplacian in Terms of the Adjacency MatrixThe graph Laplacian of the one-dimensional grid is given by

(1.7.10) L = D – A,

where D is a diagonal matrix whose ith diagonal element is equal to the correspond-ing node degree, di.

1.7.3 Connectivity Lists and Oriented Incidence Matrix

The number of links in the one-dimensional network is L = N – 1. It is useful tointroduce two L-dimensional connectivity lists, k and l, defined such that the label ofthe first node of the mth link is km and the label of the second node of the mth link islm. In the case of the one-dimensional grid presently considered, we have

(1.7.11) km = m, lm = m + 1

for m = 1, . . . , L. An N × L oriented incidence matrix can be introduced, R, definedsuch that Ri,m = 0, except that

(1.7.12) Rkm,m = –1, Rlm,m = 1.

If nodes and links are labeled sequentially, as shown in Figure 1.7.1, we obtain therectangular N × (N – 1) matrix

(1.7.13) R =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

–1 0 0 · · · 0 0 01 –1 0 · · · 0 0 00 1 –1 · · · 0 0 0...

......

. . ....

......

0 0 0 · · · –1 0 00 0 0 · · · 1 –1 00 0 0 · · · 0 1 –10 0 0 · · · 0 0 1

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦encountered previously in Section 1.5.

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20 / / AN INTRODUCTION TO GRIDS, GRAPHS, AND NETWORKS

Laplacian in Terms of the Oriented Incidence MatrixThe graph Laplacian is given by

(1.7.14) L = R · RT .

In fact, this factorization is valid for arbitrary node and link labeling and for generalhigher-dimensional graphs.

Exercise

1.7.1 Node and link labeling

Derive the connectivity lists and the oriented incidence matrix for an arbitrary nodeand link labeling scheme of your choice.

1.8 PERIODIC ONE-DIMENSIONAL GRAPHS

Shown in Figure 1.8.1 is a closed or periodic one-dimensional graph consisting of Nunique nodes connected by L = N links. The N × N periodic graph Laplacian is

(1.8.1) L =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

2 –1 0 · · · 0 0 –1–1 2 –1 · · · 0 0 00 –1 1 · · · 0 0 0...

......

. . ....

......

0 0 0 · · · 2 –1 00 0 0 · · · –1 2 –1–1 0 0 · · · 0 –1 2

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦,

where the two nonzero northeastern and southwestern corner elements implementthe periodicity condition, as discussed in Section 1.6.

i − 1i

i

Links:

Nodes:

L1 2

1

2

i + 1

N

FIGURE 1.8.1 Illustration of a periodicone-dimensional graph consisting of Nunique nodes connected by L = N links.The first and last nodes numbered 1

and N + 1 coincide.

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One-Dimensional Gr ids / / 21

The periodic Laplacian is a circulant matrix. By definition, each row of anarbitrary circulant matrix derives from the previous row by shifting each elementto the right by one place and then returning the last element to the first place, asdiscussed in Section A.4, Appendix A.

The eigenvalues of the periodic Laplacian are

(1.8.2) λn = 2 – 2 cosαn = 4 sin2(12 αn

)for n = 1, . . . ,N, where

(1.8.3) αn =n – 1

N2π .

The corresponding eigenvectors, u(n), normalized so that u(n) · u(n)∗ = 1, are

(1.8.4) u(n)i =1√N

exp(–i iαn)

for n, j = 1, . . . ,N, where i is the imaginary unit and an asterisk denotes the com-plex conjugate. The presence of a zero eigenvalue, λ1 = 0, corresponding to auniform eigenvector, confirms that the periodic Laplacian is singular. The rest ofthe eigenvectors are pure harmonic waves.

A discrete Fourier orthogonality property states that

(1.8.5)N∑j=1

exp

(i jp

N

)=

{N if p = sN,0 otherwise,

where p and s are zero or arbitrary integers. This property ensures that

(1.8.6) u(s) · u(r)∗ = δsr,

that is, the eigenvectors comprise an orthonormal set.

1.8.1 Periodic Adjacency Matrix

The N × N periodic adjacency matrix is a circulant matrix,

(1.8.7) A =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 1 0 · · · 0 0 11 0 1 · · · 0 0 00 1 0 · · · 0 0 0...

......

. . ....

......

0 0 0 · · · 0 1 00 0 0 · · · 1 0 11 0 0 · · · 0 1 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦.

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22 / / AN INTRODUCTION TO GRIDS, GRAPHS, AND NETWORKS

Two nonzero corner elements appear due to the periodicity condition. The degreesof all nodes are the same, di = 2 for i = 1, . . . ,N.

The eigenvalues of the periodic adjacency matrix are

(1.8.8) μn = 2 cos(n – 1

N2π)

for n = 1, . . . ,N.The number of steps defined in (1.7.7) are zero when s is zero or an odd integer

and nonzero when s is an even integer. When N = 13, we find that

(1.8.9) n2 = 26, n4 = 78, n6 = 260, n8 = 910, n10 = 3276.

1.8.2 Periodic Oriented Incidence Matrix

If we label nodes and links sequentially, as shown in Figure 1.8.1, we will obtain asquare N × N oriented incidence matrix,

(1.8.10) R =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

–1 0 0 · · · 0 0 11 –1 0 · · · 0 0 00 1 –1 · · · 0 0 0...

......

. . ....

......

0 0 0 · · · –1 0 00 0 0 · · · 1 –1 00 0 0 · · · 0 1 –1

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦.

The periodic Laplacian is given by

(1.8.11) L = D – A = R · RT ,

where D = 2 I is a diagonal matrix hosting the degree of the N nodes.

1.8.3 Fourier Expansions

A real periodic nodal field, ψ , can be expanded in a Fourier series so that

(1.8.12) ψi =M∑

p = –M

cp exp[– i pk(i – 1)

]for i = 1, . . . ,N, where k ≡ 2π /N is the wave number of the longest wave, i is theimaginary unit, and cp are complex Fourier coefficients. To ensure that the expandednodal field is real, we require that c–p = c∗p, where an asterisk denotes the complexconjugate. The truncation level, M, is discussed later in this section.

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One-Dimensional Gr ids / / 23

An equivalent representation in terms of sines and cosines, arising by resolv-ing the Fourier coefficients and complex exponentials into their real and imaginaryparts, is

(1.8.13) ψi =1

2a0 +

M∑p=1

(ap cos

[(i – 1)pk

]+ bp sin

[(i – 1)pk

] ),

where

(1.8.14) ap = 2(cp), bp = 2�(cp)

for p = 0, . . . ,M, with the understanding that b0 = 0, where and � denote the realand imaginary parts. Accordingly,

(1.8.15) cp = 12 (ap + i bp).

Using Fourier orthogonality properties (e.g., [35]), we find that

(1.8.16) cp =1

N

N∑i= 1

ω(i – 1)p ψi

or

(1.8.17) cp =1

N

(ψ1 + ψ2 ω

p + ψ3 ω2p + · · · + ψN ω(N – 1)p

),

where

(1.8.18) ω = exp(ik).

These formulas indicate that

(1.8.19) c0 = a0 =1

N

N∑i = 1

ψi,

that is, the Fourier constant a0 is the mean of all nodal values.When N is odd, we truncate the Fourier sum atM = (N – 1)/2 and compute cp for

p = 0, . . . ,M. When N is even, we truncate the Fourier sum at M = N/2, compute cpand bp for p = 0, . . . ,M – 1, using formula (1.8.17), and set

(1.8.20) cM =1

N

(ψ1 – ψ2 + ψ3 – · · · – ψN

).

The alternating signs arise because ωN/2 = exp(iπ ) = –1.

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24 / / AN INTRODUCTION TO GRIDS, GRAPHS, AND NETWORKS

1.8.4 Cosine Fourier Expansion

If a real periodic nodal field, ψ , is symmetric with respect to the midpoint of thenetwork, that is,

(1.8.21) ψi = ψN+2–i,

we may use the cosine Fourier expansion

(1.8.22) ψi = 12 a0 +

N – 1∑p = 1

ap cos[(i – 1)pk ],

where ap are cosine Fourier coefficients.Using Fourier orthogonality properties, we find that

(1.8.23) ap =1

N

(ψ1 + ψ2 cos(kp) + ψ3 cos(2kp) + · · · + ψN cos[(N – 1)kp]

)for p = 1, . . . ,N – 1, and

(1.8.24) a0 =2

N

N∑i= 1

ψi

that is, the Fourier constant a0 is twice the arithmetic mean of the nodal values.The associated complex Fourier series is

(1.8.25) ψi =N–1∑

p= –(N – 1)

cp exp[– i (i – 1)pk

],

where cp = 12ap.

1.8.5 Sine Fourier Expansion

If a real periodic nodal field, ψ , is antisymmetric with respect to the midpoint of thenetwork, that is,

(1.8.26) ψi = –ψN+2–i,

we may use the sine Fourier expansion

(1.8.27) ψi =N – 1∑p= 1

bp sin[(i – 1) pk ],

where ap are cosine Fourier coefficients.

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One-Dimensional Gr ids / / 25

Using Fourier orthogonality properties, we find that

(1.8.28) bp =1

N

(ψ2 sin(kp) + ψ3 sin(2kp) + · · · + ψN sin[(N – 1)kp]

)for p = 1, . . . ,N – 1.

The associated complex Fourier series is

(1.8.29) ψi =N – 1∑

p= – (N – 1)

cp exp[– i (i – 1) pk

],

where cp =12i ap for p = 1, . . . ,N – 1 and cp = 0.

Exercise

1.8.1 Link labeling

Confirm that the factorization L = R · RT is independent of link labeling.

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/ / / 2 / / / GRAPHS AND NETWORKS

A graph is broadly defined as a collection of N nodes, also calledvertices, connected by L links, also called edges (e.g., [54]). The number of nodes,N, is the order of a graph and the number of links, L, is the size of a graph. In graphtheory, a graph is typically denoted as G(V , E), where the set V contains the verticesand the set E contains the edges. In science, engineering, and other applications,a graph represents a network consisting of conductive or convective pathways, asdiscussed in Chapter 4. The terms graph and network will be used interchangeablyin our discourse.

2.1 ELEMENTS OF GRAPH THEORY

One of the most attractive features of graph theory is that nodes and links canbe labeled arbitrarily, independently, and in an uncorrelated fashion, as shown inFigure 2.1.1(a), where the nodes are marked as filled circles. Eight nodes and twelvelinks define this network, N = 8 and L = 12. Note that links numbered 2 and 7 do notcross at a node. The network shown in Figure 2.1.1(a) is reminiscent of a structuraltruss.

2.1.1 Adjacency Matrix

In graph theory, an N×N adjacency matrix is introduced,A, defined such that Aij = 1if nodes i and j are connected by a link, and Aij = 0 otherwise, where i, j = 1, . . . ,N.By convention, the diagonal elements of the adjacency matrix are zero. By construc-tion, the adjacency matrix is symmetric. For example, the adjacency matrix of thenetwork shown in Figure 2.1.1(a) is the 8 × 8 matrix shown in Figure 2.1.1(b).

The total number of links in a network, L, is equal to the number of ones in theupper or lower triangular part of the adjacency matrix,

(2.1.1) L =N–1∑i= 1

N∑j= i + 1

Aij =N∑i= 2

i–1∑j= 1

Aij =1

2

N∑i= 1

N∑j= 1

Aij.

The fraction 1/2 in front of the last double sum accounts for the inherent symmetryof A.

26

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Graphs and Networks / / 27

(a)

24 6

9 8

7

4

2

11

12

1 3

5

86

7

3

5

1

10

(b)

A =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 1 1 1 0 0 0 01 0 1 1 0 0 0 01 1 0 0 1 0 0 01 1 0 0 1 1 0 00 0 1 1 0 0 1 10 0 0 1 0 0 1 00 0 0 0 1 1 0 10 0 0 0 1 0 1 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

.

(c)

R =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

−1 0 1 0 0 0 −1 0 0 0 0 01 −1 0 −1 0 0 0 0 0 0 0 00 1 −1 0 0 1 0 0 0 0 0 00 0 0 1 −1 0 1 −1 0 0 0 00 0 0 0 1 −1 0 0 0 −1 0 10 0 0 0 0 0 0 1 −1 0 0 00 0 0 0 0 0 0 0 1 1 −1 00 0 0 0 0 0 0 0 0 0 1 −1

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

.

FIGURE 2.1.1 (a) Illustration of a typical graph consisting of N = 8

nodes, also called vertices, connected by L = 12 links, also callededges. (b) The corresponding 8 × 8 adjacency matrix and (c) thecorresponding 8 × 12 oriented incidence matrix. In this conceptual

depiction, edges are allowed to cross over without intersecting at a

node.

Spectrum of the Adjacency MatrixThe eigenvalues and eigenvectors of the node adjacency matrix, A, denoted by μifor i = 1, . . . ,N contain useful information on the structure of the graph. Althoughthe layout of the adjacency matrix depends on the node labeling, the eigenvalues areindependent of node labeling.

Since A is symmetric, it has real eigenvalues and a complete set of orthogonaleigenvectors. The sum of the eigenvalues of A is equal to the trace of A, which iszero. A necessary but not sufficient condition for two graphs to be isomorphic is thatthe spectra of the corresponding adjacency matrices are identical.

Suppose that we begin traveling on a continuous path departing from the ith nodealong a chain of s links, so that we end up at the jth node. The number of possible

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28 / / AN INTRODUCTION TO GRIDS, GRAPHS, AND NETWORKS

pathways is equal to the ij component of the matrix powerAs. The sum of the numberof pathways that return to a starting node after s steps have been made is

(2.1.2) ns =N∑i= 1

μsi .

We find that

(2.1.3) n0 = N, n1 = 0, n2 = 2L, n3 = 6T ,

where T is the number of triangles formed by the links.

2.1.2 Node Degrees

The degree of the ith node, denoted by di, is defined as the number of links attachedto the node, connecting the node to its nearest neighbors. A node and its nearestneighbors define a neighborhood.

By construction, di is equal to the number of ones in the ith row or column of theadjacency matrix. The degree of an isolated node is zero. For example, the degreesof the eight nodes comprising the network shown in Figure 2.1.1(a) are

(2.1.4)d1 = 3, d2 = 2, d3 = 3, d4 = 4,

d5 = 4, d6 = 2, d7 = 3, d8 = 2.

In the case of an infinite network consisting of a regular lattice, the vertex degreesare also called the lattice coordination number, as discussed in Section 2.6.

The sum of the degrees of all nodes in a finite network is equal to twice thenumber of all links,

(2.1.5)N∑i= 1

di = 2L.

Consequently,

(2.1.6)N

L=

2

dav,

where

(2.1.7) dav ≡ 1

N

N∑i= 1

di

is the average or mean node degree. Equation (2.1.6) is also valid for an infinitenetwork where N and L are infinite but their ratio is well defined.

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Graphs and Networks / / 29

2.1.3 The Complete Graph

By definition, each node of a complete graph is connected to every other node, asshown in Figure 2.1.2. Consequently, all off-diagonal elements of the adjacency ma-trix are equal to unity, that is, the adjacency matrix is the complement of the identitymatrix. The degree of each node is N –1, and the number of links is equal to the num-ber of elements in the strictly upper or lower triangular part of the adjacency matrix,

(2.1.8) L =1

2N(N – 1).

For N = 3, we find that L = 3, describing a triangle. For N = 5, we find that L = 10,as shown in Figure 2.1.2. Sometimes a complete graph is also called a clique. Inqualitative terms, the complete graph describes the best connected network.

2.1.4 Complement of a Graph

The union of a graph and its complement forms a complete graph. Consequently,the complement of an arbitrary graph with adjacency matrix A is another graph withadjacency matrix

(2.1.9) A′ = Ac – A,

where Ac is the adjacency matrix of the complete graph. For example, the adjacencymatrix of the complement of the graph shown in Figure 2.1.1(a) is

(2.1.10) A′ =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 0 0 0 1 1 1 10 0 0 0 1 1 1 10 0 0 1 0 1 1 10 0 1 0 0 0 1 11 1 0 0 1 1 0 01 1 1 0 0 1 0 11 1 1 1 0 0 0 01 1 1 1 0 1 0 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦.

15

12

3

6

9

10

57

34

8

42

FIGURE 2.1.2 Illustration of a complete graph consist-ing of N = 5 nodes (vertices) connected by L = 10

links (edges).

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30 / / AN INTRODUCTION TO GRIDS, GRAPHS, AND NETWORKS

In this case, the complement contains a higher number of links than the originalgraph.

2.1.5 Connectivity Lists and the Oriented Incidence Matrix

It is helpful to introduce two connectivity lists represented by the L-dimensionalvectors k and l, defined such that the label of the first node of the mth link is km andthe label of the second node of the mth link is lm, where m = 1, . . . , L. These twoconnectivity lists can be arranged into L × 2 edge list. The adjacency matrix can beextracted from the edge list, and vice versa.

For example, the 12-dimensional connectivity lists of the 12 links comprising thenetwork shown in Figure 2.1.1(a) are

(2.1.11)k = [ 1, 2, 3, 2, 4, 5, 1, 4, 6, 5, 7, 8 ],

l = [ 2, 3, 1, 4, 5, 3, 4, 6, 7, 7, 8, 5 ].

If nodes, links, or both are relabeled, the connectivity lists undergo correspondingpermutations.

In an undirected graph, discussed exclusively in this book, because the order ofthe end points is immaterial, km and lm can be switched freely for each m. This is nottrue in the case of a directed graph, also called a digraph, where an ordered part ofend points defines an arrow. In-degrees and out-degrees are defined in a digraph.

It is useful to introduce an N × L oriented incidence matrix, R, defined such thatRi, j = 0, except that

(2.1.12) Rkm,m = –1, Rlm,m = 1

for m = 1, . . . ,L. For example, the oriented incidence matrix of the network shownin Figure 2.1.1(a) is the 8 × 12 matrix shown in Figure 2.1.1(c). Typically, but notalways, the number of links is much greater than the number of nodes, L � N, andthe matrix R resembles a horizontal strip.

2.1.6 Connected and Unconnected Graphs

A graph is connected if at least one continuous path of links can be found leadingus from an arbitrary node to any other arbitrary node. If a continuous path cannot befound, the graph is unconnected. Fragments and islands consisting of isolated nodesor groups of nodes are found in an unconnected graph. The number of islands inan unconnected graph can be diagnosed from the number of zero eigenvalues of theLaplacian matrix, as discussed in Section 2.2.

2.1.7 Pairwise Distance and Diameter

A physical or abstract length or weight can be assigned to each link of a graph. Thelength of each link of an unweighed graph is set to unity by convention, whereas the

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Graphs and Networks / / 31

length of a link in an weighed graph is set to a specified link weight, as discussed inChapter 4. In both cases, length is measured in predetermined units appropriate forthe physical, engineering, or information system under consideration.

The pairwise distance between two selected nodes is the minimum length of theshortest path between these nodes. In the case of an unweighed graph, the pairwisedistance is an integer expressing the number of links along the shortest path betweenthe two nodes. The maximum pairwise distance over all pairs of nodes is the graphdiameter.

2.1.8 Trees

We saw that a complete graph describes the best connected network for a given num-ber of nodes, in that any pair of nodes is connected by a link. The number of links ina complete network scales with N2.

On the opposite part of the spectrum lies a tree network distinguished by theabsence of cyclical paths, as shown in Figure 2.1.3. The number of links in a treenetwork is less by one than the number of nodes, L = N – 1. If an arbitrary link isclipped, a connected tree network breaks up into two disconnected tree networks.Metaphorically speaking, a tree network is on the verge of disintegration.

2.1.9 Random and Real-Life Networks

A random graph with N vertices is characterized by the probability, p, that any pairnodes is connected by a link, independent of any other connections. The expectednode degree is

(2.1.13) < d >= p (N – 1),

1

3

2

5

4

6

2

1

5

6 7

3

4

FIGURE 2.1.3 Illustration of a tree network consistingof N = 7 nodes (vertices) connected by L = 6

links (edges). The absence of triangles and cycles is adistinguishing feature of a general tree network.

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32 / / AN INTRODUCTION TO GRIDS, GRAPHS, AND NETWORKS

and the expected number of links is

(2.1.14) < L >= p 12 N(N – 1).

When p = 1, we obtain a complete graph where the expected values are equal to thecorresponding actual values.

The degree distribution in a random graph is described by a binomial function,

(2.1.15) PN(d) =

(N – 1d

)pd(1 – p)N–1–k,

where d ≤ N – 1. The first large parentheses on the right-hand side denote thecombinatorial,

(2.1.16)(

mk

)≡ m !

k ! (m – k) !=

l∏=1

m – + 1

,

where l is the minimum of k and m – k. As N → ∞, the binomial distribution tendsto the Poisson distribution.

Real-Life NetworksDeterministic and random networks encountered in real life are described by nodedegree distributions that differ significantly from the binomial or Poisson distribu-tion (e.g., [32]). Node degree distributions are often skewed to the right or exhibita power-law behavior. Theoretical models of real-life networks have been proposedaccording to their indented physical, engineering, biological, sociological, or otherapplication in different specializations.

Exercises

2.1.1 Complement of a graph

Draw the complement of the graph shown in Figure 2.1.1(a).

2.1.2 Node clustering

The clustering index of the ith node is defined as κi = mi/(mi)max, where mi is thenumber of links connecting its neighbors. Show that (mi)max = 1

2di(di – 1).

2.2 LAPLACIAN MATRIX

Let D be a diagonal matrix whose ith diagonal element is equal to the degree ofthe ith node, di. The N × N graph Laplacian matrix, L, is defined in terms of theadjacency matrix, A, the degrees of the nodes encapsulated in D, and the orientedincidence matrix, R, as

(2.2.1) L = D – A

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Graphs and Networks / / 33

or

(2.2.2) L = R · RT .

By construction, the sum of the elements in each row or column of L is zero.The factorization (2.2.2) shows that the Laplacian is given by the sum of the

tensor product of L vectors,

(2.2.3) L =L∑

m=1

�(m) ⊗ �(m),

where �(m) is the mth column of R and ⊗ denotes the tensor product of two vectors.Specifically, �(m) ⊗ �(m) is an N × N matrix with components

(2.2.4) [�(m) ⊗ �(m)]ij = �(m)i �

(m)j .

We recall that the vector �(m) is filled with zeros, except that the entry correspondingto the first end node is –1 and the entry corresponding to the second end node is 1,and find that two diagonal components of the tensor product are equal to 1 and twooff-diagonal components are equal to –1. Thus, the matrix �(m) ⊗ �(m) has only fournonzero components.

For example, the 8× 8 Laplacian matrix of the network shown in Figure 2.1.1(a)is given by

(2.2.5) L =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

3 –1 –1 –1 0 0 0 0–1 3 –1 –1 0 0 0 0–1 –1 3 0 –1 0 0 0–1 –1 0 4 –1 –1 0 00 0 –1 –1 4 0 –1 –10 0 0 –1 0 2 –1 00 0 0 0 –1 –1 3 –10 0 0 0 –1 0 –1 2

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦.

Note that the sum of the elements in each row or column is zero.

2.2.1 Properties of the Laplacian Matrix

Being a real and symmetric matrix, the Laplacian matrix, L, has real eigenvalues anda complete set of mutually orthogonal eigenvectors. The eigenvalues and eigenvec-tors of L provide us with a wealth of information on the structure of the underlyingnetwork. Since the sum of the elements in each row of L is zero, a vector with equalcomponents is an eigenvector of L corresponding to the null eigenvalue.

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34 / / AN INTRODUCTION TO GRIDS, GRAPHS, AND NETWORKS

Let an N-dimensional vector, ψ , contain the nodal values of a discrete field at theN nodes of a network. We find that

(2.2.6) ψ · L · ψ =L∑

m=1

(ψkm – ψlm)2 ≥ 0,

which demonstrates that the Laplacian is positive semidefinite. Consequently, theeigenvalues of L, denoted by λi, are either zero or positive. The sum of the eigenval-ues of L is equal to the trace of L, which is equal to the trace of D, which is equal tothe sum of the degrees of all nodes.

We may assume that the eigenvalues have been ordered so that

(2.2.7) 0 = λ1 ≤ λ2 ≤ · · · ≤ λN ,

where the first eigenvalue, λ1, is always zero. Further or all other eigenvalues mayalso be zero.

The second smallest eigenvalue, λ2, is of particular interest in spectral graph the-ory. The value of λ2 is sometimes called the algebraic connectivity of the network.We know that λ2 > 0 only when the graph is not connected, that is, when the graphconsists of two or more unconnected subgraphs. This observation suggests that λ2 isa sensible measure of the contiguity of a network represented by a graph. The max-imum value λ2 = N is attained for a complete graph. More generally, the number ofzero eigenvalues of L is equal to the number of isolated nodes or clusters of nodes.

The set of eigenvalues of a graph consisting of a number of disconnected sub-graphs is the union of the eigenvalues of the constituent subgraphs, where eachsubgraph contributes a zero eigenvalue. Other properties of the Laplacian eigenvaluesare reviewed by Mohar [30].

Let u(i) be the eigenvector of the Laplacian corresponding to the ith eigenvalue.We know that the eigenvector u(1) corresponding to the zero eigenvalue, λ1 = 0, isfilled with ones. Orthogonality of the set of eigenvectors requires that u(i) · u(1) = 0for i > 1, yielding

(2.2.8)N∑j= 1

u(i)j = 0

for i > 1, which shows that the mean value of the components of any but the firsteigenvector is zero.

2.2.2 Complete Graph

All elements of the Laplacian matrix of a complete graph are equal to –1, except forthe diagonal elements that are equal to N – 1,

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Graphs and Networks / / 35

(2.2.9) L =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

N – 1 –1 –1 · · · –1 –1 –1–1 N – 1 –1 · · · –1 –1 –1–1 –1 N – 1 · · · –1 –1 –1...

......

. . ....

......

–1 –1 –1 · · · N – 1 –1 –1–1 –1 –1 · · · 1 N – 1 –1–1 –1 –1 · · · –1 –1 N – 1

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦.

One may confirm that L2 = NL and, more generally,

(2.2.10) Lk = Nk–1L,

for any positive integer, k.The first eigenvalue of the Laplacian matrix is zero, and the rest of the eigenvalues

are equal to the number of nodes, N,

(2.2.11) λ1 = 0, λn = N

for n = 2, . . . ,N.One useful set of eigenvectors, u(n), normalized so that their lengths are equal to

unity, u(n) · u(n)∗ = 1, is

(2.2.12) u(n)i =1√N

exp(–i iαn ),

for n = 1, . . . ,N, where

(2.2.13) αn =n – 1

N2π

and i is the imaginary unit, i2 = –1. Because of the pronounced multiplicity of theeigenvalues, other sets of eigenvectors can be chosen.

2.2.3 Estimates of Eigenvalues

Estimates for the magnitudes of the second smallest and largest eigenvalues of thearbitrary graph, λ2 and λN , are available (e.g., [27, 30]). For example, it can be shownthat the second eigenvalue satisfies the inequality

(2.2.14) λ2 ≤ N

N – 1mini(di).

The last eigenvalue satisfies the inequality

(2.2.15)N

N – 1maxi(di) ≤ λN ≤ max

i,j(di + dj),

for any pair of nodes, i and j, are connected by a link. We conclude that, if all nodedegrees are zero, λN = 0 and all eigenvalues are also zero.

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36 / / AN INTRODUCTION TO GRIDS, GRAPHS, AND NETWORKS

2.2.4 Spanning Trees

A spanning tree is a continuous chain of links that visit all N nodes of a network inthe absence of local loops. Kirchhoff’s spanning-tree theorem states that the numberof spanning trees in a network is

(2.2.16) nt =1

Nλ2 · · · λN .

In fact, nt is the absolute value of any minor of the graph Laplacian.

2.2.5 Spectral Expansion

An arbitrary nodal field encapsulated in a vector, ψ , can be expressed as a weighedsum of eigenvectors of the Laplacian matrix, u( j ), so that

(2.2.17) ψ =N∑j = 1

cju( j ),

where cj are appropriate coefficients. In index notation,

(2.2.18) ψi =N∑j= 1

cju( j )i

for i = 1, . . . ,N.Assume that the eigenvectors have been normalized such that their norm is unity,

u( j ) · u( j )∗ = 1, where an asterisk denotes the complex conjugate. Exploiting theorthogonality of the eigenvectors, we obtain

(2.2.19) cj = ψ · u( j )∗ .

The spectral expansion in terms of the eigenvectors shown in (2.2.17) is the dis-crete counterpart of the Fourier expansion of a continuous function in terms oftrigonometric functions or orthogonal polynomials.

2.2.6 Spectral Partitioning

We have remarked that the eigenvector corresponding to the zero eigenvalue of theLaplacian matrix, λ1, is uniform over the nodes of a network. Higher eigenvectorspartition the network into two or a higher number of pieces (spectral partitioning).To partition a network, we may group together nodes whose eigenvector componentscorresponding to a specified eigenvalue have the same sign. The eigenvalue withthe second smallest magnitude, λ2, is chosen for division into two fragments, whilehigher eigenvalues are chosen for division into a higher number of fragments.

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Graphs and Networks / / 37

The success of spectral partitioning relies on the zero-mean property expressedby (2.2.8), roughly stating that an equal number of eigenvector components with pos-itive and negative sign appear. More sophisticated partitioning methods are available(e.g., [11]).

Square NetworkAs an example, the spectral partitioning of a square network is shown in Figure 2.2.1.Positive components of an eigenvector are marked as filled circles, negative compo-nents are marked as dots, and zero components are unmarked. The network shownconsists of N = 172 = 289 nodes connected by L = 544 links. The degrees of the 4corner nodes is 2, the degrees of the 60 edge nodes is 3, and the degrees of the 225interior nodes is 4.

Exact expressions for the eigenvalues and eigenvectors of the Laplacian of thesquare network are available, as discussed in Chapter 3. The first nine eigenvaluescorresponding to the eigenvectors shown in Figure 2.2.1 are λ = 0, 0.0341 (double),0.0681, 0.1351 (double), 0.1691 (double), and 0.2701, accurate to the fourth decimalplace.

FIGURE 2.2.1 Spectral partitioning of a Cartesian network consisting of a complete setof horizontal and vertical links.

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38 / / AN INTRODUCTION TO GRIDS, GRAPHS, AND NETWORKS

2.2.7 Complement of a Graph

The Laplacian of the complement of a graph, indicated by a prime, is given by

(2.2.20) L′ = D′ – A′ = (Dc – D) – (Ac – A) = Lc – L,

where the superscript c denotes the complete graph.Let P(λ) be the characteristic polynomial of the Laplacian of a graph, L. The

characteristic polynomial of the Laplacian of the complement of the graph, L′, is

(2.2.21) P ′(λ) = (–1)N–1λ

N – λP(N – λ).

Corresponding eigenvalues are related by

(2.2.22) λ′1 = 0, λ′

i+1 = N – λN–i+1

for i = 1, . . . ,N – 1. In the case of a complete graph, λN–i+1 = N and λ′i+1 = 0.

2.2.8 Normalized Laplacian

Suppose that none of the degrees of the vertices is zero, that is, isolated nodes donot appear. A normalized incidence matrix, R, and the corresponding normalizedLaplacian, L, can be defined as

(2.2.23) R ≡ D–1/2 · R, L ≡ R · RT,

where R is the oriented incidence matrix. Subject to these definitions, we have

(2.2.24) L = R · RT = D1/2 · L · D1/2

and

(2.2.25) L = D–1/2 · L · D–1/2 = I – D–1/2 · A · D–1/2,

where I is the N × N identity matrix. By construction, all diagonal components ofthe normalized Laplacian are equal to unity, Lii = 1. The off-diagonal componentsare Lij = –1/

√didj if nodes i and j are connected by a link, and zero otherwise.

The normalized Laplacian is a positive semidefinite matrix, having one zero ei-genvalue corresponding to an eigenvector whose ith component is

√di. However, the

rest of the eigenvalues are not necessarily equal to those of the Laplacian matrix. Infact, the eigenvalues of the normalized Laplacian lie in the range [0, 2], whereas thoseof the Laplacian lie in the range [0,∞). The normalized Laplacian finds applicationsin the theory of random walks.

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Graphs and Networks / / 39

2.2.9 Graph Breakup

A graph, G, can be broken into two pieces, G1 and G2, by removing a set of links, E .By construction, one end point of each removed link belongs to G1, and the secondend point belongs to G2. We are interested in finding the smallest possible cut set, E ,that separates G into the two largest possible pieces. A measure of the quality of acut and fragility of G is the scalar

(2.2.26) h ≡ |E ||G1||G2|

,

where the vertical bars denote an appropriate magnitude (volume). Cheeger’sconstant is defined as

(2.2.27) hG ≡ maxG1

h.

Cheeger’s theorem relates Cheeger’s constant to the second eigenvalue of thenormalized Laplacian, ν2,

(2.2.28) hG ≥ √2ν2, hG ≤ 12 ν2.

As ν2 tends to zero, indicating graph fragmentation, Cheeger’s constant also tends tozero.

Exercise

2.2.1 Normalized Laplacian

Derive the normalized Laplacian of the graph shown in Figure 2.1.1(a).

2.3 CUBIC NETWORK

A three-dimensional network in physical space can be projected onto a plane forbetter visualization. For example, a cubic network consisting of N = 8 nodes andL = 12 links can be projected onto a planar network, as shown in Figure 2.3.1(a).Nodes and links are labeled arbitrarily in this illustration. The corresponding 8 × 8node adjacency matrix is shown in Figure 2.3.1(b).

The 12-dimensional connectivity lists of the 12 links defining the link end pointsare

(2.3.1)k = [ 1, 2, 3, 4, 5, 6, 7, 8, 1, 2, 3, 4 ],

l = [ 2, 3, 4, 1, 6, 7, 8, 5, 6, 7, 8, 1 ].

The 8×12 oriented incidence matrix is shown in Figure 2.3.1(c). The degree of eachnode is 3, and the graph Laplacian is

(2.3.2) L = 3 I – A = R · RT ,

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40 / / AN INTRODUCTION TO GRIDS, GRAPHS, AND NETWORKS

(a)

3 4

8

4

5

18

6

7

910

12

2

1

4

3

4

1

5

6

7

8

910

1112

3

211

21

67

5

58

7

2

3

6

(b)

A =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 1 0 1 0 1 0 01 0 1 0 0 0 1 00 1 0 1 0 0 0 11 0 1 0 1 0 0 00 0 0 1 0 1 0 11 0 0 0 1 0 1 00 1 0 0 0 1 0 10 0 1 0 1 0 1 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(c)

R =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

−1 0 0 1 0 0 0 0 −1 0 0 01 −1 0 0 0 0 0 0 0 −1 0 00 1 −1 0 0 0 0 0 0 0 −1 00 0 1 −1 0 0 0 0 0 0 0 −10 0 0 0 −1 0 0 1 0 0 0 10 0 0 0 1 −1 0 −0 1 0 0 00 0 0 0 0 1 −1 0 0 1 0 00 0 0 0 0 0 1 −1 0 0 1 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

FIGURE 2.3.1 (a) Illustration of a cubic network and its projection on theplane. (b) The adjacency matrix and (c) the oriented incidence matrix.The cubic network consists of N = 8 nodes (vertices) connected by

L = 12 links (edges). Nodes and links are labeled arbitrarily in this

example.

where I is the 8 × 8 identity matrix. Making substitutions, we find that

(2.3.3) L =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

3 –1 0 –1 0 –1 0 0–1 3 –1 0 0 0 –1 00 –1 3 –1 0 0 0 –1–1 0 –1 3 –1 0 0 00 0 0 –1 3 –1 0 –1–1 0 0 0 –1 3 –1 00 –1 0 0 0 –1 3 –10 0 –1 0 –1 0 –1 3

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦.

The eight eigenvalues of L are λ = 0, 2 (triple), 4 (triple), and 6. The corre-sponding eigenvectors are illustrated in Figure 2.3.2, where positive components aremarked with filled (green) circles, negative components are marked with hollow (red)circles, and zero components are unmarked. The sets of filled or hollow circles pro-vide us with a spectral partitioning of the cubic network. Note that in the case of the

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Graphs and Networks / / 41

FIGURE 2.3.2 Spectral partitioning based on the eigenvectors of theLaplacian matrix of a cubic network corresponding to eigenvaluesλ = 0, 2, 2, 2 (first row) and λ = 4, 4, 4, 6 (second row). Posi-

tive components are marked with filed circles, negative compo-

nents are marked with hollow circles, and zero components are

unmarked.

highest eigenvalue, λ = 6, each filled circle has three nearest hollow circles, and eachhollow circle has three nearest filled circles.

Exercises

2.3.1 Node and link labeling

Derive the Laplacian matrix for a node and link labeling scheme of your choicethat is different than that shown in Figure 2.3.1. Confirm that the eigenvalues of theLaplacian remain unchanged.

2.3.2 Diagonal link

Derive the Laplacian matrix when a diagonal link of your choice is added to a cubicnetwork.

2.4 FABRICATED NETWORKS

Finite or closed, planar or three-dimensional networks can be fabricated by the finiteelement subdivision of a parental structure or by the Delaunay triangulation based ona specified set of nodes, as discussed in this section. In the case of the finite elementtessellation, the network links are the element edges and the network nodes are theelement vertices. Each network can be mapped onto another isomorphic network thatcan be partitioned in similar ways in terms of the eigenvectors of the Laplacian. Inthe following discussion, the number of nodes with degree d is denoted as nd, wherethe sum of all nd is the number of nodes, N.

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42 / / AN INTRODUCTION TO GRIDS, GRAPHS, AND NETWORKS

2.4.1 Finite-Element Network on a Disk

A network associated with a finite element assembly of three-node triangles on adisk, generated by the successive subdivision of a hard-coded, four-element parentalstructure, is shown in Figure 2.4.1(a). The number of nodes is N = 145, the numberof lines is L = 400, and the node degree distribution is n3 = 4, n4 = 29, and n6 = 112,indicating a dominant hexagonal structure.

The first few eigenvalues of the Laplacian are λ = 0, 0.1436 (double), 0.3388,0.3523, 0.4890, 0.7154 (double), and 0.9322 (double). The multiple eigenvaluesare due to the inherent fourfold symmetry of the network. The corresponding

(a)

(b)

FIGURE 2.4.1 Spectral partitioning of a network arising from a finiteelement grid on (a) a disk and (b) a square.

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Graphs and Networks / / 43

eigenvectors implementing spectral partitioning are shown with circular symbols inFigure 2.1.1(a). In all illustrations presented in this section, positive components ofan eigenvector are marked with a filled circle, negative components are marked witha dot, and zero components are unmarked.

2.4.2 Finite-Element Network on a Square

A network arising from a finite element assembly of triangles on a square, generatedby the successive subdivision of a hard-coded, eight-element parental structure, isshown in Figure 2.4.1(b). The number of nodes is N = 172 = 289, the number oflinks is L = 800, and the node degree distribution is n3 = 8, n4 = 56, and n6 = 224,indicating a nearly hexagonal structure.

The first few eigenvalues of the Laplacian are λ = 0, 0.0564 (double), 0.1436(double), 0.2540 (double), 0.3388, 0.3597, and 0.4890. The corresponding eigen-vectors implementing spectral partitioning are shown in Figure 2.4.1(b). Comparingthis partitioning with that shown in Figure 2.2.1 for the Cartesian network revealssignificant differences in the spatial structure of high-order eigenvectors.

2.4.3 Delaunay Triangulation of an Arbitrary Set of Nodes

A network arising from the Delaunay triangulation based on an arbitrary set of nodesin the xy plane is shown in Figure 2.4.2(a). The underlying Voronoi tessellation con-sisting of polygons, performed by a Matlab function, is indicated by the dashed (red)lines. Each point inside a Voronoi cell is nearest to the corresponding central nodethan to any other node.

The network shown in Figure 2.4.2(a) consists of N = 19 nodes connected byL = 45 links arising from the Delaunay triangulation. The node degree distribution isbroad: n3 = 2, n4 = 7, n5 = 4, and n6 = 6. The first few eigenvalues of the Laplacianare λ = 0, 0.0564, 0.7950, 1.3897, 2.1478, 2.8924, 3.1845, 3.7750, 4.4834, 4.7087,and 5.1862. The corresponding eigenvectors implementing spectral partitioning areshown in Figure 2.1.2(c). We observe that the second and third eigenvectors dividethe network into two different contiguous pieces.

2.4.4 Delaunay Triangulation of a Perturbed Cartesian Grid

A network arising from Delaunay triangulation of a perturbed Cartesian grid isshown in Figure 2.4.2(b). To generate this network, nodes are distributed on a Carte-sian grid with spacings �x and �y and are then displaced randomly along the x andy axes by distances ρ�x and ρ�y, where ρ is a uniform deviate. The links are deter-mined by Delaunay triangulation performed by a Matlab function. The underlyingVoronoi tessellation is indicated by the dashed (red) lines in Figure 2.4.2(b).

The particular network shown in Figure 2.4.2(b) consists of N = 92 = 81 nodesconnected by L = 208 links. The node degree distribution is broader than that of

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44 / / AN INTRODUCTION TO GRIDS, GRAPHS, AND NETWORKS

(a) (b)

−1 −0.5 0 0.5 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

x

y

−1 −0.5 0 0.5 1

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

x

y

(c)

FIGURE 2.4.2 A network arising from the Delaunay triangulation of (a) a set ofarbitrary points or (b) a perturbed square lattice. (c) Spectral partitioning of thenetwork shown in (a). The spectral partitioning of the network shown in (b) isillustrated in Figure 2.4.3.

the corresponding Cartesian network, n2 = 2, n3 = 5, n4 = 22, n5 = 20, n6 =18, n7 = 11, and n8 = 3. The first few eigenvalues of the Laplacian matrix areλ = 0, 0.1965, 0.2050, 0.3547, 0.7003, 0.8131, 0.8908, 0.9472, 1.4048, and 1.5030.Multiple eigenvalues do not appear due to the lack of symmetry. The correspondingeigenvectors implementing spectral partitioning are shown in Figure 2.4.3.

2.4.5 Finite Element Network Descending from an Octahedron

A closed network associated with a finite element grid of triangles on a sphere,generated by the successive subdivision of an octahedral assembly, is shown in

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Graphs and Networks / / 45

FIGURE 2.4.3 Spectral partitioning of a network produced by the Delaunaytriangulation of a set of nodes deployed on a perturbed square lattice.

Figure 2.4.4(a). The number of nodes is N = 258, the number of links is L = 768, andthe node degree distribution is bimodal (n4 = 6 and n6 = 252), indicating a nearlyhexagonal structure. As seen previously, the number of links is significantly higherthan the number of nodes.

The first several eigenvalues of the Laplacian matrix are λ = 0, 0.1648 (triple),0.3946 (double), 0.5691 (triple), and 0.8253 (triple). The corresponding eigenvectorsimplementing spectral partitioning are shown in Figure 2.4.4(a).

2.4.6 Finite Element Network Descending from an Icosahedron

A closed network associated with a finite element grid of triangles on a sphere,generated by the successive subdivision of an icosahedral assembly, is shown in Fig-ure 2.4.4(b). The number of nodes is N = 162, the number of links is L = 480, andthe node degree distribution is bimodal, n5 = 12 and n6 = 150, indicating a nearlyhexagonal network.

The first few eigenvalues of the Laplacian matrix are λ = 0, 0.2643 (triple),0.7715 (quintic), and 1.3707 (triple). The corresponding eigenvectors implementingspectral partitioning are shown in Figure 2.4.4(b).

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46 / / AN INTRODUCTION TO GRIDS, GRAPHS, AND NETWORKS

(a)

(b)

FIGURE 2.4.4 Spectral partitioning of a network associ-ated with a finite element grid arising from the subdi-vision of (a) an octahedron or (b) an icosahedron on asphere.

Exercise

2.4.1 Delaunay triangulation

Generate a graph based on the Delaunay triangulation of a set of nodes of yourchoice.

2.5 LINK REMOVAL AND ADDITION

In science, engineering, and other applications, a graph typically describes a physicalor abstract network. Links can be attenuated or removed due to damage, or added toenhance the performance of the network, as discussed in Chapter 6. Link clipping

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Graphs and Networks / / 47

or addition alters the Laplacian matrix and may have a profound influence on theoverall performance of the network.

2.5.1 Single and Multiple Link

Suppose that one link numbered m is removed (clipped) from a network, where m =1, . . . , L and L is the total number of links in the pristine state. If L0 is the pristineLaplacian before clipping, then

(2.5.1) L = L0 – �(m) ⊗ �(m)

will be the altered Laplacian after clipping, where �(m) is the mth column of thepristine oriented incidence matrix before link removal, R0, and ⊗ denotes the tensorproduct of an ordered pair of vectors. The ij component of the symmetric matrix�(m) ⊗ �(m) is

(2.5.2) [�(m) ⊗ �(m)]ij = �(m)i �

(m)j .

We recall that the vector �(m) is filled with zeros, except that the entry correspondingto the first end node is –1, and the entry corresponding to the second end node is 1.The number of links, L, and the degrees of the two nodes defining the broken linkare reduced by one unit after link clipping.

Using Cauchy’s interlacing theorem, we find that the eigenvalues of the alteredmatrix, L, interlace those of the pristine matrix, L0, that is,

(2.5.3) 0 = λ1 = λ01 ≤ λ2 ≤ λ02 ≤ · · · ≤ λN ≤ λ0N ,

which means that all eigenvalues move toward zero, in agreement with physicalintuition (e.g., [15]).

One-Dimensional NetworkIn the case of a one-dimensional network discussed in Section 1.7, the pristineLaplacian is

(2.5.4) L0 =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

1 –1 0 · · · 0 0 0–1 2 –1 · · · 0 0 00 –1 2 · · · 0 0 0...

......

. . ....

......

0 0 0 · · · 2 –1 00 0 0 · · · –1 2 –10 0 0 · · · 0 –1 1

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦.

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48 / / AN INTRODUCTION TO GRIDS, GRAPHS, AND NETWORKS

After the mth link has been clipped, the altered Laplacian is

(2.5.5) L =

[L0m 00 L0

N–m

],

where L0m is the m×m pristine Laplacian and L0

N–m is the (N –m)× (N –m) pristineLaplacian.

For example, when m = 3, the altered Laplacian is

(2.5.6) L =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

1 –1 0 0 0 · · · 0 0 0 0–1 2 –1 0 0 · · · 0 0 0 00 –1 1 0 0 · · · 0 0 0 00 0 0 1 –1 0 · · · 0 0 00 0 0 –1 2 –1 · · · 0 0 00 0 0 0 –1 2 · · · 0 0 0...

......

......

.... . .

......

...0 0 0 0 0 0 · · · 2 –1 00 0 0 0 0 0 · · · –1 2 –10 0 0 0 0 0 · · · 0 –1 1

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦.

Removing one link in a one-dimensional graph results in a disconnected graph.The eigenvalues of the damaged Laplacian are

(2.5.7) λr = 4 sin2( r – 1

m

π

2

), λs = 4 sin2

( s – 1

N – m

π

2

)for r = 1, . . . ,m and s = 1, . . . ,N – m. Note that two eigenvalues are zero due tonetwork disruption. The union of these eigenvalues interlaces those of L0, as shownin Figure 2.5.1.

Multiple Link RemovalIf several links are clipped from a network, corresponding terms are subtracted fromthe right-hand side of (2.5.1). Suppose that M links are clipped, where M ≤ L. TheLaplacian after clipping is

0 0.5 1 1.5 2 2.5 3 3.5−0.2

0

0.2

λ

FIGURE 2.5.1 Eigenvalue spectrum of the Laplacian of a one-dimensionalnetwork with N = 16 nodes after the fourth link has been clipped,m = 4.The vertical lines mark the eigenvalues before clipping, and the square

and × symbols mark the eigenvalues after clipping.

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Graphs and Networks / / 49

(2.5.8) L = L0 –M∑i=1

�(mi) ⊗ �(mi),

where mi is the label of the ith clipped link and 1 ≤ mi ≤ L. If all links are clipped(M = L), the Laplacian reduces to the null matrix. A general theorem on the interlac-ing of the eigenvalues after multiple clippings is not available, except whenM = 1 orL. However, the second eigenvalue after multiple clippings, λ2, is guaranteed to beless than that before clipping [10].

The number of zero eigenvalues of the Laplacian after clipping,N0, is equal to thenumber of isolated nodes or clusters of nodes. If no links are clipped in a connectednetwork (M = 0), the number of zero eigenvalues is precisely equal to one,N0 = 1. Ifall links are clipped (M = L), the number of zero eigenvalues is equal to the numberof nodes,N0 = N. These observations suggest that the number of zero eigenvalues isa useful diagnostic of the operational state of a network.

When a small number of pL links remain intact in a randomly clipped, almostdevastated network, we obtain

(2.5.9) N0 � N – pL = N

(1 – p

L

N

),

irrespective of the network structure, where p � 0 is the percentage of active links.Higher-order terms in p depend on the network structure [37].

2.5.2 Link Addition

Suppose that one link labeled L + 1 is added to an existing graph with L links. If L0

is the Laplacian before addition, then

(2.5.10) L = L0 + �(L+1) ⊗ �(L+1)

will be the Laplacian after addition, where �(L+1) is the L + 1 column of the neworiented incidence matrix, R. The number of links, L, and the degrees of the twonodes defining the new link increase by one unit after addition. However, unlessnew nodes are introduced, the number of nodes, N, remains unchanged. Cauchy’sinterlacing theorem can be used to relate the eigenvalues of the Laplacian before andafter link addition.

Suppose that ν new links are added to an existing graph. If L0 is the Laplacianbefore addition, then

(2.5.11) L = L0 +ν∑p=1

�(L+p) ⊗ �(L+p)

will be the Laplacian after addition, where �(L+p) is the L + p column of the neworiented incidence matrix, R.

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50 / / AN INTRODUCTION TO GRIDS, GRAPHS, AND NETWORKS

Exercise

2.5.1 Periodic one-dimensional graph

Prepare the counterpart of Figure 2.5.1 for a periodic one-dimensional graphdiscussed in Section 1.8.

2.6 INFINITE LATTICES

Structured networks forming infinite lattices are convenient theoretical models forstudying the structural and transport properties of idealized states. Infinite latticesare typically visualized as crystals in physical two- or three-dimensional space. Otherisomorphic representations can be obtained by compressing, stretching, or deformingthese physical states.

The node degree of a lattice, d, is also called the lattice coordination number.When not all node degrees are equal, a mean coordination number can be defined asthe arithmetic average of all node degrees, dav. We recall from (2.1.6) that the ratioof the number of nodes to the number of links is

(2.6.1)N

L=

2

dav.

Although both N and L are infinite, the ratio N/L is well-defined, determined by themean coordination number.

2.6.1 Bravais Lattices

The node position of a two-dimensional Bravais lattice in physical state can beidentified by two indices, i1 and i2,

(2.6.2) xi1,i2 = x0,0 + i1 a1 + i2 a2,

where a1 and a2 are two corresponding base vectors. In three dimensions, threeindices are employed, i1, i2, and i3, and the nodal positions are

(2.6.3) xi1,i2,i3 = x0,0,0 + i1 a1 + i2 a2 + i3 a3,

where a1, a2, and a2 are three base vectors.Physically, a Bravais lattice appears the same, independent of the choice of the

anchor node, x0,0 in two dimensions or x0,0,0 in three dimensions.

Reciprocal LatticeA two-dimensional Bravais lattice has a reciprocal lattice whose base vectors, b1 andb2, satisfy the relation

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Graphs and Networks / / 51

(2.6.4) ai · bj = 2π δij,

where δij is Kronecker’s delta defined such that δij = 0 if i = j and δij = 1 if i = j.A three-dimensional Bravais lattice has a reciprocal lattice whose base vectors,

b1, b2, and b3, also satisfy equation (2.6.4). Using the properties of the triple mixedproduct, we find that

(2.6.5) b1 =2π

τa2 × a3, b2 =

τa3 × a1, b3 =

τa1 × a2,

where

(2.6.6) τ = a1 · (a2 × a3)

is the volume of the unit cell in the physical space. The nodes of the reciprocal latticeare located at

(2.6.7) lp1,p2,p3 = p1 b1 + p2 b2 + p3 b3,

where p1, p2, and p3 are three integers.

Periodic FunctionsConsider a function, f (x), that is repeated periodically in the direction of each basevector so that

(2.6.8) f (x) = f (x + i1 A1 + i2A2 + i3A3)

for any triplet of integers, i1, i2, and i3, where

(2.6.9) A1 = N1 a1, A2 = N2 a2, A3 = N3 a3,

and N1, N2, and N3 are specified integers determining the size of the periodic box.The periodic function can be expanded in a Fourier series,

(2.6.10) f (x) =∑

p1,p2,p3

cp1,p2,p3 exp(– i kp1,p2,p3 · x),

where cp1,p2,p3 are Fourier coefficients,

(2.6.11) kp1,p2,p3 = p1 B1 + p2 B2 + p3 B3

are wave numbers, p1, p2, and p3 are three integers, and

(2.6.12) B1 =1

N1b1, B2 =

1

N2b2, B3 =

1

N3b3

are fractions of the reciprocal base vectors. The sum in (2.6.10) is computed over afinite portion of the reciprocal lattice, called the discrete Brillouin zone or Wigner–Seitz cell.

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52 / / AN INTRODUCTION TO GRIDS, GRAPHS, AND NETWORKS

One-Dimensional LatticeIn the case of a uniform one-dimensional lattice along the x axis, we omit thesubscript 1 indicating the x direction and set

(2.6.13) a = a e, b =2π

ae, k =

Nae,

where a is the node separation and e is the unit vector along the x axis. The Fourierexpansion of a periodic function is

(2.6.14) f (x) =∑p

cp exp(– i p

N

x

a

).

Evaluating this expansion on the lattice nodes, xi = (i – 1) a, yields the Fourier series(1.8.12). The discrete Brillouin zone is discussed at the conclusion of Section 1.8.3.

Two-Dimensional Cartesian LatticeIn the case of a uniform two-dimensional Cartesian (square) lattice in the xy plane,we set

(2.6.15) a1 = a e1, a2 = a e2

and

(2.6.16) b1 =2π

ae1, b2 =

ae2,

and also define

(2.6.17) k1 =2π

N1 ae1, k2 =

N2 ae2,

where a is the common node separation in each direction and e1, e2 are unit vectorsalong the first and second directions, which can be identified with the x and y axes.Consequently,

(2.6.18) f (x, y) =∑p1,p2

cp1,p2 exp

(–i

[p1

N1

x

a+ p2

N2

y

a

]).

Evaluating this expansion at nodes defined by the grid lines

(2.6.19) xi1 = (i1 – 1)a, yi2 = (i2 – 1)a

for i1 = 1, . . . ,N1 and i2 = 1, . . . ,N2 yields the Fourier series discussed in Sec-tion 3.1.4. The discrete Brillouin zone of the two-dimensional lattice is discussednear the end of Section 3.1.4.

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Graphs and Networks / / 53

2.6.2 Archimedean Lattices

An Archimedean lattice consists of an infinite doubly periodic array regular poly-gons. In particular, each node is surrounded by the same sequence of polygons.Precisely 11 Archimedean lattices can be found, as shown in Figure 2.6.1. Thenotation (nm, kl, . . . ) signifies that each node is surrounded sequentially by m n-sided polygons, followed by l k-sided polygons and possibly other similar polygonsindicated be the three dots [13].

Square, A(44) LatticeThe Archimedean 44 lattice, also known as the square lattice, is a Bravais latticeconsisting of a doubly periodic array of empty squares, as shown in Figure 2.6.1(a),

(a) (b)

(c) (d)

(e) (f)

FIGURE 2.6.1 Illustration of the first six Archimedean lattices, including (a)the square, (b) honeycomb, (c) hexagonal, (d) kagomé (e) A(3, 122), and(f ) bathroom tile lattice. The dashed lines in (a, b, c, f ) describe the dual

lattice. The dual of the bathroom tile lattice (f ) is the Union Jack lattice

shown in Figure 2.6.2(a). (Continued on next page)

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54 / / AN INTRODUCTION TO GRIDS, GRAPHS, AND NETWORKS

(g) (h)

(i) (j)

(k)

FIGURE 2.6.1 (Continued) Illustration of the last five Archimedean lattices,including the (g) cross (h) ruby, (i) maple leaf, (j ) the A(33, 42), and (k) thepuzzle lattice. The dashed lines in (j) describe the dual lattice shown in

Figure 2.6.2(b).

where two base vectors are drawn with arrows. The notation 44 signifies that eachnode is surrounded by four squares. The lattice coordination number is d = 4 and theratio of the number of nodes to the number of links is N/L = 2/d = 1/2.

Hexagonal or Triangular, A(36) LatticeThe Archimedean 36 lattice, also known as the hexagonal or triangular lattice, is aBravais lattice consisting of two doubly periodic arrays of vacant triangles, as shownin Figure 2.6.1(b), where two base vectors are drawn with arrows. The notation 36

signifies that each node is surrounded by six triangles. The hexagonal lattice arisesfrom a sheared square lattice by introducing one family of slanted parallel links. Thelattice coordination number is d = 6 and the ratio of nodes to links is N/L = 1/3.

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Graphs and Networks / / 55

Honeycomb, A(63) LatticeThe Archimedean 63 lattice, also known as the honeycomb lattice, consists of a dou-bly periodic array of vacant hexagonal tiles, as shown in Figure 2.6.1(c). The notation63 signifies that each node is surrounded by three hexagons. The hexagonal latticeis a composite lattice consisting of two displaced triangular lattices, as discussed inSection 3.5. The lattice coordination number is d = 3 and the ratio of the number ofnodes to the number of links is N/L = 2/d = 2/3.

Kagomé, A(3, 6, 3, 6) LatticeThe Archimedean (3, 6, 3, 6) lattice, also known as the kagomé lattice, tiles theplane with triangles and hexagons, as shown in Figure 2.6.1(d). The Japanese wordkagomé means woven bamboo lattice. The notation (3, 6, 3, 6) signifies that eachnode is surrounded sequentially by one triangle, one hexagon, another triangle, andanother hexagon. The kagomé lattice is a composite Bravais lattice consisting ofthree displaced hexagonal lattices, as discussed in Section 3.6. The lattice coordina-tion number is d = 4 and the ratio of the number of nodes to the number of links isN/L = 1/2.

Star, A(3, 122) LatticeThe Archimedean (3, 122) lattice, also known as the star lattice, tiles the plane withtriangles and dodecagons (12-sided polygons), as shown in Figure 2.6.1(e). The no-tation (3, 122) signifies that each node is surrounded sequentially by one triangle andtwo dodecagons. The lattice coordination number is d = 3 and the ratio of the numberof nodes to the number of links is N/L = 2/3.

Square Octagon, Bathroom Tile, A(4, 82) LatticeThe Archimedean (4, 82) lattice, also known as the square octagon or bathroom tilelattice, covers the plane with squares and octagons, as shown in Figure 2.6.1(f ). Thenotation (4, 82) signifies that each node is surrounded sequentially by one square andtwo octagons. The lattice coordination number is d = 3 and the ratio of the numberof nodes to the number of links is N/L = 2/3. The bathroom tile lattice arises fromthe square lattice shown in Figure 2.6.1(a) by replacing alternating nodes with smalltilted squares.

Cross, A(4, 6, 12) LatticeThe Archimedean (4, 6, 12) lattice, also known as the cross lattice, tiles the planewith squares, hexagons, and dodecagons, as shown in Figure 2.6.1(g). The notation(4, 6, 12) signifies that each node is surrounded sequentially by one square, one hex-agon, and one dodecagon. The lattice coordination number is d = 3 and the ratio ofthe number of nodes to the number of links is N/L = 2/3.

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Ruby or Bounce, A(3, 4, 6, 4) LatticeThe Archimedean (3, 4, 6, 4) lattice, also known as the ruby or bounce lattice, cov-ers the plane with triangles, squares, and hexagons, as shown in Figure 2.6.1(h).The notation (3, 4, 6, 4) signifies that each node is surrounded sequentially by onetriangle, one square, one hexagon, and another square. The lattice coordination num-ber is d = 4 and the ratio of the number of nodes to the number of links isN/L = 1/2.

Maple Leaf, Snub Hexagonal, A(34, 6) LatticeThe Archimedean (34, 6) lattice, also known as the snub hexagonal or maple leaflattice, covers the plane with triangles and hexagons, as shown in Figure 2.1.1(i). Thenotation (346) signifies that each node is surrounded sequentially by four trianglesand one hexagon. The lattice coordination number is d = 5 and the ratio of thenumber of nodes to the number of links is N/L = 2/d = 2/5.

Bridge, A(33, 42) LatticeThe Archimedean (33, 42) lattice, also known as the bridge lattice, tiles the plane withtriangles and squares, as shown in Figure 2.6.1( j ). The notation (3342) signifies thateach node is surrounded sequentially by three triangles and two squares. The latticecoordination number is d = 5 and the ratio of the number of nodes to the number oflinks is N/L = 2/5.

Puzzle, Snub Square, A(32, 4, 3, 4) LatticeThe Archimedean (32, 4, 3, 4) lattice, also known as the snub square or puzzle lattice,tiles the plane with triangles and squares, as shown in Figure 2.6.1(k). The notation(32, 4, 3, 4) signifies that each node is surrounded sequentially by two triangles, onesquare, another triangle, and another square. The lattice coordination number is d = 5and the ratio of the number of nodes to the number of links is N/L = 2/d = 2/5.

2.6.3 Laves Lattices

Laves lattices, denoted by the prefix D, are the duals of the Archimedean lattices.A Laves lattice arises by introducing vertices in the middle of the tiles (faces) ofan Archimedean lattice and then connecting the vertices to cross the edges of theArchimedean lattice.

The dual of the square lattice is the same square lattice, the dual of the hexagonallattice is the honeycomb lattice, and the dual of the honeycomb lattice is the hexag-onal lattice, as shown in Figure 2.6.1(a–c). The dual lattices of the remaining eightArchimedean lattices are non-Archimedean lattices. Because all vertices do not havethe same degree, only a mean coordination number can be defined. Two examplesillustrated in Figure 2.6.2 are discussed in the remainder of this section.

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Graphs and Networks / / 57

(a) (b)

FIGURE 2.6.2 Illustration of (a) the Union Jack,D(4, 82), lattice and(b) the D(33, 42) lattice.

Union Jack, Tetrakis, Kisquadrille, D(4, 82) LatticeThe D(4, 82) Laves lattice, also called the Union Jack, tetrakis, or kisquadrille lattice,is shown in Figure 2.6.2(a). The node degrees are d = 4, 8, the mean node degree isd = 6, and the ratio of the number of nodes to the number of links is N/L = 1/3.

Pentagonal, D(33, 42) LatticeThe D(33, 42) Laves lattice, also called the pentagonal lattice, is shown in Fig-ure 2.6.2(b). The node degrees are d = 3, 4, the mean node degree is d = 10/3,and the ratio of the number of nodes to the number of links is N/L = 3/5.

2.6.4 Other Two-Dimensional Lattices

A variety of other lattices have been proposed. The martini lattice tiles the plane withtriangles and enneagons (nine-sided polygons), as shown in Figure 2.6.3(a) [13, 39].The lattice coordination number is d = 3 and the ratio of the number of nodes to thenumber of links is N/L = 2/d = 2/3. The martini lattice arises from the honeycomblattice by replacing every other junction around each hexagon with a triangle, therebyintroducing three additional edges.

(a) (b)

FIGURE 2.6.3 Illustration of (a) the martini lattice and (b) the bow-tielattice.

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58 / / AN INTRODUCTION TO GRIDS, GRAPHS, AND NETWORKS

The bow-tie lattice, shown in Figure 2.6.3(b), tiles the plane with triangles, andrectangles. The node degrees are d = 4 and 6, the mean node degree is d = 5, and theratio of the number of nodes to the number of links is N/L = 2/d = 2/5.

2.6.5 Cubic Lattices

Three Bravais cubic lattice are known, including the simple cubic lattice, the body-centered cubic (bcc) lattice, and the face-centered cubic (fcc) lattice, as shown inFigure 2.6.4.

(a)

(b)

(c)

FIGURE 2.6.4 Illustration of (a) the simple cu-bic lattice, (b) the body-centered cubic (bcc)lattice, and (c) the face-centered cubic (fcc)lattice. Links are shown as solid lines and

lattice reference lines are shown as broken

lines.

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Graphs and Networks / / 59

Simple Cubic LatticeThe simple cubic lattice is a Bravais lattice consisting of empty cubes, as shownin Figure 2.6.4(a), where the three base vectors are drawn with arrows. The latticecoordination number is d = 6 and the ratio of the number of nodes to the number oflinks is N/L = 2/d = 1/3.

Body-Centered Cubic LatticeThe body-centered cubic (bcc) lattice is a Bravais lattice consisting of two displacedsimple cubic lattices, as shown in Figure 2.6.1(b), where the three base vectors aredrawn with arrows. The lattice coordination number is d = 8 and the ratio of thenumber of nodes to the number of links is N/L = 2/d = 1/4.

Face-Centered Cubic latticeThe face-centered cubic (fcc) lattice is a Bravais lattice arising from the simple cu-bic lattice by introducing one node at the center of each square face, as shown inFigure 2.6.1(c), where the three base vectors are drawn with arrows. The latticecoordination number is d = 12 and the ratio of the number of nodes to the num-ber of links is N/L = 2/d = 1/6. The fcc lattice accommodates the densest possiblearray of spheres.

Exercise

2.6.1 Cartesian networks

Compute the reciprocal base vectors of a three-dimensional Cartesian network withbase vectors a1 = a e1, a2 = b e2, and a3 = c e3, where a, b, and c are three constantsand e1, e2, and e3, are Cartesian unit vectors.

2.7 PERCOLATION THRESHOLDS

With reference to the infinite lattices discussed in Section 2.6, now we address theimportant concept of percolation threshold determining the functional and opera-tional state of a damaged network. A distinction between the link and the nodepercolation threshold must be made at the outset according to the cause of the damageinflicted on a given pristine state.

2.7.1 Link (Bond) Percolation Threshold

Suppose that a fraction of links, qlink, are clipped randomly from a large test sectionof a pristine lattice containing L links, where 0 ≤ qlink ≤ 1. The fraction of intactlinks is plink = 1 – qlink. This means that the probability that an arbitrary link is intactis plink.

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(a) (b)

(c) (d)

FIGURE 2.7.1 Structure of a doubly periodic 16×16 square lattice afterM links have been clipped for (a)qlink =M/L = 0.1719, (b) 0.3809, (c)0.5840, and (d) 0.7754, where L is the number of links in the pristinestate. Unremoved links are shown as solid segments and removed

links are shown as broken segments.

As an example, the damaged states of a square or honeycomb doubly periodicnetwork are shown in Figures 2.7.1 and 2.7.2 for four values of qlink. Because of theinflicted damage, isolated clusters of nodes appear at sufficiently high values of thedamaged fraction, qlink.

As the sizes of the periodic boxes shown in Figures 2.7.1 and 2.7.2 increase inboth directions, the probability that a node belongs to a cluster spanning the periodicbox vanishes above a critical threshold, qlinkc . The corresponding probability,

(2.7.1) plinkc = 1 – qlinkc ,

defines the link or bond percolation threshold. Physically, as plink tends to plinkc fromlower values, the mean cluster size becomes infinite. Conversely, as plink tends toplinkc from higher values, the mean cluster size becomes finite.

In the case of a one-dimensional network consisting of an infinite or closed chainof links, the link percolation threshold is precisely zero, plinkc = 0. The reason is thatall links must be intact for a cluster spanning the entire network to appear.

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Graphs and Networks / / 61

(a) (b)

(c) (d)

FIGURE 2.7.2 Structure of a doubly periodic 16 × 16 honeycomb latticeafterM links have been clipped for (a) qlink =M/L = 0.2044, (b) 0.3893,(c) 0.5612, and (d) 0.8047, where L is the number of intact links. Intactlinks are shown as solid segments and removed links are shown as

broken segments.

2.7.2 Node Percolation Threshold

Now suppose that a fraction of nodes, qnode, are removed randomly from a large testsection of pristine lattice containing N nodes, along with the links originating fromeach node, where 0 ≤ qnode ≤ 1. The fraction of remaining nodes is pnode = 1–qnode.This means that the probability that an arbitrary node is intact is pnode. Since a linkis intact only if both end nodes are present, the corresponding probability that a linkis present is

(2.7.2) p = pnode2.

As an example, the damaged states of a square or honeycomb doubly periodic net-work are shown in Figures 2.7.3 and 2.7.4. As in the link removal problem, becauseof the damage inflicted, isolated clusters of nodes are observed at sufficiently highvalues of qnode.

As the sizes of the periodic boxes shown in Figures 2.7.3 and 2.7.4 increase inboth directions, the probability that an unremoved node belongs to a cluster spanningthe periodic box vanishes above a critical threshold, qnodec . The correspondingprobability,

(2.7.3) pnodec = 1 – qnodec ,

is the node or site percolation threshold. It is important to note that pnodec is not relatedto plinkc by (2.7.2).

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(a) (b)

(c) (d)

FIGURE 2.7.3 Structure of a doubly periodic 16 × 16 square latticeafter K nodes have been removed for (a) qnode = K /N = 0.1797,(b) 0.3945, (c) 0.5977, and (d) 0.7773, where N is the number ofnodes. Intact links are shown as solid segments, removed links are

shown as broken segments, and unremoved nodes are marked as

circles inside a period.

In the case of a one-dimensional network consisting of an infinite or closed chainof links, the node percolation threshold is zero, pnodec = 0. The reason is that all linksmust be intact for a cluster spanning the entire network to appear.

2.7.3 Computation of Percolation Thresholds

To compute the link percolation threshold, we may consider the ghost of a latticeconsisting of invisible links playing the role of nameplates. In the bond percolationproblem, functional links are gradually introduced to replace the ghost links untillong-range connectivity is established at the bond percolation threshold. In the sitepercolation problem, nodes and their associated links are introduced until long-rangeconnectivity is established at the site percolation threshold.

Exact link and bond percolation thresholds are known only for a few lattice ge-ometries [40, 43, 44, 57]. Remarkably accurate percolation thresholds have beencalculated by numerical methods for other lattices (e.g., [28, 42, 58]). A compre-hensive compilation accompanied by an extensive list of references is available at

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Graphs and Networks / / 63

(a) (b)

(c) (d)

FIGURE 2.7.4 Structure of a doubly periodic 16 × 16 honeycomb latticeafter K nodes have been removed for (a) qnode = K /N = 0.1953, (b)0.4141, (c) 0.5938, and (d) 0.7969, whereN is the number of nodes. Un-

removed links are shown as solid segments, removed links are shown

as broken segments, and unremoved nodes are marked as circles

inside a period.

the Internet site: http://en.wikipedia.org/wiki/Percolation_threshold. Results for aseveral lattices are shown in Table 2.7.1.

The link (bond) percolation threshold of an Archimedean lattice and its corre-sponding Laves lattice add up to unity. The link percolation threshold of a latticeis precisely equal to the node percolation threshold of its covering lattice. Forexample, the link percolation threshold of the honeycomb lattice is equal to the nodepercolation threshold of the kagomé lattice.

CorrelationsGraphs of the percolation thresholds against the lattice coordination number or meannode degrees, d, are shown in Figure 2.7.5. Partially successful efforts have beenmade to derive universal formulas for these thresholds in terms of the lattice coor-dination number and possibly other parameters (e.g., [53]). Of particular interestare simple formulas that provide us with easily computable estimates for use inengineering risk analysis and design.

For the link percolation problem, Vyssotsky et al. [48] proposed the approxima-tion

(2.7.4)(plinkc

)2D � 2

d,

(plinkc

)3D � 3

2

1

d,

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64 / / AN INTRODUCTION TO GRIDS, GRAPHS, AND NETWORKS

TABLE 2.7.1 (a) Link (Bond) and (b) Node (Site) Percolation Thresholds for Several Lattices

Space Lattice d d plinkc pnodec Figure

1 Open chain 2 1.0 1.0

2 Square 4 0.5a 0.59275 2.6.1(a)

2 Hexagonal 6 0.34730b 0.5c 2.6.1(b )

2 Honeycomb 3 0.65270d 0.69704 2.6.1(c )

2 Kagomé 4 0.52441 0.65270d 2.6.1(d )

2 A(3, 122) 3 0.74042 0.80790e 2.6.1(e )

2 Bathroom tile 3 0.67680 0.72972 2.6.1(f )

2 Cross 3 0.69373 0.74781 2.6.1(g)

2 Ruby 4 0.52483 0.62182 2.6.1(h)

2 Maple leaf 5 0.43431 0.57950 2.6.1(i )

2 A(33, 42) 5 0.41964 0.55021 2.6.1(j )

2 Puzzle 5 0.41414 0.55081 2.6.1(k )

2 D(33, 42) 3, 4 313

0.58035 0.64708 2.6.2(a)

2 D-Bathroom tile 4, 8 6 0.23220 0.5c 2.6.2(b )

2 Martini 3 0.70711f 0.76482g 2.6.3(a)

2 Bowtie 4, 6 5 0.404518h 0.547 2.6.3(b )

3 Simple cubic 6 0.24881 0.31160 2.6.4(a)

3 bcc 8 0.18029 0.246 2.6.4(b )

3 fcc 12 0.12016 0.19923 2.6.4(c )

3 Diamond 4 0.43

a This is an exact value [43, 44].

b The exact value is a root of the cubic equation x3 – 3x + 1 = 0, given by plinkc = 2 sin π18 [43, 44].

c This is an exact value.

d The exact value is 1 – 2 sin π18 [40]. The Kagomé lattice is the covering lattice of the honeycomb

lattice.

e The exact value is (1 – 2 sin π18 )1/2 [40].

f The exact value is 1/√2 [57].

g The threshold is a root of the quartic equation x4 – 3x3 + 1 = 0 [57].

h The threshold is a root of the quintic equation x5 – 6x3 + 6x2 + x – 1 = 0.

in two or three dimensions, where d is the lattice coordination number or mean nodedegree. These estimates, represented by the solid and broken lines in Figure 2.7.5(a),are in good agreement with known exact results.

Formula (2.7.4) for a two-dimensional lattice is consistent with the more generalformula

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Graphs and Networks / / 65

(a)

2 4 6 8 10 120

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

d

p clin

k

(b)

2 3 4 5 6 7 8 9 10 11 120

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

d

p cnode

FIGURE 2.7.5 Percolation thresholds plotted againstthe lattice coordination number or mean nodedegress, d , for (a) link and (b) node percolation. Two-

dimensional lattices are represented by circles or

crosses and three-dimensional lattices are repre-

sented by squares or × symbols. The solid and bro-

ken lines represent, respectively, the predictions of

(2.7.4) for two- or three-dimensional lattices.

(2.7.5) plinkc � 2

d=N

L,

where d is the mean node degree, N is the number of nodes, and L is the number oflinks. This approximation is motivated by the functional dependence of the numberof zero eigenvalues of the Laplacian on q [37]. Although more involved formulas forpredicting link percolation thresholds in regular lattices have been proposed (e.g.,[46]), their practical utility is called into question and their generalization to finiteand inhomogeneous networks is unclear.

Scher and Zallen [41] introduced the notion of critical node percolation density,ρc, defined with respect to the distance of a node from its nearest neighbor. Thedistance is identified with the diameter of a disk in two dimensions or a sphere in

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66 / / AN INTRODUCTION TO GRIDS, GRAPHS, AND NETWORKS

thee dimensions. If φ is the filling factor, defined as the fraction of the plane or spaceoccupied by all circles or spheres, then

(2.7.6) pnodec � 1

φρc.

It remarkable that ρc is nearly constant, equal to 0.44 for a two-dimensional latticeor 0.154 for a three-dimensional lattice.

Exercise

2.7.1 Link percolation thresholds

Verify from the results shown in Table 2.7.1 that the link (bond) percolation thresholdof an Archimedean lattice and its corresponding Laves lattice add up to unity.

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/ / / 3 / / / SPECTRA OF LATTICES

The nodes of a two- or three-dimensional regular lattice, regardedas a structured network, can be identified by two or three indices assigned to the in-dividual lattice directions. The adjacency and Laplacian matrices can be compiled byinspection and their properties can be assessed by standard analytical methods. Sev-eral lattice networks with different boundary conditions are discussed in this chapterand the spectrum of their Laplacian is delineated. The results will find applications inChapter 5 for computing of lattice Green’s functions and in Chapter 6 for analyzingthe performance of conducting networks.

3.1 SQUARE LATTICE

A network whose structure is isomorphic to that of a square lattice consists of twointersecting one-dimensional arrays of links. A rectangular patch of a square latticecontaining N1 links in the first direction, parametrized by the index i1, and N2 links inthe second direction, parametrized by the index i2, is shown in Figure 3.1.1. Isolated,periodic, doubly periodic, and other configurations of a distributed nodal field canbe envisioned.

N2

2

2

1

i2

i11 N1

FIGURE 3.1.1 Illustration of a rectangular patch of a squarenetwork containing N1 links in the first direction and N2

links in the second direction. All links are assumed to

have the same conductance.

67

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68 / / AN INTRODUCTION TO GRIDS, GRAPHS, AND NETWORKS

A theorem due to Fiedler states that the eigenvectors of the Laplacian matrix forcertain types of boundary conditions are tensor products of those of the constituentone-dimensional graphs, and the eigenvalues are the sums of the eigenvalues of theLaplacian of the constituent one-dimensional graphs. [10]. This property reflects theseparability of the discrete Laplace operator in Cartesian coordinates.

3.1.1 Isolated Network

The total number of nodes in the isolated network shown in Figure 3.1.1 is

(3.1.1) N = (N1 + 1)(N2 + 1)

and the total number of links is

(3.1.2) L = N1(N2 + 1) + (N1 + 1)N2.

Note that the number of links is significantly higher than the number of nodes.The nodal values of a nodal scalar field, ψ , can be compiled in a sequence of

horizontal layers from the bottom where i2 = 1 to the top where i2 = N2 + 1, into anN-dimensional vector

(3.1.3) ψ =

⎡⎢⎢⎢⎢⎢⎣ψ (1)

ψ (2)

...ψ (N2)

ψ (N2+1)

⎤⎥⎥⎥⎥⎥⎦,

where the subvectors

(3.1.4) ψ (1) ≡

⎡⎢⎢⎢⎣ψ1,1

ψ2,1...ψN1+1,1

⎤⎥⎥⎥⎦, . . . , ψ (N2+1) ≡

⎡⎢⎢⎢⎣ψ1,N2+1

ψ2,N2+1...ψN1+1,N2+1

⎤⎥⎥⎥⎦encapsulate horizontal profiles. The Laplacian matrix consists of N2 + 1 rows of(N1 + 1) × (N1 + 1) square tridiagonal blocks, F, E, and I, in the followingconfiguration:

(3.1.5) L =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

F –I 0 · · · 0 0 0–I E –I · · · 0 0 00 –I E · · · 0 0 0...

......

. . ....

......

0 0 0 · · · E –I 00 0 0 · · · –I E –I0 0 0 · · · 0 –I F

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦,

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Spectra of Lat t ices / / 69

where I is the (N1 + 1) × (N1 + 1) identity matrix. When N1 = 3, we have

(3.1.6) F =

⎡⎢⎢⎣2 –1 0 0–1 3 –1 00 –1 3 –10 0 –1 2

⎤⎥⎥⎦, E =

⎡⎢⎢⎣3 –1 0 0–1 4 –1 00 –1 4 –10 0 –1 4

⎤⎥⎥⎦.The two entries of F correspond to corner nodes, the three entries of F and Ecorrespond to boundary nodes, and the four entries of E correspond to interiornodes.

The eigenvalues of the Laplacian matrix are

(3.1.7) λn1, n2 = 4 sin2(12 αn1

)+ 4 sin2

(12 βn2

)or

(3.1.8) λn1, n2 = 4 – 2 cosαn1 – 2 cosαn2 ,

where

(3.1.9) αn1 =n1 – 1

N1 + 1π , βn2 =

n2 – 1

N2 + 1π

for n1 = 1, . . . ,N1 + 1 and n2 = 1, . . . ,N2 + 1.The corresponding eigenvectors, u(n1, n2), normalized so that their lengths are

equal to unity, u(n1, n2) · u(n1, n2) = 1, are

(3.1.10) un1, n2i1, i2= An1Bn2

2√(N1 + 1)(N2 + 1)

cos

[(i1 –

1

2

)αn1

]cos

[(i2 –

1

2

)βn2

]for n1, i1 = 1, . . . ,N1 + 1 and n2, i2 = 1, . . . ,N2 + 1, where An1 = 1, Bn2 = 1, exceptthat A1 = 1/

√2 and B1 = 1/

√2.

The spectral partitioning of a 17 × 17 network is shown in Figure 2.2.1. Posi-tive eigenvector components are marked with filled circles, negative components aremarked with dots, and zero components are unmarked.

3.1.2 Periodic Strip

With continued reference to the rectangular patch of the square lattice shown in Fig-ure 3.1.1, now we assume that a nodal scalar field, ψ , is periodic in the first directionso that

(3.1.11) ψ1,i2 = ψN1+1,i2

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70 / / AN INTRODUCTION TO GRIDS, GRAPHS, AND NETWORKS

for i2 = 1, . . . ,N2 + 1. The vector of unknown nodal values is

(3.1.12) ψ =

⎡⎢⎢⎢⎢⎢⎣ψ (1)

ψ (2)

...ψ (N2)

ψ (N2+1)

⎤⎥⎥⎥⎥⎥⎦,

where the subvectors

(3.1.13) ψ (1) =

⎡⎢⎢⎢⎢⎢⎣ψ1,1

ψ2,1...ψN1–1,1

ψN1,1

⎤⎥⎥⎥⎥⎥⎦, . . . , ψ (N2+1) =

⎡⎢⎢⎢⎢⎢⎣ψ1,N2+1

ψ2,N2+1...ψN1–1,N2+1

ψN1,N2+1

⎤⎥⎥⎥⎥⎥⎦encapsulate horizontal profiles. The total number of entries in the vector ψ isN = N1(N2 + 1). Note that nodal values along the left side are not included in thevector of unknowns, as they are periodic images of those along the right side of therectangular strip.

The Laplacian matrix consists of N2+1 rows of N1×N1 square, nearly tridiagonalblocks, F and E,

(3.1.14) L =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

F –I 0 · · · 0 0 0–I E –I · · · 0 0 00 –I E · · · 0 0 0...

......

. . ....

......

0 0 0 · · · E –I 00 0 0 · · · –I E –I0 0 0 · · · 0 –I F

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦,

where I is the N1 × N1 identity matrix. For example, when N1 = 5, we have

(3.1.15) F =

⎡⎢⎢⎢⎢⎣3 –1 0 0 –1–1 3 –1 0 00 –1 3 –1 00 0 –1 3 –1–1 0 0 –1 3

⎤⎥⎥⎥⎥⎦and

(3.1.16) E =

⎡⎢⎢⎢⎢⎣4 –1 0 0 –1

–1 4 –1 0 00 –1 4 –1 00 0 –1 4 –1

–1 0 0 –1 4

⎤⎥⎥⎥⎥⎦.

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Spectra of Lat t ices / / 71

The three entries of F correspond to the bottom and top edge nodes. The northeasternand southwestern one entries of F and E implement the periodicity condition.

The eigenvalues of the Laplacian matrix are

(3.1.17) λn1, n2 = 4 sin2(12 αn1

)+ 4 sin2

(12 βn2

)or

(3.1.18) λn1, n2 = 4 – 2 cosαn1 – 2 cos αn2 ,

where

(3.1.19) αn1 =n1 – 1

N12π , βn2 =

n2 – 1

N2 + 1π

for n1 = 1, . . . ,N1 and n2 = 1, . . . ,N2 + 1.The corresponding eigenvectors, un1,n2 , normalized so that their lengths are equal

to unity, un1, n2 · un1, n2∗= 1, are

(3.1.20) un1, n2i1, i2= An1

√2√

N1(N2 + 1)exp(– i i1αn1

)cos

[(i2 –

1

2

)βn2

]for n1, i1 = 1, . . . ,N1 and n2, i2 = 1, . . . ,N2 + 1, where i is the imaginary unit, anasterisk denotes the complex conjugate, and An1 = 1, except that A1 = 1/

√2.

Shown in Figure 3.1.2 is a network with N1 = 16 and N2 = 8 divisions. The firstfew eigenvalues of the Laplacian are λ = 0, 0.1206, 0.1522 (double), 0.2729 (dou-ble), 0.4679, 0.5858 (double), 0.6203 (double), 0.7064 (double), and 1.0000. Thecorresponding eigenvectors implementing spectral partitioning are also displayed.

Vertical StripWith continued reference to the square network shown in Figure 3.1.1, now weassume that the nodal scalar field, ψ , is periodic in the second direction,

(3.1.21) ψi1,1 = ψi1,N2+1

for i1 = 1, . . . ,N1. The vector of unknown nodal values is

(3.1.22) ψ =

⎡⎢⎢⎢⎢⎢⎣ψ (1)

ψ (2)

...ψ (N2–1)

ψ (N2)

⎤⎥⎥⎥⎥⎥⎦,

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FIGURE 3.1.2 Spectral partitioning of a periodic strip of a square lattice with N1 = 16

andN2 = 8 divisions. Positive eigenvector components are marked with filled circles,

negative components are marked with dots, and zero components are unmarked.

where the subvectors

(3.1.23) ψ (1) =

⎡⎢⎢⎢⎣ψ1,1

ψ2,1...ψN1+1,1

⎤⎥⎥⎥⎦ , . . . , ψ (N2) =

⎡⎢⎢⎢⎣ψ1,N2

ψ2,N2...ψN1+1,N2

⎤⎥⎥⎥⎦encapsulate horizontal profiles. The total number of entries in the vector ψ isN = (N1 + 1)N2. Note that the nodal values along the top side of the rectangularstrip are not included in the vector of unknowns.

The Laplacian consists of N2 rows of (N1 +1)× (N1 +1) square tridiagonal blocksand two negative unit corner blocks implementing the periodicity condition in thesecond direction,

(3.1.24) L =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

E –I 0 · · · 0 0 –I–I E –I · · · 0 0 00 –I E · · · 0 0 0...

......

. . ....

......

0 0 0 · · · E –I 00 0 0 · · · –I E –I–I 0 0 · · · 0 –I E

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦,

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Spectra of Lat t ices / / 73

where E is an (N1 + 1) × (N1 + 1) tridiagonal matrix and I is the (N1 + 1) × (N1 + 1)identity matrix. The northeastern and southwestern identity blocks implement theperiodicity condition in the second direction. When N1 = 4, we have

(3.1.25) E =

⎡⎢⎢⎢⎢⎣3 –1 0 0 0

–1 4 –1 0 00 –1 4 –1 00 0 –1 4 –10 0 0 –1 3

⎤⎥⎥⎥⎥⎦.The three corner entries correspond to the left and right boundary nodes.

The eigenvalues of the Laplacian matrix are

(3.1.26) λn1, n2 = 4 sin2(12 αn1

)+ 4 sin2

(12 βn2

)or

(3.1.27) λn1, n2 = 4 – 2 cosαn1 – 2 cos αn2 ,

where

(3.1.28) αn1 =n1 – 1

N1 + 1π , βn2 =

n2 – 1

N22π

for n1 = 1, . . . ,N1 + 1 and n2 = 1, . . . ,N2.The corresponding eigenvectors, un1, n2 , normalized so that their lengths are equal

to unity, un1,n2 · un1, n2∗= 1, are

(3.1.29) un1,n2i1, i2= Bn2

√2√

(N1 + 1)N2cos[(i1 – 1

2

)αn1]exp(–i i2 βn2)

for n1, i1 = 1, . . . ,N1 + 1 and n2, i2 = 1, . . . ,N2, where i is the imaginary unit andBn2 = 1, except that B1 = 1/

√2.

3.1.3 Doubly Periodic Network

With reference to the rectangular network shown in Figure 3.1.1, now we assumethat the nodal scalar field is periodic in two directions so that

(3.1.30) ψ1,i2 = ψN1+1,i2 , ψi1,1 = ψi1,N2+1.

The vector of unknown nodal values inside each period is

(3.1.31) ψ =

⎡⎢⎢⎢⎢⎢⎣ψ (1)

ψ (2)

...ψ (N2–1)

ψ (N2)

⎤⎥⎥⎥⎥⎥⎦,

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74 / / AN INTRODUCTION TO GRIDS, GRAPHS, AND NETWORKS

where the subvectors

(3.1.32) ψ (1) =

⎡⎢⎢⎢⎣ψ1,1

ψ2,1...ψN1,1

⎤⎥⎥⎥⎦, . . . , ψ (N2) =

⎡⎢⎢⎢⎣ψ1,N2

ψ2,N2...ψN1,N2

⎤⎥⎥⎥⎦encapsulate horizontal profiles. The total number of entries in the vector ψ isN = N1N2. Note that the nodal values along the right and top boundaries of therectangular strip are not included in the vector of unknowns.

The Laplacian consists of N2 rows of N1 ×N1 nearly tridiagonal blocks, E, upperand lower diagonal negative unit blocks, –I, and two negative unit corner blocksimplementing the periodicity condition in the second direction,

(3.1.33) L =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

E –I 0 · · · 0 0 –I–I E –I · · · 0 0 00 –I E · · · 0 0 0...

......

. . ....

......

0 0 0 · · · E –I 00 0 0 · · · –I E –I–I 0 0 · · · 0 –I E

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦,

where I is the N1 × N1 identity matrix. The northeastern and southwestern cor-ner blocks of L implement the periodicity condition in the second direction. WhenN1 = 5, we have

(3.1.34) E =

⎡⎢⎢⎢⎢⎣4 –1 0 0 –1

–1 4 –1 0 00 –1 4 –1 00 0 –1 4 –1

–1 0 0 –1 4

⎤⎥⎥⎥⎥⎦.

The northeastern and southwestern corner elements of E implement the periodicitycondition in the first direction.

An eigenvalue, λ, and the corresponding eigenvector, u, of the doubly periodicLaplacian, L, satisfy the equation

(3.1.35) 4 ui1, i2 – ui1–1, i2 – ui1+1, i2 – ui1, i2–1 – ui1, i2+1 = λ ui1, i2

at any node, (i1, i2).We find that the eigenvalues are given by

(3.1.36) λn1, n2 = 4 sin2(12 αn1

)+ 4 sin2

(12 βn2

)

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Spectra of Lat t ices / / 75

or

(3.1.37) λn1, n2 = 4 – 2 cosαn1 – 2 cosβn2 ,

where

(3.1.38) αn1 =n1 – 1

N12π , βn2 =

n2 – 1

N22π

for n1 = 1, . . . ,N1 and n2 = 1, . . . ,N2. We can write

(3.1.39) αn1 = (n1 – 1) k1, βn2 = (n2 – 1) k2,

where the parameters

(3.1.40) k1 =2π

N1, k2 =

N2

are directional wave numbers.The corresponding eigenvectors, un1, n2 , normalized so that their lengths are equal

to unity, un1, n2 · un1, n2∗= 1, are

(3.1.41) un1, n2i1, i2=

1√N1N2

exp[– i (i1αn1 + i2βn2)

]for n1, i1 = 1, . . . ,N1 and n2, i2 = 1, . . . ,N2, where i is the imaginary unit andan asterisk denotes the complex conjugate. The eigenvalues given in (3.1.37) canbe computed by substituting the eigenvectors given in (3.1.41) into (3.1.35) andsimplifying the resulting expression.

Block Circulant MatricesAn alternative method of deriving the eigenvalues hinges on the observation that thedoubly periodic Laplacian (3.1.33) is a block circulant matrix, that is, a circulantmatrix whose scalar elements are replaced by constituent matrices. A theorem due toFriedman [12] states that the spectrum of eigenvalues of this matrix is the union ofthe spectra of the following N1 × N1 matrices:

(3.1.42) L(n2) = – exp(–iβn2) I + E – exp(iβn2) I

or

(3.1.43) L(n2) = –2 cosβn2 I + E

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76 / / AN INTRODUCTION TO GRIDS, GRAPHS, AND NETWORKS

for n2 = 1, . . . ,N2. For example, when N1 = 4,

(3.1.44) L(n2) =

⎡⎢⎢⎣4 – 2 cos βn2 –1 0 –1

–1 4 – 2 cos βn2 –1 00 –1 4 – 2 cos βn2 –1–1 0 –1 4 – 2 cosβn2

⎤⎥⎥⎦.Using expression (1.8.2) for the eigenvalues of the one-dimensional periodic Lapla-cian, we recover the eigenvalues displayed in (3.1.37).

A doubly periodic network with N1 = 12 and N2 = 8 divisions is shownin Figure 3.1.3. The first few eigenvalues of the Laplacian matrix are λ = 0,0.2679 (double), 0.5858 (double), 0.8537 (quadruple), 0.1351 (double), 1.0000 (dou-ble), and 1.5858 (quadruple). The corresponding eigenvectors implementing spectralpartitioning of a doubly periodic field are also shown.

Fourier Expansions on a Cartesian GridA real, doubly periodic nodal field, ψ , defined over an N1 ×N2 square lattice can beexpanded into a doubly Fourier series so that

(3.1.45) ψi1, i2 =M1∑

p1=–M1

M2∑p2=–M2

cp1, p2 exp(– i[(i1 – 1) p1k1 + (i2 – 1) p2k2

] ),

where M1 and M2 are two appropriate truncation levels and cp1, p2 are complex Fou-rier coefficients. If the number of intervals, N1, is odd, we truncate the double Fouriersum at the value M1 = (N1 – 1)/2. If N1 is even, we truncate the double Fourier sumat the value M1 = N1/2. Similar truncation levels apply toM2 (e.g., [35]).

To ensure that the right-hand side of (3.1.45) is real, we require that

(3.1.46) c–p1, –p2 = c∗p1, p2 ,

where an asterisk denotes the complex conjugate. The complex Fourier coefficientsare given by

(3.1.47) cp1, p2 =1

N1N2

(q1 + ω

p11 q2 + ω

2p12 q3 + · · · + ω(N1–1)p1

1 qN1

),

where

(3.1.48) qm = ψm,1 + ωp22 ψm, 2 + ω

2p22 ψm, 3 + · · · + ω(N2–1)p2

y ψm,N2

and we have defined

(3.1.49) ω1 = exp(i k1), ω2 = exp(i k2).

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Spectra of Lat t ices / / 77

FIGURE 3.1.3 Spectral partitioning of a periodic square lattice withN1 =

12 and N2 = 8 divisions inside each period. Positive eigenvector

components are marked with filled circles, negative components are

marked with dots, and zero components are unmarked.

3.1.4 Doubly Periodic Sheared Network

The nodal field of a doubly periodic Cartesian network that is sheared along first axisaxis satisfies the periodicity conditions

(3.1.50) ψ1, j = ψN1+1, j, ψi,1 = ψi+r,N2+1,

where r is a specified integer. The Laplacian matrix is given in (3.1.33), except thatthe northeastern corner block, I, is replaced by

(3.1.51) J =

[0 IN1–r

Ir 0

],

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FIGURE 3.1.4 Spectral partitioning of a sheared periodic square lattice withN1 = 12 andN2 = 12 divisions inside each period, for shearing level r = 6.Positive eigenvector components are marked with filled circles, negative

components are marked with dots, and zero components are unmarked.

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Spectra of Lat t ices / / 79

and the southwestern corner block of L is replaced by JT , where Ip is the p × pidentity matrix. For example, when N1 = 8 and r = 5, we have

(3.1.52) J =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 0 0 0 0 1 0 00 0 0 0 0 0 1 00 0 0 0 0 0 0 11 0 0 0 0 0 0 00 1 0 0 0 0 0 00 0 1 0 0 0 0 00 0 0 1 0 0 0 00 0 0 0 1 0 0 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦.

When r = 0 or N1, the matrix J reduces to the N1 × N1 identity matrix, I.The spectral partitioning of a sheared network with N1 = 12 and N2 = 12

divisions is shown in Figure 3.1.4.

Exercises

3.1.1 Particle vibrations

The particles of a two-dimensional crystal are arranged on a square latticeparametrized by two indices, i1 and i2, in the xy plane. Small departures from theequilibrium position generate restoring forces. The motion of the (i1, i2) particle isgoverned by Newton’s law,

(3.1.53) md2xi1, i2dt2

= k(xi1+1, i2 + xi1–1, i2 + xi1, i2+1 + xi1, i2–1 – 4 xi1, i2

),

where m is the particle mass, k is a spring constant, and t stands for time. In the caseof harmonic oscillations,

(3.1.54) xi1, i2 = wi1, i2 exp(–iωt),

where i is the imaginary unit, ω is the angular frequency, and wi1, i2 is an eigendis-placement. Derive and solve an algebraic eigenvalue problem for the eigenfrequen-cies and eigendisplacements.

3.1.2 Periodic Laplacian

Confirm by direct substitution that the eigenvalues given in (3.1.36) and correspond-ing eigenvectors given in (3.1.37) satisfy equation (3.1.35).

3.2 MÖBIUS STRIPS

A section of a Cartesian strip can be twisted by 180◦ around its length into the con-figuration shown in Figure 3.2.1(a). A sequence of twisted Cartesian strips can beglued together to form a helical strip. A finite twisted strip can be bent, and thenarrow edges can be attached to yield the Möbius strip shown in Figure 3.2.1(b).

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80 / / AN INTRODUCTION TO GRIDS, GRAPHS, AND NETWORKS

(a) (b)

FIGURE 3.2.1 (a) A Cartesian strip is twisted by 180◦ along itslength to give a helical strip. (b) The helical strip can be bent andthe narrow edges can be attached to yield the Möbius strip.

3.2.1 Horizontal Strip

The nodal profile of a Möbius strip in the direction of the first index, i1, satisfies thereverse periodicity condition

(3.2.1) ψ1,i2 = ψN1+1,N2+2–i2

for i2 = 1, . . . ,N2. For example, the southwestern nodal value is equal to thenortheastern nodal value,

(3.2.2) ψ1,1 = ψN1+1,N2+1.

The vector of nodal values encapsulating N = N1(N2 + 1) unknowns is

(3.2.3) ψ =

⎡⎢⎢⎢⎢⎢⎣ψ (1)

ψ (2)

...ψ (N2)

ψ (N2+1)

⎤⎥⎥⎥⎥⎥⎦,

where

(3.2.4) ψ (1) =

⎡⎢⎢⎢⎢⎢⎣ψ1,1

ψ2,1...ψN1–1,1

ψN1,1

⎤⎥⎥⎥⎥⎥⎦, . . . , ψ (N2+1) =

⎡⎢⎢⎢⎢⎢⎣ψ1,N2+1

ψ2,N2+1...ψN1–1,N2+1

ψN1,N2+1

⎤⎥⎥⎥⎥⎥⎦ .

The Laplacian matrix consists ofN2+1 rows of N1×N1 square tridiagonal blocks,F and E, and a chain of sparse backdiagonal blocks,

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Spectra of Lat t ices / / 81

(3.2.5) L =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

F –I 0 · · · 0 0 –J–I E –I · · · 0 –J 00 –I E · · · –J 0 0...

......

. . ....

......

0 0 –J · · · E –I 00 –J 0 · · · –I E –I–J 0 0 · · · 0 –I F

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦,

where I is the N1 × N1 unit matrix. The N1 × N1 matrix J is null, expect that thenortheastern and southwestern corner elements are equal to unity. When N2 is even,the central element of L is E – J. When N1 = 4,

(3.2.6) F =

⎡⎢⎢⎣3 –1 0 0–1 3 –1 00 –1 3 –10 0 –1 3

⎤⎥⎥⎦, E =

⎡⎢⎢⎣4 –1 0 0–1 4 –1 00 –1 4 –10 0 –1 4

⎤⎥⎥⎦and

(3.2.7) I =

⎡⎢⎢⎣1 0 0 00 1 0 00 0 1 00 0 0 1

⎤⎥⎥⎦, J =

⎡⎢⎢⎣0 0 0 10 0 0 00 0 0 01 0 0 0

⎤⎥⎥⎦.The three entries of F correspond to the bottom and top edge nodes.

The eigenvalues of the Laplacian matrix are

(3.2.8) λn1, n2 = 4 sin2(12 αn1, n2

)+ 4 sin2

(12 βn2

)or

(3.2.9) λn1, n2 = 4 – 2 cosαn1, n2 – 2 cosαn2 ,

where

(3.2.10) αn1, n2 =n1 – 1 + γn2

N12π , βn2 =

n2 – 1

N2 + 1π

for n1 = 1, . . . ,N1 and n2 = 1, . . . ,N2 + 1, where γn2 = 0 if n2 is odd and γn2 = 1/2 ifn2 is even [56]. Formally, we write

(3.2.11) γn2 =1 + (–1)n2

4.

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Note that these expressions differ from those given in (3.1.19) for the periodicrectangular network only by the presence of γn2 in the first fraction.

The corresponding eigenvectors, un1, n2 , normalized so that their lengths are equalto unity, un1, n2 · un1, n2∗

= 1, are given by

(3.2.12) un1, n2i1, i2= An1

√2√

N1(N2 + 1)exp(– i i1αn1, n2) cos

[(i2 – 1

2

)βn2

]for n1, i1 = 1, . . . ,N1 and n2, i2 = 1, . . . ,N2 + 1, where i is the imaginary unit, anasterisk denotes the complex conjugate, and An1 = 1, except that A1 = 1/

√2.

A Möbius network with N1 = 16 and N2 = 8 divisions is shown in Figure 3.2.2.The first few eigenvalues of the Laplacian matrix are λ = 0, 0.1522 (double), 0.1590(double), 0.4577 (double), 0.4679, 0.5858 (double), 0.6202 (double), and 1.0095(double). The corresponding eigenvectors implementing spectral partitioning areshown in Figure 3.2.2.

FIGURE 3.2.2 Eigenvectors on aMöbius strip of a square networkwithN1 = 16 andN2 = 8 divisions. Positive eigenvector com-

ponents are marked with filled circles, negative components

are marked with dots, and zero components are unmarked.

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Spectra of Lat t ices / / 83

3.2.2 Vertical Strip

The nodal profile of the vertical Möbius strip satisfies a reverse periodicity conditionin the second direction,

(3.2.13) ψi,1 = ψN1+2–i,N2+1.

For example, the southwestern nodal value is equal to the northeastern nodal value,ψ1,1 = ψN1+1,N2+1. The vector of unknown nodal values encapsulating N = (N1+1)N2

unknowns is

(3.2.14) ψ =

⎡⎢⎢⎢⎢⎢⎣ψ (1)

ψ (2)

...ψ (N2–1)

ψ (N2)

⎤⎥⎥⎥⎥⎥⎦,

where

(3.2.15) ψ (1) =

⎡⎢⎢⎢⎢⎢⎣ψ1,1

ψ2,1...ψN1,1

ψN1+1,1

⎤⎥⎥⎥⎥⎥⎦ , . . . , ψ (N2) =

⎡⎢⎢⎢⎢⎢⎣ψ1,N2

ψ2,N2...ψN1,N2

ψN1+1,N2

⎤⎥⎥⎥⎥⎥⎦.

The Laplacian matrix consists of N2 rows of (N1 +1)× (N1 +1) tridiagonal blocksin addition to two (N1 + 1) × (N1 + 1) corner blocks,

(3.2.16) L =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

E –I 0 · · · 0 0 –J–I E –I · · · 0 0 00 –I E · · · 0 0 0...

......

. . ....

......

0 0 0 · · · E –I 00 0 0 · · · –I E –I–J 0 0 · · · 0 –I E

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦,

where I is the (N1 + 1) × (N1 + 1) unit matrix and J is the (N1 + 1) × (N1 + 1) unitback-diagonal matrix. When N1 = 4, we obtain the 5 × 5 matrices

(3.2.17) E =

⎡⎢⎢⎢⎢⎣3 –1 0 0 0

–1 4 –1 0 00 –1 4 –1 00 0 –1 4 –10 0 0 –1 3

⎤⎥⎥⎥⎥⎦, J =

⎡⎢⎢⎢⎢⎣0 0 0 0 10 0 0 1 00 0 1 0 00 1 0 0 01 0 0 0 0

⎤⎥⎥⎥⎥⎦.

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The eigenvalues of the Laplacian matrix are

(3.2.18) λn1,n2 = 4 sin2(12 αn1,n2

)+ 4 sin2

(12 βn2

)or

(3.2.19) λn1, n2 = 4 – 2 cosαn1, n2 – 2 cosαn2 ,

where

(3.2.20) αn1 =n1 – 1

N1 + 1π , βn1, ln2 =

n2 – 1 + γn1N2

for n1 = 1, . . . ,N1 + 1 and n2 = 1, . . . ,N2, where γn1 = 0 if n1 is odd and γ = 1/2 ifn1 is even.

The corresponding eigenvectors, un1, n2 , normalized so that their lengths are equalto unity, un1, n2 · un1, n2∗

= 1, are given by

(3.2.21) un1, n2i1, i2= Bn2

√2√

(N1 + 1)N2cos[(i1 – 1

2

)αn1

]exp(– i i2βn1,n2

)for n1, i1 = 1, . . . ,N1 + 1 and n2, i2 = 1, . . . ,N2, where i is the imaginary unit, anasterisk denotes the complex conjugate, and Bn2 = 1, except that B1 = 1/

√2.

3.2.3 Klein Bottle

The Klein bottle consists of two attached Möbius strips that are glued together alongone side and then folded to produce a bottle. The nodal field of the Klein bottlesatisfies the reverse periodicity condition of the Möbius strip in the first directionand the usual periodic condition in the second direction,

(3.2.22) ψ1,i2 = ψN1+1,N2+2–i2 , ψi1,1 = ψi1,N2+1

for i1 = 1, . . . ,N1 + 1 and i2 = 1, . . . ,N2 + 1. The vector of unknown nodal values is

(3.2.23) ψ =

⎡⎢⎢⎢⎢⎢⎣ψ (1)

ψ (2)

...ψ (N2–1)

ψ (N2)

⎤⎥⎥⎥⎥⎥⎦,

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where

(3.2.24) ψ (1) =

⎡⎢⎢⎢⎢⎢⎣ψ1,1

ψ2,1...ψN1–1,1

ψN1,1

⎤⎥⎥⎥⎥⎥⎦, . . . , ψ (N2+1) =

⎡⎢⎢⎢⎢⎢⎣ψ1,N2+1

ψ2,N2+1...ψN1–1,N2+1

ψN1,N2+1

⎤⎥⎥⎥⎥⎥⎦.

The Laplacian matrix consists of N2 rows of N1 × N1 tridiagonal blocks, E, a chainof sparse backdiagonal blocks, and two unit corner blocks,

(3.2.25) L =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

E –I 0 · · · 0 0 –J – I–I E –I · · · 0 –J 00 –I E · · · –J 0 0...

......

. . ....

......

0 0 0 · · · E –I 00 –J 0 · · · –I E –I

–J – I 0 0 · · · 0 –I E

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦,

where I is the N1 × N1 identity matrix. The N1 × N1 matrix J is null, except that thenortheastern and southwestern corner elements are equal to unity. When N1 = 4, wehave

(3.2.26) E =

⎡⎢⎢⎣4 –1 0 0–1 4 –1 00 –1 4 –10 0 –1 4

⎤⎥⎥⎦, J =

⎡⎢⎢⎣0 0 0 10 0 0 00 0 0 01 0 0 0

⎤⎥⎥⎦.The eigenvalues of the Laplacian matrix are

(3.2.27) λn1, n2 = 4 sin2(12 αn1,n2

)+ 4 sin2

(12 βn2

)or

(3.2.28) λn1, n2 = 4 – 2 cosαn1, n2 – 2 cosαn2 ,

where

(3.2.29) αn1,n2 =n1 – 1 + γn2

N12π , βn2 =

n2 – 1

N22π

for n1 = 1, . . . ,N1 and n2 = 1, . . . ,N2 + 1, and

(3.2.30) γn2 ={0 for n2 = 1, 2, . . . , k,12 for n2 = k + 1, . . . ,N2.

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The crossover threshold, k, is given by k = (N2 + 1)/2 if N2 is odd, or k = N2/2 if N2

is even [45, 56].The corresponding eigenvectors, un1, n2 , normalized so that their lengths are equal

to unity, un1, n2 · un1, n2∗= 1, are

(3.2.31) un1, n2i1, i2= An1

(2

N1N2

)1/2exp(– i i1αn1, n2

)cos

[ (i2 – 1

2

)βn2

]for n1, i1 = 1, . . . ,N1, i2 = 1, . . . ,N2, and n2 = 1, . . . , k, where i is the imaginaryunit, an asterisk denotes the complex conjugate, and An1 = 1, except that A1 = 1/

√2.

When n2 = k + 1, . . . ,N2, the cosine is replaced by a sine on the right-hand side of(3.2.31). When N2 is even and n2 = N2/2, the cosine yields a sawtooth wave.

Exercise

3.2.1 Möbius strips and Klein bottle

(a) Confirm the eigenvalues and eigenvectors of the horizontal Möbius strip. (b)Repeat for the vertical Möbius strip. (c) Repeat for the Klein bottle.

3.3 HEXAGONAL LATTICE

The hexagonal lattice arises from the square lattice by adding one right- or left-leaning slanted link inside each square cell, dividing it into two triangular cells. Arectangular patch of a hexagonal network consisting of N1 links in the first direction,N2 links in the second direction, and one left-leaning slanted link inside each squarecell is shown in Figure 3.3.1(a). As in the case of the square lattice, the nodes areparametrized by two indices, i1 and i2.

Natural StateThe natural state of the hexagonal lattice patch shown in Figure 3.3.1(a) consists ofarrays of equilateral triangles in the xy plane, as shown in Figure 3.3.1(b). The nodesfall on a Bravais lattice with base vectors

(3.3.1) a1 = a (1, 0), a2 = a1

2(1,

√3),

where a is the triangle edge length. The nodal positions are

(3.3.2) xi1, i2 = x1,1 + (i1 – 1)a1 + (i2 – 1)a2,

for i1 = 1, . . . ,N1 +1 and i2 = 1, . . . ,N2 +1, where x1,1 is the arbitrary position of thefirst node. The reciprocal base vectors of the hexagonal lattice, b1 and b2, satisfyingby definition ai · bj = 2πδij are given by

(3.3.3) b1 =2π

a

(1, –

1√3

), b2 =

a

(0,

2√3

),

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where δij is Kronecker’s delta. The nodes of the reciprocal lattice are located at

(3.3.4) kn1, n2 = (n1 – 1) b1 + (n2 – 1)b2,

where n1 and n2 are two integers.

3.3.1 Isolated Network

The total number of nodes in the isolated network depicted in Figure 3.3.1 is

(3.3.5) N = (N1 + 1)(N2 + 1)

and the total number of links is

(3.3.6) L = N1(N2 + 1) + (N1 + 1)N2 + N1N2.

The nodes shown in Figure 3.3.1(a) can be compiled in a sequence of horizontal lay-ers from the bottom where i2 = 1 to the top where i2 = N2 + 1. With this convention,

(a)

1 N12

N2

i2

1

2

i1(b)

1 21

2

a1

a2N2

N1i1a

x

y

i2

FIGURE 3.3.1 (a) Illustration of a hexagonal network containing N1 linksin the first direction, N2 links in the second direction, and an appropriatenumber of cross-links links. A hexagonal cell can be identified around

each interior node. (b) The network has been sheared to its physicalconfiguration consisting of stacked equilateral triangles.

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a nodal field, ψ , is hosted by the N-dimensional vector

(3.3.7) ψ =

⎡⎢⎢⎢⎢⎢⎣ψ (1)

ψ (2)

...ψ (N2)

ψ (N2+1)

⎤⎥⎥⎥⎥⎥⎦,

where

(3.3.8) ψ (1) ≡

⎡⎢⎢⎢⎣ψ1,1

ψ2,1...ψN1+1,1

⎤⎥⎥⎥⎦, . . . , ψ (N2+1) ≡

⎡⎢⎢⎢⎣ψ1,N2+1

ψ2,N2+1...ψN1+1,N2+1

⎤⎥⎥⎥⎦.The Laplacian matrix consists of N2 + 1 rows of (N1 + 1) × (N1 + 1) tridiagonal

blocks, F, E, and G, in the following configuration:

(3.3.9) L =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

F –J 0 · · · 0 0 0–JT E –J · · · 0 0 00 –JT E · · · 0 0 0...

......

. . ....

......

0 0 0 · · · E –J 00 0 0 · · · –JT E –J0 0 0 · · · 0 –JT G

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦,

where J is the (N1 + 1) × (N1 + 1) lower bidiagonal unit matrix. For example, whenN1 = 4, we have

(3.3.10) F =

⎡⎢⎢⎢⎢⎣2 –1 0 0 0

–1 4 –1 0 00 –1 4 –1 00 0 –1 4 –10 0 0 –1 3

⎤⎥⎥⎥⎥⎦, E =

⎡⎢⎢⎢⎢⎣4 –1 0 0 0–1 6 –1 0 00 –1 6 –1 00 0 –1 6 –10 0 0 –1 4

⎤⎥⎥⎥⎥⎦.and

(3.3.11) G =

⎡⎢⎢⎢⎢⎣3 –1 0 0 0–1 4 –1 0 00 –1 4 –1 00 0 –1 4 –10 0 0 –1 2

⎤⎥⎥⎥⎥⎦, J =

⎡⎢⎢⎢⎢⎣1 0 0 0 01 1 0 0 00 1 1 0 00 0 1 1 00 0 0 1 1

⎤⎥⎥⎥⎥⎦.The two entries of F and G correspond to the southwestern and northeastern cornernodes, the four entries of F and E correspond to the edge nodes, and the six entriesof E correspond to interior nodes. Analytical expressions for the eigenvalues andeigenvectors of the Laplacian are not available.

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3.3.2 Doubly Periodic Network

With continued reference to Figure 3.3.1(a), now we assume that a nodal scalar field,ψ , is periodic in the directions of both indices, i1 and i2, so that

(3.3.12) ψ1, i2 = ψN1+1, i2 , ψi1, 1 = ψi1,N2+1.

The vector of unknown nodal values encapsulating N = N1N2 unknowns inside eachperiodic is

(3.3.13) ψ =

⎡⎢⎢⎢⎢⎢⎣ψ (1)

ψ (2)

...ψ (N2–1)

ψ (N2)

⎤⎥⎥⎥⎥⎥⎦,

where

(3.3.14) ψ (1) =

⎡⎢⎢⎢⎣ψ1,1

ψ2,1...ψN1,1

⎤⎥⎥⎥⎦ , . . . , ψ (N2) =

⎡⎢⎢⎢⎣ψ1,N2

ψ2,N2...ψN1,N2

⎤⎥⎥⎥⎦.The Laplacian matrix consists of N2 rows of N1 × N1 nearly tridiagonal blocks,

E, upper and lower diagonal blocks, K, and two corner blocks implementing theperiodicity condition in the second direction,

(3.3.15) L =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

E –K 0 · · · 0 0 –KT

–KT E –K · · · 0 0 00 –KT E · · · 0 0 0...

......

. . ....

......

0 0 0 · · · E –K 00 0 0 · · · –KT E –K–K 0 0 · · · 0 –KT E

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦,

where K is the N1 × N1 lower bidiagonal unit matrix with a unit northeastern cornerelement, K(1,N1) = 1. When N1 = 5, we have

(3.3.16) E =

⎡⎢⎢⎢⎢⎣6 –1 0 0 –1

–1 6 –1 0 00 –1 6 –1 00 0 –1 6 –1

–1 0 0 –1 6

⎤⎥⎥⎥⎥⎦, K =

⎡⎢⎢⎢⎢⎣1 0 0 0 11 1 0 0 00 1 1 0 00 0 1 1 00 0 0 1 1

⎤⎥⎥⎥⎥⎦.

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The northeastern and southwestern elements of E implement the periodicity condi-tion in the first direction.

An eigenvalue, λ, of the doubly periodic Laplacian, and the correspondingeigenvector, u, satisfy the equation

(3.3.17)6 ui1, i2 – ui1–1, i2 – ui1+1, i2 – ui1, i2–1 – ui1, i2+1

– ui1–1, i2+1 – ui1+1, i2–1 = λ ui1, i2

at any node.We find that the eigenvalues are given by

(3.3.18) λn1, n2 = 4 sin2(12 αn1

)+ 4 sin2

(12 βn2

)+ 4 sin2

[12

(αn1 – βn2

) ]or

(3.3.19) λn1,n2 = 6 – 2 cosαn1 – 2 cosβn2 – 2 cos(αn1 – βn2 )

for n1 = 1, . . . ,N1 and n2 = 1, . . . ,N2, where

(3.3.20) αn1 =n1 – 1

N12π , βn2 =

n2 – 1

N22π .

The corresponding eigenvectors, un1,n2 , normalized so that their lengths are equalto unity, un1,n2 · un1,n2∗

= 1, are

(3.3.21) un1, n2i1, i2=

1√N1N2

exp[– i (i1 αn1 + i2 βn2)

]for n1, i1 = 1, . . . ,N1 and n2, i2 = 1, . . . ,N2, where i is the imaginary unitand an asterisk denotes the complex conjugate. The eigenvalues given in (3.3.18)can be derived by substituting (3.3.21) into (3.3.17) and simplifying the resultingexpression.

Block Circulant MatricesThe doubly periodic Laplacian (3.3.15) is a block circulant matrix. A theorem dueto Friedman [12] states that the spectrum of this matrix is the union of the spectra ofthe following N1 × N1 circulant matrices,

(3.3.22) L(n2) = – exp(–iβn2)KT + E – exp(iβn2)K

or

(3.3.23) L(n2) = – cosβn2 (K +KT ) + E + sinβn2 (K –KT ),

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where n2 = 1, . . . ,N2. When N1 = 4,

(3.3.24) L(n2) =

⎡⎢⎢⎣6 – 2a –1 – a – b 0 –1 – a + b

–1 – a + b 6 – 2a –1 – a – b 00 –1 – a + b 6 – 2a –1 – a – b

–1 – a – b 0 –1 – a + b 6 – 2a

⎤⎥⎥⎦,where a = cosβn2 and b = sinβn2 . Using established formulas for the eigenvaluesof circulant matrices (Section A.5, Appendix A), we find that the eigenvalues of thematrix L(n2) are given by

(3.3.25) λn1, n2 = 6 – 2a – (1 + a + b) exp(iαn1) – (1 + a – b) exp(–iαn1 )

for n1 = 1, . . . ,N1. Simplifying the right-hand side of this expression, we recover theeigenvalues shown in (3.3.19).

Natural StateWith reference to the physical lattice shown in Figure 3.3.1(b), we introduce basevectors pertaining to the periodic patch,

(3.3.26) A1 = N1a1 = N1 a(1, 0), A2 = N2 a2 = N2 a

12

(1,

√3).

The associated reciprocal base vectors are

(3.3.27) B1 =2π

N1a

(1, –

1√3

), B2 =

N2a

(0,

2√3

).

The nodes of the reciprocal lattice are

(3.3.28) kn1, n2 = (n1 – 1)B1 + (n2 – 1)B2,

where n1 and n2 are two arbitrary integers.Every link in the natural state is parallel to one of the following three consecutive

link vectors attached to an arbitrary node:

(3.3.29) �1 = a1, �2 = a2, �3 = a1 – a2.

The eigenvalues of the Laplacian matrix can be expressed in the form

(3.3.30) λn1,n2 = 6 – 23∑r=1

cos(kn1,n2 · �r),

where

(3.3.31) kn1, n2 · �1 = αn1 , kn1, n2 · �2 = βn2 , kn1, n2 · �3 = αn1 – βn2 .Expression (3.3.30) provides us with a useful mnemonic rule.

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3.3.3 Alternative Node Indexing

An alternative node indexing scheme of the hexagonal lattice is shown in Fig-ure 3.3.2(a). The base vectors of the associated network in the natural state, shownin Figure 3.3.2(b),

(3.3.32) a1 = a(1, 0), a2 = a

1

2

(– 1,

√3),

form an angle of 120◦. The corresponding reciprocal base vectors are

(3.3.33) b1 =2π

a

(1, 1√

3

), b2 =

a

(0,

2√3

).

We may readily confirm that a1 · b1 = 2π , a2 · b2 = 2π , and a1 · b2 = 0, as required.An eigenvalue, λ, of the doubly periodic Laplacian, and the corresponding

eigenvector, u, satisfy the equation

(3.3.34)6 ui1, i2 – ui1–1, i2 – ui1+1, i2 – ui1, i2–1 – ui1, i2+1

– ui1+1, i2+1 – ui1–1, i2–1 = λ ui1, i2

at any node, (i1, i2). The Laplacian matrix is given in (3.3.9), except that the matrixK is the transpose of that described after equation (3.3.9).

(a)

N1

2

1

N2

i2

i11 2

(b)

211

2

a1

N1

N2

i1

x

y

a2

a

i2

FIGURE 3.3.2 (a) Alternative node indexing of the hexagonal network con-taining N1 links in the first direction, N2 links in the second direction, andan appropriate number of cross links. (b) The network has been deformedto demonstrate the natural state consisting of stacked equilateral triangles.The angle between the two base vectors, a1 and a2, is 120◦.

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FIGURE 3.3.3 Periodic spectral partitioning of the hexagonalnetwork based on an eigenvector.

The eigenvectors of the Laplacian are given in (3.3.21). The correspondingeigenvalues are

(3.3.35) λn1, n2 = 4 sin2(12 αn1

)+ 4 sin2

(12 βn2

)+ 4 sin2

[12

(αn1 + βn2

) ]or

(3.3.36) λn1, n2 = 6 – 2 cosαn1 – 2 cosβn2 – 2 cos(αn1 + βn2).

Note that these expressions differ from those shown in (3.3.19) only by the plus signin the argument of the last cosine. In spite of this change in sign, the spectrum ofthe Laplacian remains unchanged. A typical spectral partitioning of the hexagonalnetwork is shown in Figure 3.3.3.

Exercises

3.3.1 Alternative node indexing for an isolated network

(a) Deduce the structure of the Laplacian matrix of the isolated network for the nodeindexing scheme described in Figure 3.3.2. (b) Confirm expression (3.3.30).

3.3.2 Particle vibrations

The particles of a two-dimensional crystal are arranged on a hexagonal lattice.Small departures from the equilibrium position generate restoring forces. Deriveand solve an algebraic eigenvalue problem for the eigenfrequencies and eigendisplacements [7].

3.4 MODIFIED UNION JACK LATTICE

A rectangular patch of a modified Union Jack lattice containing N1 links in the firstdirection, N2 links in the second direction, and two slanted links inside each cell is

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1 N1

N2

2

21

i1

i2

FIGURE 3.4.1 Illustration of the modified Union Jack lattice con-taining N1 links in the first direction, N2 links in the seconddirection, and two noncrossing transverse links inside each cell.

shown in Figure 3.4.1. The network nodes are marked as filled circles. Note that theslanted links do not intersect at a node inside each cell but rather bypass one another.If they intersected, the modified Union Jack lattice shown in Figure 3.4.1 wouldreduce to the regular Union Jack lattice, which is a Laves lattice, as discussed in Sec-tion 2.6.3. The coordination number of the modified Union Jack lattice is uniform,d = 8.

3.4.1 Isolated Network

The total number of nodes in an isolated patch is

(3.4.1) N = (N1 + 1)(N2 + 1)

and the total number of links is

(3.4.2) L = N1(N2 + 1) + (N1 + 1)N2 + 2N1N2.

As in the case of the rectangular and hexagonal networks discussed previously inthis chapter, the nodes can be labeled in a sequence of layers from the bottom wherei2 = 1 to the top where i2 = N2 + 1.

A nodal scalar field, ψ , is encapsulated in the N-dimensional vector

(3.4.3) ψ =

⎡⎢⎢⎢⎢⎢⎣ψ (1)

ψ (2)

...ψ (N2)

ψ (N2+1)

⎤⎥⎥⎥⎥⎥⎦,

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where

(3.4.4) ψ (1) ≡

⎡⎢⎢⎢⎣ψ1,1

ψ2,1...ψN1+1,1

⎤⎥⎥⎥⎦, . . . , ψ (N2+1) ≡

⎡⎢⎢⎢⎣ψ1,N2+1

ψ2,N2+1...ψN1+1,N2+1

⎤⎥⎥⎥⎦.The Laplacian consists of N2 + 1 rows of (N1 + 1) × (N1 + 1) tridiagonal blocks,

F and E, in the following configuration:

(3.4.5) L =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

F –J 0 · · · 0 0 0–JT E –J · · · 0 0 0

0 –JT E · · · 0 0 0...

......

. . ....

......

0 0 0 · · · E –J 00 0 0 · · · –JT E –J0 0 0 · · · 0 –JT F

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦,

where J is the (N1 + 1)× (N1 + 1) tridiagonal unit matrix. For example, when N1 = 4,

F =

⎡⎢⎢⎢⎢⎣3 –1 0 0 0–1 5 –1 0 00 –1 5 –1 00 0 –1 5 –10 0 0 –1 3

⎤⎥⎥⎥⎥⎦, E =

⎡⎢⎢⎢⎢⎣5 –1 0 0 0

–1 8 –1 0 00 –1 8 –1 00 0 –1 8 –10 0 0 –1 5

⎤⎥⎥⎥⎥⎦,

J =

⎡⎢⎢⎢⎢⎣1 1 0 0 01 1 1 0 00 1 1 1 00 0 1 1 10 0 0 1 1

⎤⎥⎥⎥⎥⎦.(3.4.6)

The three entries of F correspond to corner nodes, the five entries of F and Ecorrespond to boundary nodes, and the eight entries of E correspond to interiornodes.

3.4.2 Doubly Periodic Network

Assume that the nodal scalar field of an infinite modified Union Jack lattice, ψ , isperiodic in two directions, so that

(3.4.7) ψ1, i2 = ψN1+1, i2 , ψi1,1 = ψi1,N2+1.

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The vector of unknown nodal values inside each period is

(3.4.8) ψ =

⎡⎢⎢⎢⎢⎢⎣ψ (1)

ψ (2)

...ψ (N2–1)

ψ (N2)

⎤⎥⎥⎥⎥⎥⎦,

where

(3.4.9) ψ (1) =

⎡⎢⎢⎢⎣ψ1,1

ψ2,1...ψN1,1

⎤⎥⎥⎥⎦, . . . , ψ (N2) =

⎡⎢⎢⎢⎣ψ1,N2

ψ2,N2...ψN1,N2

⎤⎥⎥⎥⎦.The vector ψ encapsulates N = N1N2 unique unknowns.

The Laplacian consists of N2 rows of N1 × N1 nearly tridiagonal blocks and twocorner blocks implementing the periodicity condition in the second direction, in thefollowing configuration:

(3.4.10) L =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

E –K 0 · · · 0 0 –KT

–KT E –K · · · 0 0 00 –KT E · · · 0 0 0...

......

. . ....

......

0 0 0 · · · E –K 00 0 0 · · · –KT E –K–K 0 0 · · · 0 –KT E

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦,

where K is the N1 × N1 nearly tridiagonal unit matrix implementing the periodicitycondition in the second direction. For example, when N1 = 5,

(3.4.11) E =

⎡⎢⎢⎢⎢⎣8 –1 0 0 –1–1 8 –1 0 00 –1 8 –1 00 0 –1 8 –1–1 0 0 –1 8

⎤⎥⎥⎥⎥⎦, K =

⎡⎢⎢⎢⎢⎣1 1 0 0 11 1 1 0 00 1 1 1 00 0 1 1 11 0 0 1 1

⎤⎥⎥⎥⎥⎦.

The northeastern and southwestern elements implement the periodicity condition inthe first direction.

An eigenvalue, λ, of the doubly periodic Laplacian, and the correspondingeigenvector, u, satisfy the equation

(3.4.12)8 ui1, i2 – ui1–1, i2 – ui1+1, i2 – ui1, i2–1 – ui1, i2+1

– ui1–1, i2+1 – ui1+1, i2–1 – ui1+1, i2+1 – ui1–1, i2–1 = λ ui1, i2

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Spectra of Lat t ices / / 97

at any node, (i1, i2). The eigenvalues of the Laplacian are given by

(3.4.13)λn1, n2 = 4 sin2

(12αn1

)+ 4 sin2

(12βn2

)+ 4 sin2

[12

(αn1 – βn2

) ]+ 4 sin2

[12

(αn1 + βn2

) ],

which can be restated as

(3.4.14) λn1, n2 = 8 – 2 cosαn1 – 2 cosβn2 – 2 cos(αn1 + βn2) – 2 cos(αn1 – βn2)

or

(3.4.15) λn1,n2 = 8 – 2 cosαn1 – 2 cos βn2 – 4 cosαn1 cosβn2 ,

where

(3.4.16) αn1 =n1 – 1

N12π , βn2 =

n2 – 1

N22π

for n1 = 1, . . . ,N1 and n2 = 1, . . . ,N2.The corresponding eigenvectors, un1, n2 , normalized so that their lengths are equal

to unity, un1,n2 · un1,n2∗= 1, are

(3.4.17) un1, n2i1, i2=

1√N1N2

exp[– i(i1 αn1 + i2 βn2 )

]for n1, i1 = 1, . . . ,N1 and n2, i2 = 1, . . . ,N2, where i is the imaginary unit and anasterisk denotes the complex conjugate. Note that the eigenvectors are identical tothose of the square and hexagonal networks.

Block Circulant MatricesTo derive the eigenvalues, we observe that the doubly periodic Laplacian (3.4.10) isa block circulant matrix. A theorem due to Friedman [12] states that the spectrum ofthis matrix is the union of the spectra of the following N1 × N1 circulant matrices:

(3.4.18) L(n2) = – exp(–iβn2 )KT + E – exp(iβn2)K

for n2 = 1, . . . ,N2. Rearranging, we obtain

(3.4.19) L(n2) = – cosβn2(K +KT ) + E.

When N1 = 4, we find that

(3.4.20) L(n2) =

⎡⎢⎢⎣8 – 2a –1 – 2a 0 –1 – 2a–1 – 2a 8 – 2a –1 – 2a 0

0 –1 – 2a 8 – 2a –1 – a–1 – 2a 0 –1 – 2a 8 – 2a

⎤⎥⎥⎦,

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98 / / AN INTRODUCTION TO GRIDS, GRAPHS, AND NETWORKS

where a = cosβn2 . Using established expressions for the eigenvalues of circu-lant matrices (Section A.5, Appendix A), we find that eigenvalues of L(n2) aregiven by

(3.4.21) λn1,n2 = 8 – 2a – (1 + 2a) exp(iαn1) – (1 + 2a) exp(–iαn1)

for n1 = 1, . . . ,N1. Simplifying the right-hand side, we recover (3.4.15).

Exercise

3.4.1 Circulant matrices

Derive the eigenvalues of the matrix L(n2) shown in (3.4.21).

3.5 HONEYCOMB LATTICE

A patch of a honeycomb network consisting of hexagonal cells with side length ainscribed in a circle of radius b is shown in Figure 3.5.1(a). The nodes are arrangedon two different Bravais lattices with common base vectors,

(3.5.1) a1 = a(1, 0

), a2 = a 1

2

(1,

√3),

where a =√3 b is the distance of a node from its second nearest neighbor.

Nodes in the first lattice, designated as lattice A, are shown as open circles con-nected by dashed lines, and nodes in the second lattice, designated as lattice B, areshown as filled circles connected by dotted lines in Figure 3.5.1(a). Nodes on lat-tice A are parametrized by a pair of indices, (iA1 , i

A2 ), and nodes on lattice B are

parametrized by another pair of indices, (iB2 , iB2 ), where i

A1 , i

B1 = 1, . . . ,N1 + 1 and

iA2 , iB2 = 1, . . . ,N2 + 1.The position of nodes on lattice A is described by

(3.5.2) xiA1 , iA2= xA1,1 +

(iA1 – 1

)a1 +

(iA2 – 1

)a2,

and the position of nodes on lattice B is described by

(3.5.3) xiB1 , iB2= xA1,1 + η +

(iB1 – 1

)a1 +

(iB2 – 1

)a2,

where

(3.5.4) η ≡ xB1,1 – xA1,1 = a

12

(1,

1√3

)is the inner displacement of the two constituent lattices.

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Spectra of Lat t ices / / 99

(a)

11 2

a1

21

N2 + 1

N1 + 1

N1

N1

N1 + 1

N2 + 1

N2+1

2N1 2N1 + 2

N2

N2

i1A

i1B

i2A

i2B

1

b

x

y

a

a2

(b)

1 2 3 4

1

2

FIGURE 3.5.1 (a) Illustration of a honeycomb network containingN1 cells inthe first direction and N2 cells in the second direction. (b) Equivalent iso-morphic representation where the network collapses vertically into a brickwall. In the example shown, N1 = 4 and N2 = 3. Links are drawn with

solid lines.

3.5.1 Isolated Network

The numbers of nodes and links in an isolated network are twice those of thecorresponding square network,

(3.5.5) N = 2 (N1 + 1)(N2 + 1)

and

(3.5.6) L = 2N1(N2 + 1) + 2 (N1 + 1)N2.

The nodes of each constituent Bravais lattice can be counted in a sequence of hori-zontal layers from the bottom where i2 = 1 to the top where i2 = N2 +1, as in the caseof the square and hexagonal networks. A scalar nodal field, ψ , can be accommodatedinto an N-dimensional vector

(3.5.7) ψ =

[ψA

ψB

], ψA =

⎡⎢⎢⎢⎢⎢⎢⎣ψ

(1)A

ψ(2)A

...

ψ(N2)A

ψ(N2+1)A

⎤⎥⎥⎥⎥⎥⎥⎦, ψB =

⎡⎢⎢⎢⎢⎢⎢⎣ψ

(1)B

ψ(2)B

...

ψ(N2)B

ψ(N2+1)B

⎤⎥⎥⎥⎥⎥⎥⎦,

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100 / / AN INTRODUCTION TO GRIDS, GRAPHS, AND NETWORKS

where

(3.5.8) ψ (1)A,B ≡

⎡⎢⎢⎢⎣ψ1,1

ψ2,1...ψNx+1,1

⎤⎥⎥⎥⎦A,B

, . . . , ψ(N2+1)A,B ≡

⎡⎢⎢⎢⎣ψ1,N2+1

ψ2,N2+1...ψNx+1,N2+1

⎤⎥⎥⎥⎦A,B

are horizontal profiles of the constituent lattices A or B.The Laplacian matrix consists of four (N1 + 1) × (N2 + 1) square blocks, in the

following configuration:

(3.5.9) L =

[A –B–BT C

],

where A is a diagonal matrix consisting of N2 + 1 blocks of N1 + 1 elements in thefollowing order:

(3.5.10) 1, 2, . . . , 2, 2, 2, 3, . . . , 3, 3, . . . , 2, 3, . . . , 3, 3, 2, 3, . . . , 3, 3,

and C is another diagonal matrix consisting of N2 + 1 blocks of Nx + 1 elements inthe following order:

(3.5.11) 3, 3, . . . , 3, 2, 3, 3, . . . , 3, 2, . . . , 3, 3, . . . , 3, 2, 2, 2, . . . , 2, 1.

Note that the sequence (3.5.11) is the reverse of the sequence (3.5.10). The matrix Bhas the block lower bidiagonal form

(3.5.12) B =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

J 0 0 · · · 0 0 0I J 0 · · · 0 0 00 I J · · · 0 0 0...

......

. . ....

......

0 0 0 · · · J 0 00 0 0 · · · I J 00 0 0 · · · 0 I J

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦,

where I is the (N1 + 1) × (N1 + 1) identity matrix and J is the (N1 + 1) × (N1 + 1)lower bidiagonal unit matrix containing ones along the diagonal and lower diagonal.When N1 = 4, we obtain the 5 × 5 matrix

(3.5.13) J =

⎡⎢⎢⎢⎢⎣1 0 0 0 01 1 0 0 00 1 1 0 00 0 1 1 00 0 0 1 1

⎤⎥⎥⎥⎥⎦.

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Spectra of Lat t ices / / 101

3.5.2 Brick Representation

In the illustration shown in Figure 3.5.1(b), the network displayed in Figure 3.5.1(a)has been compressed vertically into a brick wall. Dashed lines in Figure 3.5.1(b) passthrough type A nodes marked as hollow circles, and dotted lines pass through typeB nodes marked as filled circles. The nodes are identified by an index i1 that rangesfrom 1 to 2N1 + 2 in the first direction and an index i2 that ranges from 1 to N2 + 1 inthe second direction.

The nodes of the brick network can be compiled in a sequence of horizontallayers from bottom where i1 = 1 to top where i2 = N2 + 1, as in the case of therectangular and hexagonal networks. A scalar nodal field, ψ , can be arranged into anN-dimensional vector

(3.5.14) ψ =

⎡⎢⎢⎢⎢⎢⎣ψ (1)

ψ (2)

...ψ (N2)

ψ (N2+1)

⎤⎥⎥⎥⎥⎥⎦,

where

(3.5.15) ψ (1) ≡

⎡⎢⎢⎢⎣ψ1,1

ψ2,1...ψ2N1+2,1

⎤⎥⎥⎥⎦ , . . . , ψ (N2+1) ≡

⎡⎢⎢⎢⎣ψ1,N2+1

ψ2,N2+1...ψ2N1+2,N2+1

⎤⎥⎥⎥⎦are (2N1 + 2)-dimensional blocks.

The Laplacian matrix of the isolated network consists of N2 + 1 rows of (2N1 +2) × (2N1 + 2) blocks, in the following configuration:

(3.5.16) L =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

F –K 0 · · · 0 0 0–KT E –K · · · 0 0 00 –KT E · · · 0 0 0...

......

. . ....

......

0 0 0 · · · E –K 00 0 0 · · · –KT E –K0 0 0 · · · 0 –KT G

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦.

The tridiagonal blocks, F,E, and G, display the node degrees along the diagonal andimplement horizontal links. For example, when N1 = 1, we obtain the 4 × 4 blocks

(3.5.17) F =

⎡⎢⎢⎣1 –1 0 0–1 3 –1 00 –1 2 –10 0 –1 3

⎤⎥⎥⎦, E =

⎡⎢⎢⎣2 –1 0 0–1 3 –1 00 –1 3 –10 0 –1 3

⎤⎥⎥⎦,

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102 / / AN INTRODUCTION TO GRIDS, GRAPHS, AND NETWORKS

and

(3.5.18) G =

⎡⎢⎢⎣2 –1 0 0–1 3 –1 00 –1 3 –10 0 –1 3

⎤⎥⎥⎦.The upper diagonal blocks implement upward links from type B nodes, while thelower diagonal blocks implement downward links from type A nodes. For example,when N1 = 2, we obtain the 6 × 6 blocks

(3.5.19) K =

⎡⎢⎢⎢⎢⎢⎢⎢⎣

0 0 0 0 0 01 0 0 0 0 00 0 0 0 0 00 0 1 0 0 00 0 0 0 0 00 0 0 0 1 0

⎤⎥⎥⎥⎥⎥⎥⎥⎦.

Every other element along the lower diagonal is zero.The eigenvalues of the Laplacian matrix (3.5.16) are identical to those of the

Laplacian matrix (3.5.9).

3.5.3 Doubly Periodic Network

Assume that a scalar nodal field, ψ , deployed over an infinite honeycomb lattice isperiodic in the direction of each base vector, so that

(3.5.20) ψA1, i2 = ψ

AN1+1, j, ψA

i1,1 = ψAi1,N2+1

for the constituent lattice A and

(3.5.21) ψB1, i2 = ψ

BN1+1, j, ψB

i1,1 = ψBi1,N2+1

for the constituent lattice B. The vector of unique unknown nodal values inside eachperiod, encapsulating N = 2N1N2 unknowns, is

(3.5.22) ψ =

[ψA

ψB

], ψA =

⎡⎢⎢⎢⎢⎢⎢⎣ψ

(1)A

ψ(2)A

...

ψ(N2–1)A

ψ(N2)A

⎤⎥⎥⎥⎥⎥⎥⎦, ψB =

⎡⎢⎢⎢⎢⎢⎢⎣ψ

(1)B

ψ(2)B

...

ψ(N2–1)B

ψ(N2)B

⎤⎥⎥⎥⎥⎥⎥⎦,

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Spectra of Lat t ices / / 103

where

(3.5.23) ψ (1)A,B ≡

⎡⎢⎢⎢⎢⎢⎣ψ1,1

ψ2,1...ψN1–1,1

ψN1,1

⎤⎥⎥⎥⎥⎥⎦A,B

, . . . , ψ(N2)A,B ≡

⎡⎢⎢⎢⎢⎢⎣ψ1,N2

ψ2,N2...ψN1–1,N2

ψNk ,N2

⎤⎥⎥⎥⎥⎥⎦A,B

for N1 ≥ 1 and N2 ≥ 2.The Laplacian matrix consists of two diagonal blocks hosting the lattice coordi-

nation number, along with two off-diagonal square blocks,

(3.5.24) L =

[3 IM –B–BT 3 IM

],

where IM is the M × M identity matrix and M = N1N2 is half the number ofunique nodes. The M×M matrix B has the following nearly lower bidiagonal blockstructure:

(3.5.25) B =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

K 0 0 · · · 0 0 II K 0 · · · 0 0 00 I K · · · 0 0 0...

......

. . ....

......

0 0 0 · · · K 0 00 0 0 · · · I K 00 0 0 · · · 0 I K

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦,

where I is the N1 × N1 identity matrix and K is the N1 × N1 lower bidiagonalunit matrix, except that the northeastern element is set to unity. For example, whenN1 = 5,

(3.5.26) K =

⎡⎢⎢⎢⎢⎣1 0 0 0 11 1 0 0 00 1 1 0 00 0 1 1 00 0 0 1 1

⎤⎥⎥⎥⎥⎦.

The northeastern block of L implements the periodicity condition in the second di-rection, and the northeastern element of K implements the periodicity in the firstdirection. The sum of the elements of B in each row or column is equal to the latticecoordination number, 3.

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An eigenvalue, λ, of the doubly periodic Laplacian, and the correspondingeigenvector, u, satisfy the equations

(3.5.27) 3 uAi1, i2 – uBi1, i2 – u

Bi1–1, i2 – u

Bi1, i2–1 = λ u

Ai1, i2

and

(3.5.28) 3 uBi1, i2 – uAi1, i2 – u

Ai1+1, i2 – u

Ai1, i2+1 = λ u

Bi1, i2 .

From the second equation, we find that

(3.5.29) uBi1, i2 =1

3 – λ

(uAi1, i2 + u

Ai1+1, i2 + u

Ai1, i2+1

).

Substituting this expression into (3.5.27) to eliminate nodal values on lattice B infavor of those on lattice A, we obtain

(3.5.30)(3 – λ)2 uAi1, i2 = 3uAi1, i2 + u

Ai1+1, i2 + u

Ai1–1, i2 + u

Ai1, i2+1 + u

Ai1, i2–1

+ uAi1–1, i2+1 + uAi1+1, i2–1.

Now substituting

(3.5.31) u = exp[– i(i1αn1 + i2βn2)

],

we obtain a quadratic equation,

(3.5.32) (λ – 3)2 = 3 + 2 cosαn1 + 2 cos βn2 + 2 cos(αn1 – βn2),

where i is the imaginary unit and

(3.5.33) αn1 =n1 – 1

N12π , βn2 =

n2 – 1

N22π .

The roots of the quadratic equation are given by

(3.5.34) λ±n1,n2 = 3 ± [ 3 + 2 cosαn1 + 2 cosβn2 + 2 cos(αn1 – βn2)

]1/2.

The sum of two conjugate eigenvalues is

(3.5.35) λ+n1, n2 + λ–n1, n2 = 6.

The product of two conjugate eigenvalues is

(3.5.36) λ+n1, n2λ–n1, n2 = 6 – 2 cosαn1 – 2 cosβn2 – 2 cos(αn1 – βn2 ).

Accordingly,

(3.5.37)1

λ+n1, n2+

1

λ–n1, n2=

3

3 – cos αn1 – cosβn2 – cos(αn1 – βn2).

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Spectra of Lat t ices / / 105

EigenvectorsThe eigenvectors of the Laplacian matrixL, normalized so that their Euclidean normsare equal to unity, consist of appropriate arrangements of the following nodal fieldon the constituent Bravais lattice A:

(3.5.38) (un1, n2i1, i2)A =

1√2N1N2

exp[– i (i1 αn1 + i2 βn2)

]for n1, i1 = 1, . . . ,N1, and the following nodal field on the constituent Bravaislattice B:

(3.5.39)(un1, n2i1, i2

)B=

1

3 – λ±n1, n2

[ (un1, n2i1, i2

)A+(un1, n2i1+1, i2

)A+(un1, n2i1, i2+1

)A ],

or

(3.5.40)(un1, n2i1, i2

)B=

1

3 – λ±n1, n2

(un1, n2i1, i2

)A (1 + e–iαn1 + e–iαn2

)for n2, i2 = 1, . . . ,N2, where i is the imaginary unit. The spectral partitioning of aperiodic honeycomb lattice with N1 = 9 and N2 = 4 is shown in Figure 3.5.2.

Brick RepresentationIn the illustration shown in Figure 3.5.1(b), the periodic patch of the honeycombnetwork displayed in Figure 3.5.1(a) has been compressed vertically into a brickwall. The periodicity condition requires that

(3.5.41) ψ1, i2 = ψN1+1, i2 , ψ2, i2 = ψN1+2, i2 , ψi1, 1 = ψi1,N2+1

for i1 = 1, . . . , 2N1 + 2 and i2 = 1, . . . ,N2 + 1. The nodes of the brick network can becompiled in a sequence of horizontal layers from bottom where i2 = 1, to top wherei2 = N2, as in the case of the periodic rectangular and hexagonal lattices. A scalarnodal field, ψ , is encapsulated in an N-dimensional vector

(3.5.42) ψ =

⎡⎢⎢⎢⎢⎢⎣ψ (1)

ψ (2)

...ψ (N2–1)

ψ (N2)

⎤⎥⎥⎥⎥⎥⎦,

where

(3.5.43) ψ (1) ≡

⎡⎢⎢⎢⎣ψ1,1

ψ2,1...ψ2N1,1

⎤⎥⎥⎥⎦, . . . , ψ (N2) ≡

⎡⎢⎢⎢⎣ψ1,N2

ψ2,N2...ψ2N1,N2

⎤⎥⎥⎥⎦are horizontal profiles.

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106 / / AN INTRODUCTION TO GRIDS, GRAPHS, AND NETWORKS

FIGURE 3.5.2 Spectral partitioning of a periodic honey-comb lattice with N1 = 9 and N2 = 4 divisions insideeach period in the natural state. Positive eigenvector

components are marked with filled circles, negative

components are marked with dots, and zero compo-

nents are unmarked.

The Laplacian matrix consists of N2 rows of (2N1) × (2N1) blocks,

(3.5.44) L =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

E –K 0 · · · 0 0 –KT

–KT E –K · · · 0 0 00 –KT E · · · 0 0 0...

......

. . ....

......

0 0 0 · · · E –K 00 0 0 · · · –KT E –K–K 0 0 · · · 0 –KT E

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦.

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Spectra of Lat t ices / / 107

The nearly tridiagonal blocks, E, display the lattice coordination number 3 alongthe diagonal and implement horizontal links. The upper diagonal blocks implementupward links originating from type B nodes. The lower diagonal blocks implementdownward links originating from type A nodes. The corner blocks implement theperiodicity condition in the second direction.

Detailed inspection reveals that

(3.5.45) E =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

A –J 0 · · · 0 0 –JT

–JT A –J · · · 0 0 00 –JT A · · · 0 0 0...

......

. . ....

......

0 0 0 · · · A –J 00 0 0 · · · –JT A –J–J 0 0 · · · 0 –JT A

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦and

(3.5.46) K =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

J 0 0 · · · 0 0 00 J 0 · · · 0 0 00 0 J · · · 0 0 0...

......

. . ....

......

0 0 0 · · · J 0 00 0 0 · · · 0 J 00 0 0 · · · 0 0 J

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦,

where

(3.5.47) A =

[3 –1

–1 3

], J =

[0 01 0

].

For example, when N1 = 2, we obtain the 4 × 4 blocks

(3.5.48) E =

⎡⎢⎢⎣3 –1 0 –1–1 3 –1 00 –1 3 –1–1 0 –1 3

⎤⎥⎥⎦, K =

⎡⎢⎢⎣0 0 0 01 0 0 00 0 0 00 0 1 0

⎤⎥⎥⎦.An eigenvalue, λ, of the doubly periodic Laplacian (3.5.44), and the correspond-

ing eigenvector, u, satisfy the equation

(3.5.49) 3ui1, i2 – ui1+1, i2 – ui1–1, i2 – ui1, i2+1 = λui1, i2

at any node. The eigenvalues and eigenvectors are identical to those of the Laplacianmatrix (3.5.24) discussed in the preceding section. An alternative derivation relies onthe block circulant structure of the Laplacian.

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Block Circulant MatricesThe doubly periodic Laplacian (3.5.44) is a block circulant matrix. A theorem dueto Friedman [12] states that the spectrum of this matrix is the union of the spectra ofthe following 2N1 × 2N1 circulant matrices:

(3.5.50) L(n2) = – exp(–iβn2)KT + E – exp(iβn2)K

or

(3.5.51) L(n2) = – cos βn2(K +KT ) + E – i sin βn2(K –KT ),

where

(3.5.52) βn2 =n2 – 1

N22π

for n2 = 1, . . . ,N2. For example, when N1 = 2,

(3.5.53) L(n2) =

⎡⎢⎢⎣3 –1 – c 0 –1

–1 – c∗ 3 –1 00 –1 3 –1 – c–1 0 –1 – c∗ 3

⎤⎥⎥⎦,where c = exp(–iβn2).

More generally,

(3.5.54) L(n2) =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

S –J 0 · · · · · · 0 –JT

–JT S –J · · · · · · 0 00 –JT S · · · · · · 0 0...

......

. . ....

......

0 0 0 · · · S –J 00 0 0 · · · –JT S –J–J 0 0 · · · 0 –JT S

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦is a block circulant matrix, where

(3.5.55) S = –cJT + A – c∗J =

[3 –1 – c

–1 – c∗ 3

].

The spectrum of L(n2) is the union of the spectra of the following 2×2 Hermitianmatrices:

(3.5.56) �(n1, n2) = – exp(–iαn1) JT + S – exp(iαn1) J

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Spectra of Lat t ices / / 109

where

(3.5.57) αn1 =n1 – 1

N12π

for n1 = 1, . . . ,N1. Explicitly,

(3.5.58) �(n1, n2) =

[3 –1 – (c + d)

–1 – (c + d)∗ 3

],

where d = exp(–iαn1). The eigenvalues are the roots of the characteristic polynomialof �(n1,n2) satisfying the quadratic equation (3.5.32), given in (3.5.38). The sum of

FIGURE 3.5.3 Spectral partitioning of a periodic brick (hon-eycomb) lattice with N1 = 9 and N2 = 4 divisionsinside each period. Positive eigenvector components

are marked with filled circles, negative components are

marked with dots, and zero components are unmarked.

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two conjugate eigenvalues is the trace of �(n1,n2), that is, the sum of the diagonalelements. The product of two conjugate eigenvalues is the determinant of �(n1,n2).The spectral partitioning of a periodic brick lattice with N1 = 9 and N2 = 4 is shownin Figure 3.5.3.

3.5.4 Alternative Node Indexing

In an alternative representation, the nodes of the honeycomb lattice are identified bytwo indices, i1 and i2, corresponding to two base vectors, a1 and a2, that form anangle of 120◦, as shown in Figure 3.5.4(a), given by

(3.5.59) a1 = a (1, 0), a2 = a 12 (1,

√3).

Nodes on lattice A are drawn as open circles connected by dashed lines, and nodeson lattice B are drawn as filled circles connected by dotted lines. The position ofnodes at lattice A is

(3.5.60) xiA1 , iA2= xA1,1 + (iA1 – 1) a1 + (iA2 – 1) a2,

and the position of nodes at lattice B is

(3.5.61) xiB1 , iB2= xA1,1 + η + (iB1 – 1) a1 + (iB2 – 1) a2,

where

(3.5.62) η ≡ xB1,1 – xA1,1 = a

1

2

(–1,

1√3

)is the inner displacement of the two constituent lattices.

The Laplacian matrix is given by (3.5.24)–(3.5.26), provided that the matrix Kdefined in (3.5.26) is replaced by its transpose. The eigenvalues of the new Laplacianare

(3.5.63) λ±n1,n2 = 3 ± [ 3 + 2 cosαn1 + 2 cos βn2 + 2 cos(αn1 + βn2)

]1/2.The eigenvectors on lattice A are given in (3.5.38) and the eigenvectors on lattice Bare given by

(3.5.64)(un1, n2i1, i2

)B=

1

3 – λ±n1, n2

[(un1, n2i1, i2

)A+(un1, n2i1–1, i2

)A+(un1, n2i1, i2+1

)A]or

(3.5.65)(un1, n2i1, i2

)B=

1

3 – λ±n1, n2

(un1, n2i1, i2

)A (1 + eiαn1 + e–iαn2

).

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Spectra of Lat t ices / / 111

(a)

21

1 21

i2B

i1B

i1A

i2A

N2 + 1

N2 + 1

2N1 + 22N1

N1

N1 N1 + 1

N1 + 1

N2 + 1

N2

N2

a

1

aa

b

y

x

1

2

(b)

43211

2

FIGURE 3.5.4 (a) Alternative indexing of a honeycomb network containingN1 cells in the first direction and N2 cells in the second direction. (b) Al-ternative representation where the network collapses vertically into a brickwall. For the configuration shown, N1 = 4 and N2 = 3. Nodes on lattice

A are shown as open circles connected by dashed lines, and nodes on

lattice B are shown as filled circles connected by dotted lines.

The equivalent brick representation is shown in Figure 3.5.4(b). The Laplacianmatrix is given in (3.5.44), provided that the matrixK is replaced by the transpose ofthan shown in (3.5.46).

Exercise

3.5.1 Particle vibrations

Assume that the particles of a two-dimensional crystal are arranged on a honey-comb lattice. Small departures from the equilibrium position generate restoringforces. Derive and solve an algebraic eigenvalue problem for the eigenfrequenciesand eigendisplacements [6, 7].

3.6 KAGOMÉ LATTICE

A rectangular patch of a kagomé lattice is shown in Figure 3.6.1(a). Although thelattice coordination number is identical to that of the square lattice, d = 4, thetwo lattices are distinct. Three families of nodes falling on different Bravais lattices,

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(a)

1

2

1

2

N2

N2

1 2 i1 N1

N2+1N2+1

i2i2

N1+11 2 N1 N1+1i1

ABC

(b)

a1

a2

A

C Bx

y

FIGURE 3.6.1 (a) Illustration of a kagomé network involving three familiesof nodes. (b) The network has been sheared into its natural state where alllinks have the same length. Links are drawn as bold lines. Type A nodes

are shown as hollow circles, type B nodes are shown as solid circles,

and type C nodes are shown as solid squares.

identified as families A, B, and C, are shown with hollow circles, filled circles, orfilled squares in Figure 3.6.1(a). Each family is parametrized by a pair of indices,i1 and i2, where i1 = 1, . . . ,N1 + 1, i2 = 1, . . . ,N2 + 1, and N1,N2 are the patchdimensions. Type A nodes lie at the intersection of vertical and horizontal solid lines,type B nodes lie at the intersection of solid and dotted lines, and type C nodes lie atthe intersection of solid and dashed lines.

The corresponding natural state of the network is illustrated in Figure 3.1.1(b).Each family of nodes falls on a Bravais lattice with two base vectors a1 and a2 thatare identical to those of the hexagonal lattice.

3.6.1 Isolated Network

The total number of nodes in the isolated network shown in Figure 3.6.1 is

(3.6.1) N = 3(N1 + 1)(N2 + 1)

and the total number of links is

(3.6.2) L = 6N1N2 + 2N1 + 2N2 + 1.

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Spectra of Lat t ices / / 113

The nodes are compiled in a sequence of horizontal layers from the bottom wherei2 = 1 to the top where i2 = N2 + 1. A nodal field, ψ , can be arranged in an N-dimensional vector

(3.6.3) ψ =

⎡⎢⎢⎢⎢⎢⎣ψ (1)

ψ (2)

...ψ (N2)

ψ (N2+1)

⎤⎥⎥⎥⎥⎥⎦,

where the constituent vectors

(3.6.4) ψ (1) ≡

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

ψA1,1ψB1,1ψC1,1

...ψAN1+1,1ψBN1+1,1ψCN1+1,1

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦, . . . , ψ (N2+1) ≡

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

ψA1,N2+1ψB1,N2+1ψC1,N2+1

...ψAN1+1,N2+1ψBN1+1,N2+1ψCN1+1,N2+1

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦.

consist of ordered triplets of A, B, C nodes.The Laplacian is a block tridiagonal matrix consisting of N2 + 1 rows of 3(N1 +

1) × 3(N1 + 1) symmetric tridiagonal blocks, F, E, and G, along with sparse lowerand upper diagonal blocks, in the following configuration:

(3.6.5) L =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

F –J 0 · · · 0 0 0–JT E –J · · · 0 · · · 0

0 –JT E · · · 0 0 0...

......

. . ....

......

0 0 0 · · · E –J 00 0 0 . . . –JT E –J0 0 0 · · · 0 –JT G

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦,

where J is a 3(N1 + 1) × 3(N1 + 1) sparse matrix.For example, when N1 = 2, we obtain the 9 × 9 matrices

(3.6.6) F =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

2 –1 –1 0 0 0 0 0 0–1 3 –1 –1 0 0 0 0 0–1 –1 3 0 0 0 0 0 00 –1 0 3 –1 –1 0 0 00 0 0 –1 3 –1 –1 0 00 0 0 –1 –1 4 0 0 00 0 0 0 –1 0 3 –1 –10 0 0 0 0 0 –1 2 –10 0 0 0 0 0 –1 –1 4

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦,

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114 / / AN INTRODUCTION TO GRIDS, GRAPHS, AND NETWORKS

(3.6.7) E =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

3 –1 –1 0 0 0 0 0 0–1 4 –1 –1 0 0 0 0 0–1 –1 3 0 0 0 0 0 00 –1 0 4 –1 –1 0 0 00 0 0 –1 4 –1 –1 0 00 0 0 –1 –1 4 0 0 00 0 0 0 –1 0 4 –1 –10 0 0 0 0 0 –1 2 –10 0 0 0 0 0 –1 –1 4

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦,

(3.6.8) G =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

3 –1 –1 0 0 0 0 0 0–1 4 –1 –1 0 0 0 0 0–1 –1 2 0 0 0 0 0 00 –1 0 4 –1 –1 0 0 00 0 0 –1 4 –1 –1 0 00 0 0 –1 –1 2 0 0 00 0 0 0 –1 0 4 –1 –10 0 0 0 0 0 –1 2 –10 0 0 0 0 0 –1 –1 2

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦,

(3.6.9) J =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 01 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 1 0 1 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 1 0 1 0 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦,

and

(3.6.10) JT =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 0 1 0 0 0 0 0 00 0 0 0 0 1 0 0 00 0 0 0 0 0 0 1 00 0 0 0 0 1 0 0 00 0 0 0 0 0 0 0 10 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 10 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦.

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Spectra of Lat t ices / / 115

The two and three entries along the diagonal lines of F, E, andG, correspond to edgeand corner nodes. The unit elements of J correspond to upward links from type Cto type A and B nodes. The nonzero elements of JT correspond to downward linksfrom type A or B to type C nodes.

When N1 = 2 and N2 = 1, the first component of the product φ ≡ L · ψcorresponding to the (1, 1)A node reads

(3.6.11)

φ1 = 2 × ψA1,1 – 1 × ψB

1,1 – 1 × ψC1,1

+ 0 × ψA2,1 + 0 × ψB

2,1 + 0 × ψC2,1 + 0 × ψA

3,1 + 0 × ψB3,1 + 0 × ψC

3,1

+ 0 × ψA1,2 + 0 × ψB

1,2 + 0 × ψC1,2 + 0 × ψA

2,2 + 0 × ψB2,2 + 0 × ψC

2,2

+ 0 × ψA3,2 + 0 × ψB

3,2 + 0 × ψC3,2,

the second component corresponding to the (1, 1)B node reads

(3.6.12)

φ2 = – 1 × ψA1,1 3 × ψB

1,1 – 1 × ψC1,1

– 1 × ψA2,1 + 0 × ψB

2,1 + 0 × ψC2,1 + 0 × ψA

3,1 + 0 × ψB3,1 + 0 × ψC

3,1

+ 0 × ψA1,2 + 0 × ψB

1,2 + 0 × ψC1,2 + 0 × ψA

2,2 + 0 × ψB2,2 + 0 × ψC

2,2

+ 0 × ψA3,2 + 0 × ψB

3,2 + 0 × ψC3,2,

and the third component corresponding to the (1, 1)C node reads

(3.6.13)

φ3 = – 1 × ψA1,1 – 1 × ψB

1,1 3 × ψC1,1

+ 0 × ψA2,1 + 0 × ψB

2,1 + 0 × ψC2,1 + 0 × ψA

3,1 + 0 × ψB3,1 + 0 × ψC

3,1

– 1 × ψA1,2 + 0 × ψB

1,2 + 0 × ψC1,2 + 0 × ψA

2,2 + 0 × ψB2,2 + 0 × ψC

2,2

+ 0 × ψA3,2 + 0 × ψB

3,2 + 0 × ψC3,2.

The coefficients are consistent with the entries of the matrices (3.6.6)–(3.6.10).

3.6.2 Doubly Periodic Network

Assume that the nodal scalar field of an infinite kagomé network, ψ , is periodic inthe direction of each base vector, so that

(3.6.14) ψA1, i2 = ψ

AN1+1, j, ψA

i1, 1 = ψAi1,N2+1

for the constituent lattice A

(3.6.15) ψB1, i2 = ψ

BN1+1, j, ψA

i1, 1 = ψBi1,N2+1

for the constituent lattice B, and

(3.6.16) ψC1,i2

= ψCN1+1, j

, ψCi1,1

= ψCi1,N2+1

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116 / / AN INTRODUCTION TO GRIDS, GRAPHS, AND NETWORKS

for the constituent lattice C. The nodal field, ψ , can be accommodated in a vector ψincorporating 3N1N2 unknowns,

(3.6.17) ψ =

⎡⎢⎢⎢⎢⎢⎣ψ (1)

ψ (2)

...ψ (N2–1)

ψ (N2)

⎤⎥⎥⎥⎥⎥⎦,

where

(3.6.18) ψ (1) ≡

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

ψA1,1ψB1,1ψC1,1

...ψAN1,1ψBN1,1ψCN1,1

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦, . . . , ψ (N2) ≡

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

ψA1,N2

ψB1,N2

ψC1,N2

...ψAN1,N2

ψBN1,N2

ψCN1,N2

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦.

Subject to these definitions, the Laplacian takes the form of a nearly tridiagonalblock circulant matrix consisting of N2 rows of 3N1 × 3N1 blocks, in the followingconfiguration:

(3.6.19) L =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

E –J 0 · · · 0 0 –JT

–JT E –J · · · 0 0 00 –JT E · · · 0 0 0...

......

. . ....

......

0 0 0 · · · E –J 00 0 0 · · · –JT E –J–J 0 0 · · · 0 –JT E

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦.

The nearly tridiagonal blocks, E, display the lattice coordination number 4 alongthe diagonal. The northeastern and southwestern corner blocks implement theperiodicity condition in the second direction.

The matrix E takes the block circulant form

(3.6.20) E =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

A –B 0 · · · 0 0 –BT

–BT A –B · · · 0 0 00 –BT A · · · 0 0 0...

......

. . ....

......

0 0 0 · · · A –B 00 0 0 · · · –BT A –B–B 0 0 · · · 0 –BT A

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦,

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Spectra of Lat t ices / / 117

where

(3.6.21) A =

⎡⎣ 4 –1 –1–1 4 –1–1 –1 4

⎤⎦, B =

⎡⎣ 0 0 01 0 00 0 0

⎤⎦ .

The matrix J takes the lower bidiagonal block circulant form

(3.6.22) J =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

C 0 0 · · · 0 0 DT

DT C 0 · · · 0 0 00 DT C · · · 0 0 0...

......

. . ....

......

0 0 0 · · · C 0 00 0 0 · · · DT C 00 0 0 · · · 0 DT C

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦,

where

(3.6.23) C =

⎡⎣ 0 0 00 0 01 0 0

⎤⎦, D =

⎡⎣ 0 0 00 0 10 0 0

⎤⎦,For example, when N1 = 3, we obtain the 9 × 9 matrices

(3.6.24) E =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

4 –1 –1 0 0 0 0 –1 0–1 4 –1 –1 0 0 0 0 0–1 –1 4 0 0 0 0 0 00 –1 0 4 –1 –1 0 0 00 0 0 –1 4 –1 –1 0 00 0 0 –1 –1 4 0 0 00 0 0 0 –1 0 4 –1 –1–1 0 0 0 0 0 –1 4 –10 0 0 0 0 0 –1 –1 4

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦and

(3.6.25) J =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 01 0 0 0 0 0 0 1 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 1 0 1 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 0 0 0 0 00 0 0 0 1 0 1 0 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦.

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An eigenvalue of the doubly periodic Laplacian, λ, and the correspondingeigenvector, u, satisfy the equations

(3.6.26)

(4 – λ)uAi1,i2 – uBi1–1,i2 – u

Bi1,i2 – u

Ci1,i2 – u

Ci1,i2–1 = 0,

(4 – λ)uBi1,i2 – uCi1+1,i2–1

– uCi1,i2 – uAi1+1,i2 – u

Ai1,i2 = 0,

(4 – λ)uCi1,i2 – uAi1,i2 – u

Ai1,i2+1 – u

Bi1–1,i2+1 – u

Bi1,i2 = 0.

The eigenvalues can be calculated by eliminating the lattice B and C nodes in favorof the lattice A nodes, and then setting

(3.6.27) u = exp[– i(i1αn1 + i2βn2)

],

where i is the imaginary unit and

(3.6.28) αn1 =n1 – 1

N12π , βn2 =

n2 – 1

N22π .

Simplifying, we derive a cubic equation.In an essentially equivalent approach, we note that the doubly periodic Lapla-

cian (3.6.19) is a block circulant matrix. A theorem due to Friedman [12] states thatthe spectrum of this matrix is the union of the spectra of the following 3N1 × 3N1

circulant matrices:

(3.6.29) L(n2) = – exp(–iβn2)JT + E – exp(iβn2) J

or

(3.6.30) L(n2) = – cosβn2(J + JT ) + E – i sinβn2(J – JT ),

where

(3.6.31) βn2 =n2 – 1

N22π

for n2 = 1, . . . ,N2. Making substitutions, we find that

(3.6.32) L(n2) =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

P –Q 0 · · · 0 0 –QA

–QA P –Q · · · 0 0 00 –QA P · · · 0 0 0...

......

. . ....

......

0 0 0 · · · P –Q 00 0 0 · · · –QA P –Q–Q 0 0 · · · 0 –QA P

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦,

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Spectra of Lat t ices / / 119

where the superscript A denoted the matrix adjoint defined as the complex conjugateof the transpose,

(3.6.33) P = –cCT + A – c∗C =

⎡⎣ 4 –1 –1 – c–1 4 –1

–1 – c∗ –1 4

⎤⎦,

(3.6.34) Q = –cD + B =

⎡⎣ 0 0 01 0 –c0 0 0

⎤⎦,and c ≡ exp(–iβn2). The spectrum of L(n2) is the union of the spectra of the following3 × 3 Hermitian matrices:

(3.6.35) �(n1, n2) = – exp(–iαn1)QA + P – exp(iαn1)Q,

where

(3.6.36) αn1 =n1 – 1

N12π

for n1 = 1, . . . ,N1, and the superscript A denotes the matrix adjoint. Explicitly,

(3.6.37) �(n1,n2) =

⎡⎣ 4 –1 – d –1 – c–1 – d∗ 4 –1 – cd∗–1 – c∗ –1 – c∗d 4

⎤⎦,where d = exp(–iαn1).

The trace of �(n1,n2) is

(3.6.38) T ≡ trace(�(n1,n2)

)= λn1,n2 + λ

+n1,n2 + λ

–n1,n2 = 12,

where λ◦n1, n2 , λ

+n1, n2 , and λ

–n1, n2 are the three eigenvalues.

The determinant of �(n1, n2) is

(3.6.39) D ≡ det(�(n1, n2)

)= λ◦

n1, n2 × λ+n1, n2 × λ–n1, n2 .

Performing the calculations, we find that

(3.6.40) D = 36 – 12 cosαn1 – 12 cosβn2 – 12 cos(αn1 – βn2).

The negative of the characteristic polynomial of �(n1, n2) is

(3.6.41) Pn1, n2(λ) ≡ det(λ I –�(n1, n2)

)= (λ – λ◦

n1,n2)(λ – λ+n1, n2)(λ – λ

–n1, n2 ).

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Carrying out the multiplications, we obtain

(3.6.42) Pn1, n2(λ) = λ3 – T λ2 + E λ – D,

where I is the 3 × 3 identity matrix and

(3.6.43) E = λ◦n1, n2λ

+n1, n2 + λ

+n1, n2λ

–n1, n2 + λ

–n1, n2λ

◦n1, n2 .

We find that

(3.6.44)

E = det

([4 –1 – d

–1 – d∗ 4

])+ det

([4 –1 – c

–1 – c∗ 4

])+ det

([4 –1 – cd∗

–1 – c∗d 4

]).

Computing the three 2 × 2 determinants and consolidating the sum, we find that

(3.6.45) E = 42 – 2 cosαn1 – 2 cosβn2 – 2 cos(αn1 – βn2).

Accordingly,

(3.6.46)1

λ◦n1, n2

+1

λ+n1, n2+

1

λ–n1, n2=

ED =

21 – cosαn1 – cosβn2 – cos(αn1 – βn2)

18 – 6 cosαn1 – 6 cosβn2 – 6 cos(αn1 – βn2).

For convenience, we denote

(3.6.47)a = –T = –12,b = E = 42 – 2 cos αn1 – 2 cosβn2 – 2 cos(αn1 – βn2),c = –D = –36 + 12 cosαn1 + 12 cos βn2 + 12 cos(αn1 – βn2 ).

The roots of the characteristic polynomial can be found using Cardano’s formula,yielding

(3.6.48) λ◦n1, n2 = –

a

3+ d cos

χ

3, λ±

n1, n2 = –a

3– d cos

χ ± π3

,

where

(3.6.49) d = 2(13|p|)1/2

, χ = arccos(–

q

2 (|p|/3)3/2

),

and

(3.6.50)p = b –

1

3a2 = –2

[23 + cosαn1 + cosβn2 + cos(αn1 – βn2)

],

q = c +2

27a3 –

1

3ab = 4

[1 + cosαn1 + cosβn2 + cos(αn1 – βn2 )

].

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Spectra of Lat t ices / / 121

As an example, the spectral partitioning of a kagomé network with N1 = 6 andN2 = 5 divisions is shown in Figure 3.6.2.

Exercise

3.6.1 Particle vibrations

Assume that the particles of a two-dimensional crystal are arranged on a kagomélattice. Small departures from the equilibrium position generate restoring forces.Derive and solve an algebraic eigenvalue problem for the eigen-frequencies andeigendisplacements [7].

FIGURE 3.6.2 Spectral partitioning of a periodic kagomé lattice with N1 =

6 and N2 = 5 divisions inside each period in the natural state. Positive

eigenvector components are marked with filled circles, negative com-

ponents are marked with dots, and zero components are unmarked.

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3.7 SIMPLE CUBIC LATTICE

Our analysis for the square lattice in Section 3.1 can be extended directly to thesimple cubic lattice associated with a Cartesian grid with N1, N2, and N3 divisions,as shown in Figure 3.7.1. The coordination number of the simple cubic lattice isd = 6.

Isolated NetworkThe number of nodes in an isolated network is

(3.7.1) N = (N1 + 1)(N2 + 1)(N3 + 1)

and the number of links is

(3.7.2) L = N1N2(N3 + 1) + N1(N2 + 1)N3 + (N1 + 1)N2N3.

The eigenvalues of the Laplacian matrix are

(3.7.3) λn1, n1, n3 = 4 sin2(12 αn1

)+ 4 sin2

(12 βn2

)+ 4 sin2

(12 γn3

)or

(3.7.4) λn1, n1, n3 = 6 – 2 cosαn1 – 2 cosβn2 – 2 cos γn3 ,

11

N2+1

N1+1

N3+1

i3

i2

1

i1

FIGURE 3.7.1 Illustration of a rectangular slab ofa simple cubic network containing N1 links inthe first direction, N2 links in the second direc-tion, and N3 links in the third direction. All linksare assumed to have the same conductance. In

the configuration shown, N1 = 2, N2 = 2, and

N3 = 1.

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Spectra of Lat t ices / / 123

where

(3.7.5) αn1 =n1 – 1

N1 + 1π , βn2 =

n2 – 1

N2 + 1π , γn3 =

n3 – 1

N3 + 1π

for n1 = 1, . . . ,N1 + 1, n2 = 1, . . . ,N2 + 1, and n3 = 1, . . . ,N3 + 1.The corresponding eigenvectors, un1, n2, n3 , normalized so that their lengths are

equal to unity, un1, n2, n3 · un1, n2, n3 = 1, are

(3.7.6)un1, n2, n3i1, i2, i3

= An1Bn2Cn323/2√

(N1 + 1)(N2 + 1)(N3 + 1)

× cos[(i1 – 1

2

)αn1

]cos[(i2 – 1

2

)βn2

]cos[(i3 – 1

2

)γn3

]for n1, i1 = 1, . . . ,N1 + 1, n2, i2 = 1, . . . ,N2 + 1, and n3, i3 = 1, . . . ,N3 + 1, whereAn1 = 1, Bn2 = 1, and Cn3 = 1, except that A1 = 1/

√2, B1 = 1/

√2, and C1 = 1/

√2.

Triply Periodic NetworkThe eigenvalues of the triply periodic Laplacian matrix are

(3.7.7) λn1, n2, n3 = 4 sin2(12 αn1

)+ 4 sin2

(12 βn2

)+ 4 sin2

(12 γn3

)or

(3.7.8) λn1, n2, n3 = 6 – 2 cosαn1 – 2 cosβn2 – 6 cos γn3 ,

where

(3.7.9) αn1 =n1 – 1

N12π , βn2 =

n2 – 1

N22π , γn3 =

n3 – 1

N32π

for n1 = 1, . . . ,N1, n2 = 1, . . . ,N2, and n3 = 1, . . . ,N3. We can write

(3.7.10) αn1 = (n1 – 1)k1, βn2 = (n2 – 1)k2, γn3 = (n3 – 1)k3,

where the parameters

(3.7.11) k1 =2π

N1, k2 =

N2, k3 =

N3

are directional wave numbers.The corresponding eigenvectors, normalized so that their lengths are equal to

unity, un1,n2,n3 · un1,n2,n3∗= 1, are

(3.7.12) un1, n2, n3i1, i2, i3=

1√N1N2N3

exp[– i(i1αn1 + i2βn2 + i3βn3)

]

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124 / / AN INTRODUCTION TO GRIDS, GRAPHS, AND NETWORKS

for n1, i1 = 1, . . . ,N1, n2, i2 = 1, . . . ,N2, and n3, i3 = 1, . . . ,N3, where i is theimaginary unit and an asterisk denotes the complex conjugate.

Exercise

3.7.1 Periodic cubic lattice

Derive the eigenvalues and eigenvectors of the simple cubic lattice subject to (a) theperiodicity condition in the first direction and (b) the periodicity condition in the firstand second directions.

3.8 BODY-CENTERED CUBIC (BCC) LATTICE

The nodes of the body-centered cubic (bcc) lattice can be parametrized by threeindices, i1, i2, and i3, as shown in Figure 3.8.1. The lattice coordination number isd = 8. In the Cartesian coordinates defined in Figure 3.8.1, the base vectors of theassociated Bravais lattice are

(3.8.1)a1 = a 1

2 (–ex + ey + ez), a2 = a 12 (ex – ey + ez),

a3 = a 12 (ex + ey – ez),

where ex, ey, and ex are unit vectors along the x, y, and z axes, respectively. Thereciprocal lattice base vectors are

(3.8.2)b1 = 2π

a (ey + ez), b2 = 2πa (ez + ex),

b3 = 2πa (ex + ey).

The reciprocal lattice of the bcc lattice defines the face-centered cubic (fcc) latticediscussed in Section 3.9.

i1

i2

i3

x

z

y

a

FIGURE 3.8.1 Node indexing of the body-centered cubic (bcc)network in terms of three indices, i1, i2, and i3.

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Spectra of Lat t ices / / 125

Triply Periodic NetworkThe eigenvectors of the triply periodic Laplacian matrix, normalized so that theirlengths are equal to unity, un1, n2, n3 · un1, n2, n3∗

= 1, are

(3.8.3) un1, n2, n3i1, i2, i3=

1√N1N2N3

exp[– i (i1αn1 + i2βn2 + i3βn3)

]for n1, i1 = 1, . . . ,N1, n2, i2 = 1, . . . ,N2, and n3, i3 = 1, . . . ,N3, where i is theimaginary unit and an asterisk denotes the complex conjugate.

The equation defining the eigenvalues, λ, and associated eigenvectors, u, speci-fies that

(3.8.4)8 ui1, i2, i3 – ui1–1, i2, i3 – ui1+1, i2, i3 – ui1, i2–1, i3 – ui1, i2+1, i3

–ui1, i2, i3–1 – ui1, i2, i3+1 – ui1+1, i2+1, i3+1 – ui1–1, i2–1, i3–1 = λ ui1, i2, i3 .

Substituting the expression given in (3.8.3) and simplifying, we obtain

(3.8.5)λn1, n2, n3 = 8 – 2 cosαn1 – 2 cos βn2 – 2 cos γn3

– 2 cos(αn1 + βn2 + γn3),

where

(3.8.6) αn1 =n1 – 1

N12π , βn2 =

n2 – 1

N22π , γn3 =

n3 – 1

N32π .

It is useful to introduce three new variables, ϕ1, ϕ2, and ϕ3, defined such that

(3.8.7)αn1 = –ϕ1 + ϕ2 + ϕ3, βn2 = ϕ1 – ϕ2 + ϕ3,

γn3 = ϕ1 + ϕ2 – ϕ3.

Conversely,

(3.8.8) ϕ1 = 12 (βn2 + γn3), ϕ2 =

12 (γn3 + αn1), ϕ3 =

12 (αn1 + βn2).

Substituting expressions (3.8.7) into (3.8.5) and simplifying, we obtain

(3.8.9) λn1, n2, n3 = 8(1 – cosϕ1 cos ϕ2 cos ϕ3

).

The eigenvectors given in (3.8.3) can be expressed in the form

(3.8.10) uϕ1,ϕ2,ϕ3i′1, i′2, i′3

=1√

N1N2N3exp[– i (i′1ϕ1 + i′2ϕ2 + i′3ϕ3)

],

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126 / / AN INTRODUCTION TO GRIDS, GRAPHS, AND NETWORKS

where

(3.8.11) i′1 = –i1 + i2 + i3, i′2 = i1 – i2 + i3, i′3 = i1 + i2 – i3.

Conversely,

(3.8.12) i1 = 12 (i

′2 + i

′3), i2 = 1

2 (i′3 + i

′1), i3 = 1

2 (i′1 + i

′2).

Note that the indices i′1, i′2, and i′3, are not independent. For example, if i′1 is odd oreven, then i′2 is also odd or even.

The relative position of the nodes in physical space is

(3.8.13) xi1, i2, i3 – x0,0,0 = i1a1 + i2a2 + i3a3 = i′1 a

′1 + i

′2 a2 + i

′3 a3,

where

(3.8.14)a′1 =

12 (a2 + a3) = 1

2 a ex, a′2 =

12 (a3 + a1) = 1

2 a ey,

a′3 =

12 (a1 + a2) = 1

2 a ez

are Cartesian base vectors associated with the primed indices.When N1 = N2 = N3 = N , we obtain

(3.8.15) ϕ1 =m1 – 1

N 2π , ϕ2 =m2 – 1

N 2π , ϕ3 =m3 – 1

N 2π ,

where

(3.8.16) m1 = 12 (n2 + n3), m2 = 1

2 (n3 + n1), m3 = 12 (n1 + n2).

Exercise

3.8.1 Base vectors

Confirm that the base vectors shown in (3.8.2) are the reciprocal of those shown in(3.8.1).

3.9 FACE-CENTERED CUBIC (FCC) LATTICE

The nodes of the face-centered cubic (fcc) lattice can be parametrized by three in-dices, i1, i2, and i3, as shown in Figure 3.9.1. The lattice coordination number isd = 12. In the Cartesian coordinates defined in Figure 3.9.1, the base vectors of theassociated Bravais lattice are

(3.9.1)a1 = a 1

2 (ey + ez), a2 = a 12 (ez + ex),

a3 = a 12 (ex + ey),

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Spectra of Lat t ices / / 127

i3

i2

i1

a

x

y

z

FIGURE 3.9.1 Node indexing of the face-centered cubic (fcc) network interms of three indices, i1, i2, and i3. Links are drawn as solid lines.

Nodes are located at the intersection of two dashed lines or two dotted

lines. The indices, i1, i2, and i3 vary in the directions of the three base

vectors.

where ex, ey, and ez, are unit vectors along the x, y, and z axes, respectively. Thereciprocal lattice base vectors are

(3.9.2)b1 =

a(–ex + ey + ez), b2 =

a(ex – ey + ez),

b3 =2π

a(ex + ey – ez).

The reciprocal lattice defines the body-centered cubic (bcc) lattice discussed inSection 3.8.

Triply Periodic NetworkThe eigenvectors of the triply periodic Laplacian matrix, normalized so that theirlengths are equal to unity, un1,n2,n3 · un1,n2,n3∗

= 1, are

(3.9.3) un1,n2,n3i1,i2,i3=

1√N1N2N3

exp[– i(i1αn1 + i2βn2 + i3βn3)

]for n1, i1 = 1, . . . ,N1, n2, i2 = 1, . . . ,N2, and n3, i3 = 1, . . . ,N3, where i is theimaginary unit and an asterisk denotes the complex conjugate.

The equation defining the eigenvalues, λ, and associated eigenvectors, u, speci-fies that

(3.9.4)12 ui1, i2, i3 – ui1–1, i2, i3 – ui1+1, i2, i3 – ui1, i2–1, i3 – ui1, i2+1, i3–ui1, i2+1, i3–1 – ui1, i2–1, i3+1 – ui1–1, i2, i3+1 – ui1+1, i2, i3–1–ui1+1, i2–1, i3+1 – ui1–1, i2+1, i3–1 = λ ui1, i2, i3 .

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128 / / AN INTRODUCTION TO GRIDS, GRAPHS, AND NETWORKS

Substituting the eigenvectors given in (3.9.3) and simplifying, we obtain

(3.9.5)λn1, n2, n3 = 12 – 2 cos αn1 – 2 cos βn2 – 2 cos γn3

– 2 cos(αn1 – βn2) – 2 cos(βn2 – γn3) – 2 cos(γn3 – αn1 ),

where

(3.9.6) αn1 =n1 – 1

N12π , βn2 =

n2 – 1

N22π , γn3 =

n3 – 1

N32π .

It is useful to introduce three new variables, ϕ1, ϕ2, and ϕ3, such that

(3.9.7) αn1 = ϕ2 + ϕ3, βn2 = ϕ3 + ϕ1, γn3 = ϕ1 + ϕ2.

Conversely,

(3.9.8)ϕ1 = 1

2 (–αn1 + βn2 + γn3), ϕ2 = 12 (αn1 – βn2 + γn3),

ϕ3 = 12 (αn1 – βn2 – γn3).

Substituting expressions (3.9.7) into (3.9.5) and simplifying, we obtain

(3.9.9) λn1, n2, n3 = 4(3 – cos ϕ1 cos ϕ2 – cosϕ2 cos ϕ3 – cosϕ3 cos ϕ1

).

The eigenvectors given in (3.9.3) can be expressed in the form

(3.9.10) uϕ1,ϕ2,ϕ3i′1, i′2, i′3

=1√

N1N2N3exp[– i (i′1ϕ1 + i′2ϕ2 + i′3ϕ3)

],

where

(3.9.11) i′1 = i2 + i3, i′2 = i3 + i1, i′3 = i1 + i2.

Conversely,

(3.9.12)i1 = 1

2 (–i′1 + i

′2 + i

′3), i2 = 1

2 (i′1 – i

′2 + i

′3),

i3 = 12 (–i

′1 + i

′2 – i

′3).

Note that the indices i′1, i′2, and i′3, are not independent. For example, if i′2 = 0 andi′3 = 0, the index i′1 is even.

The distance of a node from a designated zero node in physical space is

(3.9.13) xi1, i2, i3 – x0,0,0 = i1 a1 + i2 a2 + i3 a3 = i′1 a

′1 + i

′2 a2 + i

′3 a3,

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Spectra of Lat t ices / / 129

where

(3.9.14)a′1 =

12 (–a1 + a2 + a3) = 1

2 a ex, a′2 =

12 (a1 – a2 + a3) = 1

2 a ey,

a′3 =

12 (a1 + a2 – a3) = 1

2 a ez

are Cartesian base vectors associated with the primed indices.When N1 = N2 = N3 = N , we obtain

(3.9.15) ϕ1 =m1 – 1

N 2π , ϕ2 =m2 – 1

N 2π , ϕ3 =m3 – 1

N 2π ,

where

(3.9.16)m1 =

12(–n1 + n1 + n2), m2 =

12 (n1 – n2 + n3),

m3 = 12 (n1 + n2 – n3).

Exercise

3.9.1 Base vectors

Confirm that the base vectors shown in (3.9.2) are the reciprocal of those shown in(3.9.1).

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/ / / 4 / / / NETWORK TRANSPORT

In science, engineering, biological, and other applications, agraph describes a physical or abstract, conductive, convective, or mechanical net-work (e.g, [32]). Heat, electricity, mass, or any other suitable transported entity canbe supplied, generated, or consumed at the nodes. The rate of a transported entitythrough a link is typically determined by a driving potential according to a convectiveor conductive law involving the link conductance. Introducing an appropriate trans-port law provides us with a complete set of governing equations that determines theoperational state of the network. The basic concepts involved and the pertinent math-ematical framework are discussed in this chapter with emphasis on linear networksoperating at steady state.

4.1 TRANSPORT LAWS AND CONVENTIONS

Consider a transported entity, such as heat, associated with a scalar field, ψ, such atemperature, over an arbitrary network, as shown in Figure 4.1.1. If ψ is electricalvoltage, the transported quantity is electricity through an electrical grid. If ψ is pres-sure, the transported quantity is volume or mass of a transported gas or liquid along apipeline. Other abstract scalar fields pertinent, for example, to information exchangeare possible.

4.1.1 Isolated and Embedded Networks

Selected nodes of a network can be connected to external nodes where the potential,ψ , is held at a specified value in lieu of a Dirichlet boundary condition, as shown inFigure 4.1.1. For convenience, these external nodes will be called Dirichlet nodes.

It is important to note that Dirichlet nodes are included neither in the networkconfiguration nor in the graph describing the network, but are regarded as exterioranchor points. In the absence of Dirichlet nodes, we obtain an isolated network. If atleast one Dirichlet node is present, we obtain an embedded network.

130

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Network Transport / / 131

2

3

4

6

1

58

9

10

11

12

1

Dirichlet node

Dirichlet node

8

5

Dirichlet node

6

3

2

74

7

FIGURE 4.1.1 Illustration of a conducting network consisting of N = 8

nodes connected by L = 12 links. Three selected nodes of this net-

work, labeled 2, 3, and 6, are connected to external Dirichlet nodes

where the potential ψ associated with a transported entity is held at

a constant value.

Embedding MatrixIt will be convenient to introduce an N×N diagonal matrix, J, called the embeddingmatrix, that is filled with zeros, except that Jii = 1 if the ith node of the network isconnected to a peripheral Dirichlet node for i = 1, . . . ,N. For the network shown inFigure 4.1.1 where nodes 2, 3, and 6 are connected to Dirichlet nodes,

(4.1.1) J =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 0 0 0 0 0 0 00 1 0 0 0 0 0 00 0 1 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 1 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦.

In the case of an isolated network, the matrix J is filled with zeros. The diagonalvector of the matrix J, denoted by j, will be employed in the analysis of the network.

4.1.2 Nodal Sources

A transported entity associated with a scalar nodal field, ψ , can be supplied, con-sumed, removed, or dissipated at all or selected nodes of a network at a rate that isdenoted by si, where i = 1, . . . ,N. By convention, si, is positive in the case of supplyor generation and negative in the case of removal or dissipation. In the case of anisolated network, steady state is possible only if the sum of all nodal sources andsinks is zero. If this condition is not met, accumulation will take place.

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4.1.3 Linear Transport

The rate of linear transport along the mth link of a network defined by two orderedend-nodes labeled k and l is

(4.1.2) qm = cm (ψk – ψl),

where cm is the real or complex link conductance.In the case of electricity, ψi is the node voltage, qm is the electrical current, and

cm is the electrical conductance, which is the inverse of the electrical resistance.In the case of heat transport through a network of rods or wires, ψ is the

temperature and

(4.1.3) cm =kA

L,

where k is the thermal conductivity of the rod material, A is the rod cross-sectionalarea, and L is the rod length.

In the case of fluid flow through a network of pipes or tubes, qm is the vol-umetric flow rate, ψ is the pressure, p, and cm is the hydraulic conductance. UsingPoiseuille’s law, we find that, in the case of transport through a circular tube of radiusa and length L,

(4.1.4) cm =πa4

8μL,

whereμ is the fluid viscosity (e.g., [36]). The higher the fluid viscosity, the longer thetube length, and the smaller the tube diameter, the lower the conductance. Poiseuille’slaw applies under a restricted set of conditions ensuring laminar flow. A nonlinearlaw must be employed to describe unsteady turbulent flow.

The difference in the driving potential between the second and first node of themth link

(4.1.5) �ψm ≡ ψl – ψk,

can be expressed in terms on the oriented incidence matrix, R, as

(4.1.6) �ψm = Rl,mψl + Rk,mψk =N∑j=1

Rj,mψj.

Stacking all these differences in an L-dimensional vector, we obtain

(4.1.7) �ψ = RT · ψ ,

where the vector ψ encapsulates the nodal values of the potential and the superscriptT denotes the matrix transpose.

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Network Transport / / 133

4.1.4 Nonlinear Transport

In the case of nonlinear transport, the link conductance itself depends on the drivingpotential. A nonlinear transport law may prescribe that

(4.1.8) qm = cm (ψk – ψl)q,

where q is a positive exponent that is different than unity. If linear transport ispossible only in one direction but cannot occur in the opposite direction, we maywrite

(4.1.9) qm =

{cm (ψk – ψl) if ψk – ψl > 0,0 otherwise.

Concisely,

(4.1.10) qm = cm (ψk – ψl) H(ψk – ψl),

where H(w) is the Heaviside function defined such that H(w) = 1 if w > 0 andH(w) = 0 if w < 0. In the remainder of this book, we discuss exclusively linearnetworks.

Exercises

4.1.1 Electrical and optical conductances

Discuss (a) the electrical conductance of a copper cable and (b) the opticalconductance of a fiber-optic cable.

4.1.2 Nonlinear transport

Discuss a natural or engineering system where a nonlinear transport law should beemployed.

4.1.3 Embedding matrix

What is the structure of the embedding matrix, J, when each node of a network isconnected to an external Dirichlet node?

4.2 UNIFORM CONDUCTANCES

It is instructive to consider the idealized case of a network with uniform linkconductances, c. The simplified setting serves as a convenient point of departurefor introducing basic concepts and deriving governing equations for more generalnetworks.

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134 / / AN INTRODUCTION TO GRIDS, GRAPHS, AND NETWORKS

4.2.1 Isolated Networks

Balancing the rates of transport at the ith node of an isolated network in the absenceof link dissipation, attrition, supply, or removal yields the balance equation

(4.2.1)∑m

Qm = si,

where the index m ranges over all links sharing the ith node,

(4.2.2) Qm = cm (ψi – ψj),

and j is the label of the second node of the mth link. Substituting into (4.2.1) thisliner transport law and compiling all N equations, we obtain a linear system for thenodal values of ψ encapsulated in a vector, ψ ,

(4.2.3) L · ψ =1

cs.

Because the Laplacian matrix, L, is singular, its inverse does not exist and a solutionof the linear system either is not possible or can be found up to an arbitrary constant.

Multiple solutions differing by a constant exist only when the right-hand side ofthe linear system (4.2.3) is orthogonal to the eigenvector ε corresponding to the nulleigenvalue,

(4.2.4) s · ε = 0,

where the N-dimensional vector ε is filled with ones. This condition requires thatthe sum of all nodal sources and sinks is precisely zero. Physically, when the sinksare balanced by sources, an isolated network does not have a point of reference foranchoring the nodal field of a transported quantity at steady state.

4.2.2 Embedded Networks

In the case of an embedded network, we balance the rates of transport at each nodein the possible presence of a nodal source or sink and obtain the linear system

(4.2.5) L · ψ = φ +1

cs,

where

(4.2.6) L ≡ L + J

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Network Transport / / 135

is the modified Laplacian matrix and J is the embedding matrix defined in (4.1.1).The vector φ on the right-hand side is null, except that φi is the value of ψ at theDirichlet node connected to the ith network node. In the absence of Dirichlet nodes,J = 0 and φ = 0.

For example, the modified Laplacian matrix of the embedded network shown inFigure 4.1.1 is

(4.2.7) L =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

3 –1 –1 –1 0 0 0 0–1 4 –1 –1 0 0 0 0–1 –1 4 0 –1 0 0 0–1 –1 0 4 –1 –1 0 00 0 –1 –1 4 0 –1 –10 0 0 –1 0 3 –1 00 0 0 0 –1 –1 3 –10 0 0 0 –1 0 –1 2

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦.

The sum of the elements in each row or column is not necessarily zero.It is important to note that, unless the embedding matrix J is null, the modified

Laplacian matrix, L, is nonsingular and the solution of the linear system (4.2.6) isunique.

Let ε be an N-dimensional vector filled with ones and j be the diagonal vector ofJ. For the network shown in Figure 4.1.1, we have

(4.2.8) j =[0, 1, 1, 0, 0, 1, 0

].

Since L · ε = 0, we have

(4.2.9) L · ε = J · ε = j,

which confirms that, unless j is null, ε is not an eigenvector of L corresponding tothe null eigenvalue.

Exercise

4.2.1 Modified Laplacian

Confirm that the modified Laplacian matrix displayed in (4.2.7) is nonsingular.

4.3 ARBITRARY CONDUCTANCES

A generalization is necessary in the case of arbitrary link conductances. In the caseof capillary blood flow, hydraulic conductances may differ because of the differentlengths and diameters of the individual capillary segments. In the case of informa-tion network transport, the link conductances may be adjusted to reflect preferred orundesirable transmission venues.

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4.3.1 Scaled Conductance Matrix

It is convenient to introduce a reference conductance, c, and express the conductanceof the mth link as

(4.3.1) cm = c σm

for m = 1, . . . , L, where σm are dimensionless zero or positive coefficients, called thescaled link conductance, link weight, or edge weight, and L is the number of links.

For future reference, we formulate an L×L diagonal matrix, , called the scaledconductance matrix, whose mth diagonal element is equal to σm,

(4.3.2) =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

σ1 0 0 · · · 0 0 00 σ2 0 · · · 0 0 00 0 σ3 · · · 0 0 0...

......

. . ....

......

0 0 0 · · · σL–2 0 00 0 0 · · · 0 σL–1 00 0 0 · · · 0 0 σL

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦.

The average scaled link conductance is

(4.3.3) σ ≡ 1

L

L∑m=1

σm =1

Ltrace().

In the case of uniform conductances, is the identity matrix and σ = 1.The adjacency matrix, node degrees, and Laplacian matrix must be generalized

to incorporate the edge weights, σm.

4.3.2 Weighed Adjacency Matrix

The N×N weighed adjacency matrix, �, is defined such that �ij = σm if nodes i andj are connected by a link labeled m, and �ij = 0 otherwise. If all conductances areequal to c, the weighed adjacency matrix reduces to the adjacency matrix containingones and zeros.

For the network shown in Figure 4.1.1 consisting of N = 8 nodes and L = 12links, the 8 × 8 weighed adjacency matrix is

(4.3.4) � =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 σ1 σ3 σ7 0 0 0 0σ1 0 σ2 σ4 0 0 0 0σ3 σ2 0 0 σ6 0 0 0σ7 σ4 0 0 σ5 σ8 0 00 0 σ6 σ5 0 0 σ10 σ12

0 0 0 σ8 0 0 σ9 00 0 0 0 σ10 σ9 0 σ11

0 0 0 0 σ12 0 σ11 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦.

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We emphasize that Dirichlet nodes, if present, are excluded from the network.

4.3.3 Weighed Node Degrees

The weighed degree of the ith node, also called the strength of the node, is definedas

(4.3.5) δi =∑m

σm,

where the sum is over all links sharing the ith node. Consequently, δi is equal tothe sum of all nonzero elements in the ith row or column of the weighed adjacencymatrix, �.

For the network shown in Figure 4.1.1, we have

δ1 = σ1 + σ3 + σ7, δ2 = σ1 + σ2 + σ4, δ3 = σ3 + σ2 + σ6,

δ4 = σ4 + σ5 + σ7 + σ8, δ5 = σ5 + σ6 + σ10 + σ12,

δ6 = σ8 + σ9, δ7 = σ9 + σ10 + σ11, δ8 = σ11 + σ12.(4.3.6)

The individual weighed degrees, δi, can be arranged along the diagonal line of anotherwise null N × N matrix

(4.3.7) � =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

δ1 0 0 · · · 0 0 00 δ2 0 · · · 0 0 00 0 δ3 · · · 0 0 0...

......

. . ....

......

0 0 0 · · · δN–2 0 00 0 0 · · · 0 δN–1 00 0 0 · · · 0 0 δN

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦.

The average scaled link conductance is

(4.3.8) σ ≡ 1

L

L∑m=1

σm =1

2Ltrace(�),

where the factor of two in the denominator arises because each link belongs to twonodes.

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4.3.4 Kirchhoff Matrix

The N × N weighed graph Laplacian matrix, also called the Kirchhoff matrix or theadmittance matrix, is given by

(4.3.9) K = � – �,

which reveals that K is a symmetric matrix. For the network shown in Figure 4.1.1,we have

(4.3.10) K =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

δ1 –σ1 –σ3 –σ7 0 0 0 0–σ1 –δ2 –σ2 –σ4 0 0 0 0–σ3 –σ2 δ3 0 –σ6 0 0 0–σ7 –σ4 0 δ4 –σ5 –σ8 0 00 0 –σ6 –σ5 δ5 0 –σ10 σ12

0 0 0 –σ8 0 δ6 –σ9 00 0 0 0 –σ10 –σ9 δ7 –σ110 0 0 0 –σ12 0 –σ11 δ8

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦.

By construction, the sum of all elements in each row or column of the Kirchhoffmatrix is zero.

The Kirchhoff matrix for a one-dimensional network takes a tridiagonal form, asshown in Figure 4.3.1. The Kirchhoff matrix for a periodic one-dimensional networktakes a nearly tridiagonal circulant form, as shown in Figure 4.3.2.

FIGURE 4.3.1 Illustration of a one-dimensional isolated net-work consisting of N nodes connected by L = N – 1 linksand the associated Kirchhoff matrix.

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FIGURE 4.3.2 Illustration a periodic one-dimensional network con-sisting of N unique nodes connected by L = N links and theassociated Kirchhoff matrix.

4.3.5 Weighed Oriented Incidence Matrix

An alternative representation of the Kirchhoff matrix is

(4.3.11) K = R · · RT ≡ � ·�T ,

where R is the N × L oriented incidence matrix and

(4.3.12) � ≡ R · 1/2

is a modified oriented incidence matrix defined with respect to the edge weights. Thediagonal elements of the square root, 1/2, are the square roots of , while the restof the elements are zero.

4.3.6 Properties of the Kirchhoff Matrix

The Kirchhoff matrix, K, shares many of the properties of the Laplacian matrix, L,discussed in Section 2.2. Let an N-dimensional vector, ψ, contain the nodal values of

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a function at the N nodes of an arbitrary network. For any nodal field encapsulatedin a vector, ψ, we find that

(4.3.13) ψ · K · ψ =L∑

m=1

σm (ψkm – ψlm)2 ≥ 0,

where km and lm are the end-nodes of the mth link. Since σm ≥ 0, K is positivesemidefinite. Consequently, the eigenvalues of K, denoted by λi, are either zero orpositive. The sum of the eigenvalues is equal to the trace of K, which is equal to thetrace of �, which is equal to sum of the degrees of all nodes.

We may assume that the eigenvalues of K are ordered so that

(4.3.14) 0 = λ1 ≤ λ2 ≤ · · · ≤ λN .

Note that the first eigenvalue, λ1, is always zero. Further eigenvalues may also bezero.

A vector filled with ones, denoted by ε, is an eigenvector of K corresponding tothe null eigenvalue,

(4.3.15) K · ε = 0,

independent of the link weights. The reason is that the sum of the elements in anyrow of K is zero.

A network can be partitioned into two or more pieces based on the eigenvectorsof the Kirchhoff matrix, as discussed in Section 2.2.5. The link conductances havean important effect on the resulting subgraphs.

Weyl’s TheoremWeyl’s theorem states that increasing the conductance of any one link does not de-crease the magnitude of the eigenvalues of the Kirchhoff matrix. The double negativein this statement means that the magnitude of each eigenvalue either increases orstays constant when the conductance of any one link is increased. Conversely, de-creasing the scaled conductance of any one link does not increase the magnitude ofthe eigenvalues. This behavior is in agreement with physical intuition concerning theeffect of the individual links on the overall performance of a network.

4.3.7 Normalized Kirchhoff Matrix

In the absence of unconnected nodes with zero degrees, a normalized weighted in-cidence matrix, �, and the corresponding normalized Kirchhoff matrix, K, can beintroduced:

(4.3.16) � ≡ �–1/2�, K = � · �T.

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Subject to these definitions, we have

(4.3.17) K = � ·�T = �1/2 · K ·�1/2

and

(4.3.18) K = �–1/2 · K ·�–1/2 = I –�–1/2 · � ·�–1/2,

where I is the N × N identity matrix. All diagonal components of the normalizedKirchhoff matrix are equal to unity, Kii = 1. The off-diagonal components are

(4.3.19) Kij = –1√δiδj

if nodes i and j are connected by a link, and zero otherwise.

4.3.8 Summary of Notation

We have discussed networks with uniform and varying conductances and introducedparallel concepts and corresponding notation. Terms and definitions are summarizedin Table 4.3.1. For a network where all links have the same conductance, c, the Kirch-hoff matrix, K, reduces to the Laplacian matrix, L. Correspondingly, the modifiedKirchhoff matrix, K, reduces to the modified Laplacian matrix, L.

TABLE 4.3.1 Notation and Definitions for Networks with Nonuniform and Uniform ConductancesConsisting ofN Nodes and L Linksa

Size Nonuniform Uniform

Reference link conductance c c

Scaled link conductance σm 1

Scaled Dirichlet-link conductance τm 1

Weighed node degree δi diWeighed adjacency matrix N ×N � A

Weighed degree matrix (diagonal) N ×N � D

Kirchhoff matrix N ×N K = � – � L = D –A

Weighed oriented incidence matrix N ×N � R

Link conductance matrix (diagonal) L× L I

Weighed embedding matrix (diagonal) N ×N T J

ModiÞed Kirchhoff matrix N ×N K = K + T L = L + J

aIn the last column, L is the Laplacian matrix and I is the identity matrix. When all links

have the same conductance, c, the Kirchhoff matrix, K, reduces to the Laplacian matrix, L.

Correspondingly, the modified Kirchhoff matrix, K, reduces to the modified Laplacian matrix, L.

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Exercise

4.3.1 Normalized Kirchhoff matrix

Derive the normalized Kirchhoff matrix of the network shown in Figure 4.1.1.

4.4 NODAL BALANCES IN ARBITRARY NETWORKS

Systems of linear equations for the nodal values of a potential, ψ , in a linear networkwith arbitrary link conductances can be derived by compiling the balance equationsat the individual nodes. The procedure is analogous to that discussed in Section 4.2for networks with uniform link conductances.

4.4.1 Isolated Networks

In the case of an isolated networks with arbitrary link conductances, we obtain thelinear system

(4.4.1) K · ψ =1

cs,

where the vector s incorporates theN nodal sources, si. Because the Kirchhoff matrix,K, is singular, a solution exists only when the right-hand side is orthogonal to theeigenvector ε corresponding to the null eigenvalue,

(4.4.2) s · ε = 0,

where the N-dimensional vector ε is filled with ones. When this condition is met, thesolution is defined up to arbitrary constant, independent of the link conductances.

4.4.2 Embedded Networks and the Modified Kirchhoff Matrix

In the presence of Dirichlet nodes, it is convenient to introduce an N × N diagonalmatrix, T, that is filled with zeros, except that

(4.4.3) Tii = τi

if the ith network node is connected to a Dirichlet node with an external link withconductance cτi, where summation is not implied over the repeated index, i. In theabsence of Dirichlet nodes, the matrix T is null. We refer to the matrix T as theweighed embedding matrix.

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For the network shown in Figure 4.1.1 where nodes 2, 3, and 6 are connected toDirichlet nodes, we obtain

(4.4.4) T =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 0 0 0 0 0 0 00 τ2 0 0 0 0 0 00 0 τ3 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 0 0 0 τ6 0 00 0 0 0 0 0 0 00 0 0 0 0 0 0 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦.

If all diagonal elements are zero, we obtain an isolated network.The nodal values of a transported field in the presence of Dirichlet nodes satisfy

the linear system

(4.4.5) K · ψ = T · φ +1

cs,

where

(4.4.6) K ≡ K + T

is the modified Kirchhoff matrix. The vector φ is null, except that φi is the valueof ψ at the Dirichlet node connected to the ith network node. For illustration, themodified Kirchhoff matrix of a one-dimensional network involving three Dirichletnodes, labeled 1, 3, and N, is shown in Figure 4.4.1.

It is important to remember that, unless the matrix T is null, the modifiedKirchhoff matrix, K, is nonsingular.

4.4.3 Properties of the Modified Kirchhoff Matrix

Let the N-dimensional vectorψ contain the nodal values of a potential at the N nodesof an embedded network. We find that

(4.4.7) ψ · K · ψ =L∑

m=1

σm(ψkm – ψlm)2 +

N∑i=1

τiψ2i ≥ 0,

which reveals that K is positive semidefinite. Consequently, the eigenvalues of K,denoted by λi, are zero or positive.

We may assume that the eigenvalues are ordered so that

(4.4.8) 0 ≤ λ1 ≤ λ2 ≤ · · · ≤ λN .

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FIGURE 4.4.1 Illustration of a one-dimensional embedded net-work consisting of N nodes connected by L = N – 1 links andthe associated modified Kirchhoff matrix,K.

In the absence of Dirichlet nodes, K is singular and λ1 = 0.Let ε be an N-dimensional vector filled with ones. Since K · ε = 0, we find that

(4.4.9) K · ε = T · ε = τ ,where τ is the diagonal vector of T. In the case of an isolated network, τ is filledwith zeros.

Spectral ExpansionIt is useful to introduce the diagonal matrix of eigenvalues of the modified Kirchhoffmatrix, K,

(4.4.10) � =

⎡⎢⎢⎢⎢⎢⎣λ1 0 · · · 0 00 λ2 · · · 0 0...

.... . .

......

0 0 · · · λN–1 00 0 · · · 0 λN

⎤⎥⎥⎥⎥⎥⎦,

and formulate the matrix of the corresponding eigenvectors, u(i),

(4.4.11) U =

⎡⎢⎢⎣↑ ↑ ↑ ↑ ↑u(1) u(2)

... u(N–1) u(N)

↓ ↓ ↓ ↓ ↓

⎤⎥⎥⎦,

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where each eigenvector is normalized so that its norm is equal to unity, u(m)· u(m)∗ = 1for m = 1, . . . ,N, and an asterisk denotes the complex conjugate. By definition, wehave

(4.4.12) K · u(m) = λmu(m),

and thus

(4.4.13) K · U = U ·�.

A set of orthonormal eigenvectors can be chosen so that

(4.4.14) U–1 = UA,

where the superscript A denotes the matrix adjoint, defined as the complex conjugateof the transpose. Accordingly, we obtain

(4.4.15) K = U ·� · UA,

representing the spectral expansion of the modified Kirchhoff matrix.

Exercise

4.4.1 Modified Kirchhoff matrix of a periodic network

Derive the modified Kirchhoff matrix of a one-dimensional periodic network.

4.5 LATTICES

In Chapter 3, we studied the properties of infinite structured networks with uni-form conductances associated with regular lattices. The results can be extended ina straightforward fashion to lattices with nonuniform conductances.

4.5.1 Square Lattice

Consider a rectangular patch of a square network, as shown in Figure 4.5.1, andassume that the conductance of all horizontal links is ς1c and the conductance of allvertical links is ς2c, where c is a reference conductance and ς1, ς2 are two arbitrarydimensionless constants. The eigenvalues and eigenvectors of the Kirchhoff matrixcan be computed explicitly in terms of ς1 and ς2 for isolated, singly periodic, anddoubly periodic configurations. When ς1 = ς2 = ς , the Kirchhoff matrix is K = ς L,where L is the Laplacian matrix, and the spectrum of the Kirchhoff matrix is thesame as that of the Laplacian matrix.

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1 N1

N2

2

2

1

i2

i1

FIGURE 4.5.1 Illustration of a rectangular section of asquare network containing N1 links in the first directionand N2 links in the second direction. The conductance

of all horizontal links is ς1c and the conductance

of all vertical links is ς2c, where c is a reference

conductance.

Isolated NetworkThe Kirchhoff matrix corresponding to the Laplacian matrix of an isolated networkshown in (3.5.16) is

K =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

ς1� + ς2I –ς2I 0 · · ·–ς2I ς1� + 2ς2I –ς2I · · ·0 –ς2I ς1� + 2ς2I · · ·...

......

. . .0 0 0 · · ·0 0 0 · · ·0 0 0 · · ·

–→

–→

· · · 0 0 0· · · 0 0 0· · · 0 0 0...

......

...· · · ς1� + 2ς2I –ς2I 0· · · –ς2I ς1� + 2ς2I –ς2I· · · 0 –ς2I ς1� + ς2I

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦,(4.5.1)

where I is the (N1 + 1) × (N1 + 1) identity matrix and � is the Laplacian matrix of aone-dimensional isolated network with N1 + 1 nodes. When N1 = 3, we have

(4.5.2) � =

⎡⎢⎢⎣1 –1 0 0

–1 2 –1 00 –1 2 –10 0 –1 2

⎤⎥⎥⎦.

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The eigenvalues of the Kirchhoff matrix are

(4.5.3) λn1, n2 = 4 ς1 sin2(

12 αn1

)+ 4 ς2 sin

2(

12 βn2

)or

(4.5.4) λn1, n2 = 2(ς1 + ς2) – 2 ς1 cosαn1 – 2 ς2 cos βn2 ,

where

(4.5.5) αn1 =n1 – 1

N1 + 1π , βn2 =

n2 – 1

N2 + 1π

for n1 = 1, . . . ,N1 + 1 and n2 = 1, . . . ,N2 + 1. The corresponding eigenvectors aregiven in (3.1.10).

Periodic StripThe Kirchhoff matrix corresponding to the Laplacian matrix of the periodic networkgiven in (3.1.14) is shown in (4.5.1), where I is the N1 ×N1 identity matrix and � isthe Laplacian of a one-dimensional periodic network with N1 unique nodes. WhenN1 = 4, we have

(4.5.6) � =

⎡⎢⎢⎣2 –1 0 –1

–1 2 –1 00 –1 2 –1

–1 0 –1 2

⎤⎥⎥⎦.The northeastern and southwestern corner elements implement the periodicitycondition.

The eigenvalues of the Kirchhoff matrix are

(4.5.7) λn1, n2 = 4 ς1 sin2(

12 αn1

)+ 4 ς2 sin

2(

12 βn2

)or

(4.5.8) λn1, n2 = 2 (ς1 + ς2) – 2 ς1 cosαn1 – 2 ς2 cos βn2 ,

where

(4.5.9) αn1 =n1 – 1

N12π , βn2 =

n2 – 1

N2 + 1π

for n1 = 1, . . . ,N1 + 1 and n2 = 1, . . . ,N2 + 1. The corresponding eigenvectors aregiven in (3.1.20).

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Doubly Periodic NetworkThe Kirchhoff matrix corresponding to the Laplacian matrix of a doubly periodicnetwork shown in (3.1.33) is given by

K =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

ς1� + ς2I –ς2I 0 · · ·–ς2I ς1� + 2ς2I –ς2I · · ·0 –ς2I ς1� + 2ς2I · · ·...

......

. . .0 0 0 · · ·0 0 0 · · ·

–ς2I 0 0 · · ·

–→

–→

· · · 0 0 –ς2I· · · 0 0 0· · · 0 0 0...

......

...· · · ς1� + 2ς2I –ς2I 0· · · –ς2I ς1� + 2ς2I –ς2I· · · 0 –ς2I ς1� + ς2I

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦,(4.5.10)

where I is the N1×N1 identity matrix and� is the Laplacian of a one-dimensional pe-riodic network with N1 unique nodes inside each period. For example, when N1 = 4,we have

(4.5.11) � =

⎡⎢⎢⎣1 –1 0 –1

–1 2 –1 00 –1 2 –1

–1 0 –1 2

⎤⎥⎥⎦.The northeastern and southwestern corner elements implement the periodicitycondition.

The eigenvalues of the Kirchhoff matrix are given by

(4.5.12) λn1, n2 = 4 ς1 sin2(

12 αn1

)+ 4 ς2 sin

2(

12 βn2

)or

(4.5.13) λn1, n2 = 2(ς1 + ς2) – 2 ς1 cosαn1 – 2 ς2 cosαn2 ,

where

(4.5.14) αn1 =n1 – 1

N12π , βn2 =

n2 – 1

N22π

for n1 = 1, . . . ,N1 and n2 = 1, . . . ,N2. The corresponding eigenvectors are given in(3.1.41).

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4.5.2 Möbius Strip

The Kirchhoff matrix corresponding to the Laplacian matrix of the Möbius stripshown in (3.2.5) is

K =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

ς1� + ς2I –ς2I 0 · · ·–ς2I ς1� + 2ς2I –ς2I · · ·0 –ς2I ς1� + 2ς2I · · ·...

......

. . .0 0 0 · · ·0 0 0 · · ·

–ς1J 0 0 · · ·

–→

–→

· · · 0 0 –ς1J· · · 0 0 0· · · 0 0 0...

......

...· · · ς1� + 2ς2I –ς2I 0· · · –ς2I ς1� + 2ς2I –ς2I· · · 0 –ς2I ς1� + ς2I

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦,(4.5.15)

where I is the N1 × N1 identity matrix and � is the Laplacian of an isolated one-dimensional network with N1 nodes. The N1 × N1 matrix J is null, except that thenortheastern and southwestern corner elements are equal to unity. For example, whenN1 = 3, we have

(4.5.16) � =

⎡⎢⎢⎣2 –1 0 0

–1 2 –1 00 –1 2 –10 0 –1 2

⎤⎥⎥⎦, J =

⎡⎢⎢⎣0 0 0 10 0 0 00 0 0 01 0 0 0

⎤⎥⎥⎦.The eigenvalues of the Kirchhoff matrix are given by

(4.5.17) λn1, n2 = 4 ς1 sin2(

12 αn1,n2

)+ 4 ς2 sin

2(

12 βn2

)or

(4.5.18) λn1, n2 = 2 (ς1 + ς2) – 2 ς1 cosαn1, n2 – 2 ς2 cosαn2 ,

where

(4.5.19) αn1 =n1 – 1 + γ

N1π , βn2 =

n2 – 1

N2 + 1π

for n1 = 1, . . . ,N1 and n2 = 1, . . . ,N2 + 1, where γ = 0 if n2 is odd and γ = 1/2 if n2is even [56]. The corresponding eigenvectors are given in (3.2.12).

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1 2

1

2

a1

a2

N1a i1

N2

i2

FIGURE 4.5.2 Illustration of a rectangular section of a hexagonal network containing N1

links in the first direction,N2 links in the second direction, and one inclined link inside eachcell. The conductance of all links in the first direction is ς1c, the conductance of all

links in the second direction is ς2c, and the conductance of all other links is ς3c, where

c is a reference conductance.

4.5.3 Hexagonal Lattice

Consider a periodic patch of a hexagonal network, as shown in Figure 4.5.2, andassume that the conductance of all links in the first direction is ς1c, the conductanceof all links in the second direction is ς2c, and the conductance of all other links isς3c, where c is a reference conductance.

The eigenvalues of the Kirchhoff matrix in the doubly periodic configuration are

(4.5.20) λn1, n2 = 4 ς1 sin2(

12 αn1

)+ 4 ς2 sin2

(12 βn2

)+ 4 ς3 sin2

[12 (αn1 – βn2 )

]or

(4.5.21) λn1, n2 = 2 (ς1 + ς2 + ς3) – 2 ς1 cosαn1 – 2 ς2 cos βn2 – 2ς3 cos(αn1 – βn2)

for n1 = 1, . . . ,N1 and n2 = 1, . . . ,N2, where

(4.5.22) αn1 =n1 – 1

N12π , βn2 =

n2 – 1

N22π .

The corresponding eigenvectors are same as those of the doubly periodic Laplacian,given in (3.3.21).

4.5.4 Modified Union Jack Lattice

Consider a periodic patch of a modified Union Jack lattice, as shown in Figure 4.5.3,and assume that the conductances of all links in the first direction is ς1c, the con-ductances of all links in the second direction is ς2c, the conductances of all linksinclined toward the first direction is ς3c, and the conductances of all links inclined

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Network Transport / / 151

N1

N2

2

1

i2

i121

FIGURE 4.5.3 Illustration of a modified Union Jack lattice containingN1 links in the first direction, N2 links in the second direction, andtwo noncrossing transverse links inside each cell. The conduct-

ance of all links in the first direction is ς1c, the conductance of

all links in the second direction is ς2c, the conductance of all links

inclined toward the first direction is ς3c, and the conductance of

all links inclined toward the second direction is ς4c.

toward the second direction is ς4c, where c is a reference conductance and ς1 – ς4are dimensionless coefficients.

The eigenvalues of the doubly periodic Kirchhoff matrix are given by

(4.5.23)λn1, n2 = 4 ς1 sin2

(12 αn1

)+ 4 ς2 sin2

(12 βn2

)+ 4 ς3 sin2

[12

(αn1 – βn2

) ]+ 4 ς4 sin2

[12

(αn1 + βn2

) ]or

(4.5.24)λn1,n2 = 2 (ς1 + ς2 + ς3 + ς4) – ς1 cosαn1 – ς2 cosβn2 – ς3 cos(αn1 – βn2)

– ς4 cos(αn1 + βn2 ),

where

(4.5.25) αn1 =n1 – 1

N12π , βn2 =

n2 – 1

N22π

for n1 = 1, . . . ,N1 and n2 = 1, . . . ,N2. The corresponding eigenvectors are same asthose of the doubly periodic Laplacian, given in (3.4.17).

4.5.5 Simple Cubic Lattice

Consider a simple cubic network whose nodes are arranged on a Cartesian grid, asshown in Figure 4.5.4. The conductance of all links in the first direction is ς1c, theconductance of all links in the second direction is ς2c, and the conductance of alllinks in the third direction is ς3c, where c is a reference conductance and ς1, ς2, andς3 are three arbitrary dimensionless constants.

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1i11

N2 + 1

N3 + 1

N1 + 1

i2

i3

FIGURE 4.5.4 Illustration of a rectangular slab of a

simple cubic network containing N1 links in the first

direction, N2 links in the second direction, and N3

links in the third direction. For the configuration

shown, we have N1 = 2, N2 = 2, and N3 = 1. The

conductance of all links in the first direction is ς1c,the conductance of all links in the second direction

is ς2c, and the conductance of all links in the third

direction is ς3c, where c is a reference conductance.

Isolated NetworkThe eigenvalues of the Kirchhoff matrix for an isolated network are

(4.5.26) λn1, n2, n3 = 4 ς1 sin2(

12 αn1

)+ 4 ς2 sin

2(

12 βn2

)+ 4 ς3 sin

2(

12 γn3

)or

(4.5.27) λn1, n2, n3 = 2 (ς1 + ς2 + ς3) – 2 ς1 cosαn1 – 2 ς2 cos βn2 – 2 ς3 cos γn3 ,

where

(4.5.28) αn1 =n1 – 1

N1 + 1π , βn2 =

n2 – 1

N2 + 1π , γn3 =

n3 – 1

N3 + 1π

for n1 = 1, . . . ,N1 + 1, n2 = 1, . . . ,N2 + 1, and n3 = 1, . . . ,N3 + 1, The correspondingeigenvectors are given in (3.7.6).

Triply Periodic NetworkThe eigenvalues of the Kirchhoff matrix for a triply periodic simple cubic networkare given by

(4.5.29) λn1, n2, n3 = 4 ς1 sin2(

12 αn1

)+ 4 ς2 sin

2(

12 βn2

)+ 4 ς3 sin

2(

12 γn3

)

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Network Transport / / 153

or

(4.5.30) λn1, n2, n3 = 2 (ς1 + ς2 + ς3) – 2 ς1 cosαn1 – 2 ς2 cos βn2 – 2 ς3 cos γn3 ,

where

(4.5.31) αn1 =n1 – 1

N12π , βn2 =

n2 – 1

N22π , γn3 =

n3 – 1

N32π

for n1 =1, . . . ,N1, n2 =1, . . . ,N2, and n3 =1, . . . ,N3. The corresponding eigenvectorsare given in (3.1.41).

Exercise

4.5.1 Cubic lattices

(a) Derive the eigenvalues and eigenvectors of the triply periodic Kirchhoff matrixassociated with the body-centered cubic (bcc) lattice. (b) Repeat (a) for the face-centered cubic (fcc) lattice.

4.6 FINITE DIFFERENCE GRIDS

In Section 1.1, we saw that one-dimensional graphs and their Laplacian arise fromuniform finite difference grids for solving the Laplace or Poisson equation in onedimension. Two- and higher-dimensional graphs and their Laplacian arise fromcorresponding Cartesian or curvilinear grids.

As an example, we consider the Poisson equation in the xy plane for an unknownfunction f (x, y),

(4.6.1) ∇2f =∂2f

∂x2+∂2f

∂y2+ g(x, y) = 0,

where g(x, y) is a specified source term and

(4.6.2) ∇2 =∂2

∂x2+∂2

∂y2

is the Laplacian operator expressed in Cartesian coordinates.

Cartesian GridTo implement the finite difference method, we introduce a Cartesian grid with uni-form grid spacings, �x and �y, is shown in Figure 4.6.1. Applying the Poissonequation at the (i, j) node and approximating the second partial derivatives withcentral differences,

(4.6.3)(∂2f∂x2

)i,j

� fi–1, j – 2fi, j + fi+1, j�x2

+ O(�x2),

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154 / / AN INTRODUCTION TO GRIDS, GRAPHS, AND NETWORKS

i

j

Δxx

y

Δy

FIGURE 4.6.1 A Cartesian finite difference grid used tosolve the Poisson equation in two dimensions.

and

(4.6.4)(∂2f∂y2

)i, j

� fi, j–1 – 2fi, j + fi, j+1�y2

+ O(�y2),

we obtain the difference equation

(4.6.5)fi–1, j – 2fi, j + fi+1, j

�x2+fi, j–1 – 2fi, j + fi, j+1

�y2+ gi,j = 0.

Rearranging, we obtain

(4.6.6) 2(1 + β)fi, j – fi+1, j – fi–1, j – βfi, j–1 – βfi, j+1 = �x2gi, j,

where β = (�y/�x)2. Compiling all difference equations and implementing speci-fied boundary or periodicity conditions provides us with a linear system involvingthe Kirchhoff or modified Kirchhoff matrix for the square lattice, as discussed inChapter 1 for the corresponding problem in one dimension.

Isolated networks arise when the Neumann boundary condition is specifiedaround the solution domain, and embedded networks arise when the Dirichletboundary condition is entirely or partially employed.

Interpolated FieldWhen �x = �y = a, corresponding to β = 1, we obtain the interior differenceequation

(4.6.7) 4fi, j – fi+1, j – fi–1, j – fi, j –1 – fi, j+1 = a2gi, j.

Compiling all difference equations and implementing the boundary or periodic-ity conditions, we obtain system (4.2.3) or (4.2.5) with the Laplacian or modifiedLaplacian of the square lattice.

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Network Transport / / 155

(a) (b) (c)

23

4

5 6

1

a

01

3

3

1

2

13

4

0

a

2

2

a

0

FIGURE 4.6.2 Computational stencils of the Laplacian on (a) a square,(b) a honeycomb, (c) and a hexagonal lattice.

In fact, a detailed error analysis of the square computational stencil illustrated inFigure 4.6.2(a) with �x = �y = a reveals that

(4.6.8)(∇2f

)0 � –

1

a2(4f0 – f1 – f2 – f3 – f4

)+

1

12

(∂4f∂x4

+∂4f

∂y4

)0a2

(e.g., [35], p. 508). This means that the discrete (network) solution describes exactlylinear, quadratic, and cubic continuous fields constructed by interpolation.

Honeycomb GridThe computational stencil of the Laplacian on a honeycomb lattice is shown inFigure 4.6.2(b). When the x axis is aligned with the first link, as shown in theillustration, we find that

(4.6.9) (∇2f )0 � –4

3a2(3f0 – f1 – f2 – f3

)–1

6

(∂3f∂x3

– 3∂3f

∂x∂y2

)0a

(e.g., [3], p. 507; [35], p. 511). Similar approximations can be written when the firstlink is aligned with the y axis. Compiling all difference equations and implement-ing boundary or periodicity conditions, we obtain system (4.2.3) or (4.2.5) with theLaplacian or modified Laplacian of the honeycomb lattice.

Hexagonal GridIn the case of the hexagonal lattice illustrated in Figure 4.6.2(c) where each node isshared by six links, we obtain

(4.6.10) (∇2f )0 � –2

3a2

(6 f0 –

6∑i=1

fi)–

1

16(∇4f )0 a

2 + · · · ,

where ∇4 = ∇2∇2 is the biharmonic operator (e.g., [35], p. 511). Compiling alldifference equations and implementing boundary or periodicity conditions, we obtainsystem (4.2.3) or (4.2.5) with the Laplacian or modified Laplacian of the hexagonallattice.

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Helmholtz EquationOther differential equations can be solved by finite difference methods. Consider theHelmholtz equation in two dimensions,

(4.6.11) ∇2f =∂2f

∂x2+∂2f

∂y2+ kf = 0,

where k is a real or complex constant. The counterpart of the difference equation(4.6.7) on a square grid is

(4.6.12) (4 + k) fi, j – fi+1, j – fi–1, j – fi, j –1 – fi, j +1 = 0.

A generalized equation is

(4.6.13) t fi, j – 12 γ(fi+1, j + fi–1, j + fi, j –1 + fi, j+1

)= 0,

where t and γ are arbitrary unrelated coefficients. Compiling all difference equationsand implementing boundary or periodicity conditions, we obtain a linear system thatis similar to (4.2.3) or (4.2.5).

Exercise

4.6.1 Finite difference discretization

Formulate a linear system for solving the Poisson equation on a uniform Cartesianlattice when the Neumann boundary condition is specified around the four edges ofa rectangular solution domain.

4.7 FINITE ELEMENT GRIDS

The finite element method provides us with a venue for deriving systems of algebraicequations for the nodal values of an unknown function that satisfies a given ordinaryor partial differential equation (e.g., [34]). The nodes define segments in one di-mension or geometrical elements with various shapes in two and three dimensions.The algebraic equations can be derived by various methods, including the method ofGalerkin projection and the method of least squares minimization.

4.7.1 One-Dimensional Grid

Consider a one-dimensional finite element grid consisting of straight segments,called finite elements, as shown in Figure 4.7.1. Our objective is to compute anumerical solution of the one-dimensional Laplace equation,

(4.7.1)d2f

dx2= 0,

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Network Transport / / 157

1

2

L

N1 ii−1 +1i

iElements

Nodes

x

FIGURE 4.7.1 A one-dimensional finite element grid consisting of a chain ofstraight segments connected at nodes.

subject to suitable boundary conditions. Applying the Galerkin finite element methodunder the assumption that the finite element solution varies linearly across the lengtheach element, we obtain an algebraic equation associated with the ith interior node,

(4.7.2) –1

hi–1fi–1 +

(1

hi–1+

1

hi

)fi –

1

hifi+1 = 0,

where hi = xi+1 – xi is the element length (e.g., [34]).The finite element grid can be regarded as a one-dimensional network, and equa-

tion (4.7.2) can be regarded as a nodal balance involving links with conductances

(4.7.3) ci =1

hi=a

hi

1

a= σi c, σi =

a

hi, c =

1

a,

and a is a reference length. The conductance is inversely proportional to the elementsize, in agreement with physical intuition. The Laplacian matrix can be assembledby collecting the finite element equations at each node.

4.7.2 Two-Dimensional Grid

Next, we consider a two-dimensional finite element grid consisting of three-nodetriangles (e.g., [34]). An example of a grid generated by Delaunay triangulation based

(a () b)

DE

F

A

B

G

C

H

FIGURE 4.7.2 (a) A two-dimensional finite element grid consisting ofthree-node triangles with straight edges generated by Delaunay tri-angulation based on a specified set of nodes. (b) A neighborhoodof the finite element grid where element edges are interpreted asnetwork links. The dotted lines describe the underlying Voronoi

tessellation.

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on a specified set of nodes is shown in Figure 4.7.2(a). Our objective is to computea numerical solution of the two-dimensional Laplace equation,

(4.7.4)∂2f

∂x2+∂2f

∂y2= 0,

subject to suitable boundary conditions.A typical neighborhood of a finite element grid is shown in Figure 4.7.2(b). Node

A is connected with element edges to six adjacent nodes, B–G. We denote the areaof a triangular element formed by three vertices, X, Y, and Z, by AXYZ, the squareof the length of a straight segment connecting nodes X and Y by 2XY, and the vectorconnecting an oriented pair of nodes X and Y by �XY. By definition,

(4.7.5) 2XY = �XY · �XY.

Applying the Galerkin finite element method under the assumption that the finiteelement solution varies linearly over each triangular element with respect to x and y,we obtain an algebraic equation associated with the interior node labeled A,

(4.7.6)∑

X=A, B, . . . , G

αX fX = 0,

where the scalar coefficients αX are given by

(4.7.7)

αA =2BC

AABC+2CD

AACD+2DE

AADE+2EF

AAEF+2FG

AAFG+2GB

AAGB,

αB =�AG · �GBAAGB

+�AC · �CBAABC

, αC =�AB · �BCAABC

+�AD · �DCAACD

,

αD =�AC · �CDAACD

+�AE · �EDAADE

, αE =�AD · �DEAADE

+�AF · �FEAAEF

,

αF =�AE · �EFAAEF

+�AG · �GFAAFG

, αG =�AF · �FGAAFG

+�AB · �BGAAGB

.

Using elementary geometry, we confirm that

(4.7.8) αA = –∑

X=B, . . . , H

αX,

and thus

(4.7.9)∑

X=A, . . . , H

αX = 0.

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Network Transport / / 159

The finite element grid may thus be regarded as a two-dimensional network, andequation (4.7.5) may be regarded as a nodal balance involving links originating frompoint A with conductances

(4.7.10) cAX = –αX.

for X = B, . . . , H.As an exercise, we consider the hexagonal finite element assembly shown in

Figure 4.6.2(c). The area of each triangular element is A =√34 a2. The preceding

formulas give

(4.7.11) αX =a2

A=

4√3

for X = B, . . . , H, consistent with the finite difference derivation.The interpretation of the finite element edges as network links hinges on the in-

dependence of the link conductance on the node where the finite element equation isapplied. To demonstrate this subtlety, we apply the finite element equation at pointC in Figure 4.7.1(b), which is connected by element edges to four nodes labeled A,B, H, and D, and obtain the equation

(4.7.12)∑

X=C,A,B,H,D

βX fX = 0,

where

(4.7.13)

βC =2AB

ACAB+2BH

ACBH+2HD

ACHD+2DA

ACDA,

βA =�CD · �DAACDA

+�CB · �BAACAB

, βB =�CA · �ABACAB

+�CH · �HBACBH

,

βH =�CB · �BHACBH

+�CD · �DHACHD

, βD =�CH · �HDACHD

+�CA · �ADACDA

.

Using elementary geometry, we confirm that

(4.7.14) βC = –∑

X=A,B,H,D

βX.

Equation (4.7.12) can be regarded as a nodal balance involving links originating frompoint C with conductances

(4.7.15) cCX = –βX

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160 / / AN INTRODUCTION TO GRIDS, GRAPHS, AND NETWORKS

θC

θD

D

C

B

A

FIGURE 4.7.3 The conductance of a linkconnecting nodes A and B in a two-dimensional finite element grid for solv-ing Laplace’s equation is defined in termsof the angles θC and θD according to(4.7.17).

for X = A, B, H, D. The aforementioned interpretation of element edges as networklinks with unique conductance hinges on the observation that

(4.7.16) cAC = cCA = –αC = –βA.

These results indicate that the conductance of an edge connecting nodes A and Bin a triangular finite element grid, as shown in Figure 4.7.2, is given by

(4.7.17) cAB = –�AC · �CBAABC

–�AD · �DBAADB

= cot∣∣θC | + cot| θD

∣∣,where the angles θC and θD are defined in Figure 4.7.3. Conversely, when the linkconductivities are computed from (4.7.17), the nodal field of the underlying net-work that is consistent with a finite element grid represents a solution of Laplace’sequation.

Similar conclusions are reached in the analysis of three-dimensional tetrahedralfinite element grids (e.g., [34]).

Exercise

4.7.1 Three-dimensional grid

Derive an expression for the link conductance of a three-dimensional finite elementgrid consisting of tetrahedral elements (e.g., [34]).

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/ / / 5 / / / GREEN’S FUNCTIONS

In the context of networks, a Green’s function represents thenodal field established when a source is applied at a specified node. In discussingnetwork Green’s functions, it is imperative to make a distinction between embeddednetworks connected to their environment through Dirichlet nodes and isolated net-works distinguished by the absence of Dirichlet nodes, as discussed in Section 4.1.Infinite lattices are special realizations of embedded networks. Because of the singu-lar nature of the Laplace or Kirchhoff matrix, the Moore–Penrose Green’s function,also called a generalized Green’s function associated with a matrix pseudo-inverse,must be employed in the case of isolated networks.

Regular and generalized Green’s functions are elementary mathematical devicesuseful in theoretical analysis and practical applications. The networks discussed inthis chapter are assumed to be connected, that is, to be devoid of fragments, islands,and isolated nodes, unless stated otherwise.

5.1 EMBEDDED NETWORKS

When selected nodes of a network are attached to Dirichlet nodes, the modifiedKirchhoff matrix introduced in Section 4.4, given by

(5.1.1) K = K + T,

is invertible, where K is the Kirchhoff matrix. We recall that the N × N matrix T isfilled with zeros, except that Tii = τi if the ith node is connected to a Dirichlet nodeby an external link with conductance cτi, where c is a reference conductance.

The Green’s function vector associated with the ith node, denoted by g(i), is theN-dimensional nodal field satisfying the equation

(5.1.2) K · g(j) = e(j),

where the unit vector e(j) is filled with zeros, except that the jth element is equal tounity, e(j)j = 1. Since K is invertible, a unique solution can be found, given by

(5.1.3) g(j) = K–1 · e(j)

161

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for j = 1, . . . ,N. Physically, the nodal field associated with the Green’s function isestablished when a unit source is applied at the jth node, while the potential associ-ated with the transported field is held at the reference value of zero at the Dirichletnodes supporting the network. For example, in the case of fluid flow through a net-work or capillary tubes, fluid is injected at one node, while the pressure is held at thereference value of zero at peripheral Dirichlet nodes.

5.1.1 Green’s Function Matrix

The N×N Green’s function matrix contains in its columns all nodal Green’s functionvectors,

(5.1.4) Gij ≡ g(j)i .

By definition,

(5.1.5) K · G = I, G = K–1,

where I is the N×N identity matrix. Thus, the Green’s function matrix is simply theinverse of the modified Kirchhoff matrix.

Because the modified Kirchhoff matrix is symmetric, the Green’s function matrixis also symmetric,

(5.1.6) Gij = Gji.

Thus, the N × N Green’s function matrix encompasses in its columns or rows allnodal Green’s function vectors.

In terms of the Green’s function matrix, the solution of the linear system (4.4.5)governing network transport,

(5.1.7) K · ψ = T · φ +1

cs,

is given by

(5.1.8) ψ = G · χ ,

where

(5.1.9) χ ≡ T · φ +1

cs,

and the vector s encompasses the nodal sources. These expressions are consistentwith the definition g(j) = G · e(j).

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Green’s Funct ions / / 163

Spectral ExpansionUsing (5.1.5) and the spectral expansion of the modified Kirchhoff matrix stated in(4.4.15), we find that the Green’s function admits the spectral expansion

(5.1.10) G = U ·�–1 · UA,

where the superscript A denotes the matrix adjoint, that is, the complex conjugate ofthe transpose of the underlying matrix. The corresponding sum representation is

(5.1.11) G =N∑s=1

1

λsu(s) ⊗ u(s)

∗,

where u(s) are the eigenvectors of the augmented Kirchhoff matrix, normalized sothat

(5.1.12) u(s) · u(s)∗ = 1,

and an asterisk denotes the complex conjugate. In index notation,

(5.1.13) Gij =N∑s=1

1

λsu(s)i u

(s)∗j .

Since all eigenvalues are nonzero, the sum is well-defined. This representation is alsovalid in the case of multiple eigenvalues supporting an orthonormal set of distincteigenvectors.

5.1.2 Normalized Green’s Function

The nodal field due to a point source responsible for the Green’s function can benormalized so that it takes the reference value of zero at the application point. Thecorresponding normalized Green’s function, indicated by a tilde, is defined as

(5.1.14) Gij ≡ Gij – Gjj,

where summation is not implied over the repeated index, j. By definition, thediagonal elements of ˜G are zero:

(5.1.15) Gjj = 0.

Using (5.1.13), we find that the spectral expansion of the normalized Green’sfunction is

(5.1.16) Gij =N∑s= 1

1

λs

(u(s)i – u(s)j

)u(s)

∗j .

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It is important to note that the normalized Green’s function is not necessarilysymmetric, that is,

(5.1.17) Gij = Gji,

in general. The reason is that Gii is not necessarily equal to Gjj. An exception occursin the case of an infinite regular lattice.

Exercise

5.1.1 One-dimensional network

Compute the Green’s function of the one-dimensional network shown in Figure 4.4.1for uniform link conductances, σi = 1, τ1 = 1, τ3 = 1, and τN = 1.

5.2 ISOLATED NETWORKS

In the absence of Dirichlet nodes, T = 0, the modified Kirchhoff matrix, K, reducesto the Kirchhoff matrix, K, which is singular due to the presence of a zero eigen-value with a corresponding uniform eigenvector, ε, even in the absence of networkfragments and islands, as presently assumed.

Let the N-dimensional vector ε be filled with ones. Because the right-hand sideof the linear system defining a Green’s function vector,

(5.2.1) K · g(j) = e(j),

does not satisfy the compatibility condition e(j) · ε = 0, a solution cannot be foundand the Green’s function of an isolated network is poorly defined.

5.2.1 Moore–Penrose Green’s Function

To circumvent this difficulty, we introduce a nodal field, h(j), established when a unitsource is applied at the jth node, while a uniform distribution of sinks is simultane-ously applied at all nodes, so that the total strength of the point source and sinks iszero. By definition, the nodal field h(j) satisfies the linear system

(5.2.2) K · h(j) = e(j) –1

Nε.

Since ε · ε = N,

(5.2.3)(e(j) –

1

)· ε = 0,

the compatibility condition (4.4.2) is fulfilled, and the solution of the linear system(5.2.2) can be found up to an arbitrary uniform nodal field.

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Green’s Funct ions / / 165

To render the solution unique, we may specify that

(5.2.4) h(j) · ε = 0,

that is, we may stipulate that the N elements of h(j) add up to zero.Next, we put the individual nodal fields h(j) for j = 1, . . . ,N, at the columns of a

matrix H satisfying the equation

(5.2.5) K · H = H · K = I ,

where

(5.2.6) I ≡ I –1

Nε ⊗ ε,

I is the N×N identity matrix, and all components of the N-dimensional vector ε andN × N matrix ε ⊗ ε are equal to unity (e.g., [26]). Explicitly,

(5.2.7) I =1

N

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

N – 1 –1 –1 · · · –1 –1 –1–1 N – 1 –1 · · · –1 –1 –1–1 –1 N – 1 · · · –1 –1 –1...

......

. . ....

......

–1 –1 –1 · · · N – 1 –1 –1–1 –1 –1 · · · 1 N – 1 –1–1 –1 –1 · · · –1 –1 N – 1

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦.

It will be noted that the matrix on the right-hand side is the Laplacian of a completegraph.

The imposed condition (5.2.4) requires that the sum of the elements in each rowor column of H is zero:

(5.2.8) H · ε = ε · H = 0.

Consequently and because ε · ε = N, we have

K · ε = 0, H · ε = 0, I · ε = 0,K · I = K, H · I = H.(5.2.9)

In fact, the matrix H is the Moore–Penrose inverse of the Kirchhoff matrixsatisfying the equations

(5.2.10) K · H · K = K, H · K · H = H.

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For any vector c that is orthogonal to the uniform eigenvector ε, satisfying c · ε = 0,we have

(5.2.11) K · d = c,

where

(5.2.12) d ≡ H · c

and the vector d is also orthogonal to ε.To compute the generalized Green’s function H, we note that

(5.2.13) (K + ε ⊗ ε) ·(H +

1

N2ε ⊗ ε

)= I

and set

(5.2.14) H = (K + ε ⊗ ε)–1 –1

N2ε ⊗ ε.

The inverse of the deflated Kirchhoff matrix on the right-hand side of (5.2.13),(K + ε ⊗ ε)–1, is well-defined [26].

Network transport is governed by the linear system (4.4.1):

(5.2.15) K · ψ =1

cs.

Multiplying this system by H and using (5.2.5), we obtain the solution

(5.2.16) ψ =1

cH · s + 1

N(ψ · ε) ε.

The second term on the right-hand side contributes an inconsequential uniform nodalfield.

5.2.2 Spectral Expansion

Using (5.2.12), we find that if λ1 = 0 and λs are the eigenvalues of the Kirchhoffmatrix, K, for s = 2, . . . ,N, then 1 = 0 and s = 1/λs for s = 2, . . . ,N are eigenval-ues of the generalized Green’s function H, and the corresponding eigenvectors areidentical. By definition, we have

(5.2.17) H · U = U ·�0,

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Green’s Funct ions / / 167

where

(5.2.18) �0 =

⎡⎢⎢⎢⎢⎢⎣0 0 · · · 0 00 1/λ2 · · · 0 0...

.... . .

......

0 0 · · · 1/λN–1 00 0 · · · 0 1/λN

⎤⎥⎥⎥⎥⎥⎦is a regularized matrix of inverse eigenvalues, excluding the troublesome infiniteinverse eigenvalue.

A set of eigenvectors can be chosen so that

(5.2.19) U–1 = UA,

where the subscript A denotes the matrix adjoint, that is, the complex conjugate ofthe transpose. The spectral expansion of H is

(5.2.20) H = U ·�0 · UA.

Explicitly,

(5.2.21) H =N∑s= 2

1

λsu(s) ⊗ u(s)

or

(5.2.22) Hij =N∑s= 2

1

λsu(s)i u

(s)∗j ,

where u(s) are the eigenvectors of the Kirchhoff matrix normalized so thatu(s) · u(s)∗ = 1, and an asterisk denotes the complex conjugate. Note that summationbegins at s = 2.

5.2.3 Normalized Moore–Penrose Green’s Function

The nodal field due to a point source responsible for the Moore–Penrose Green’sfunction can be normalized to take the reference value of zero at the application point,yielding the corresponding normalized Moore–Penrose Green’s function, indicatedby a tilde

(5.2.23) Hij ≡ Hij – Hjj.

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By definition, we have

(5.2.24) Hjj = 0,

where summation is not implied over the repeated index, j. The spectral expan-sion of the normalized generalized Green’s function follows from the representation(5.2.22):

(5.2.25) Hij =N∑s= 2

1

λs

(u(s)i – u(s)j

)u(s)

∗j .

It is important to note that the normalized Moore–Penrose Green’s function is notnecessarily symmetric.

5.2.4 One-Dimensional Network

As an example, we consider an isolated one-dimensional network consisting of Nnodes and L = N – 1 links with same conductance, c, as shown in Figure 5.2.1(a).Using the eigenvalues and eigenvectors of the Laplacian matrix given in (1.7.2) and(1.7.4), we obtain the Moore–Penrose Green’s function

(5.2.26) Hij =1

2N

N∑s= 2

1

sin2(12αs

) cos[ (i – 12

)αs

]cos[ (

j – 12

)αs

],

where αs = (s – 1)π /N. Note that summation begins at s = 2. The elements of H areplotted in Figure 5.2.1(b).

(a)

1

2Nodes:

Links: L

N1 i + 1i − 1 i

i

(b) (c)

2 4 6 8 10 12 14 16510

15−6−4−2

0246

i

j

i

j

Hij

Hij

2 4 6 8 10 12 14 16510

15−8−7−6−5−4−3−2−1

01

FIGURE 5.2.1 (a) Illustration of a one-dimensional isolated network consisting of N nodes con-nected by L = N –1 links. Graph of (b) the Moore–Penrose Green’s function and (c) the normalizedMoore–Penrose Green’s function illustrating the loss of symmetry forN = 16.

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Green’s Funct ions / / 169

An alternative representation is

(5.2.27) Hij =1

2N

N – 1∑p=1

1

sin2(12pk) cos

[(i –

1

2

)pk

]cos[ (j – 1

2

)pk],

where k = π /N is the fundamental wave number and p ≡ s – 1. Note that summationbegins at p = 1.

The normalized Moore–Penrose Green’s function is given by the correspondingsum representation

(5.2.28)Hij =

1

2N

N∑s= 2

1

sin2(12αs

)×(cos[ (i – 1

2

)αs

]– cos

[ (j – 1

2

)αs

])cos[ (j – 1

2

)αs

],

which can be rearranged into

(5.2.29)Hi,j =

1

2N

N – 1∑p = 1

1

sin2(12pk)

×(cos[ (i – 1

2

)pk]– cos

[ (j – 1

2

)pk] )

cos[(j – 1

2)pk].

The elements of ˜H are plotted in Figure 5.2.1(c), demonstrating the absence ofsymmetry.

5.2.5 Periodic One-Dimensional Network

As another example, we consider a one-dimensional network with N periodicallyrepeated nodes and L = N links of the same conductance, as shown in Figure 5.2.2(a).Using the eigenvalues and eigenvectors of the periodic Laplacian given in (1.8.2) and(1.8.4), we obtain the generalized Green’s function

(5.2.30) Hij =1

4N

N∑s = 2

exp[–i (i – j)αs

]sin2

(12αs

) ,

where αs = (s – 1)2π /N and i is the imaginary unit.Alternative representations are

(5.2.31) Hij =1

4N

N – 1∑p =1

cos[(i – j) pk

]sin2

(12pk) =

1

4N

N – 1∑p=1

cos[(l – 1)pk

]sin2

(12pk) ,

where k = 2π /N is the fundamental wave number, p ≡ s – 1, and l = i – j + 1.

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(a)i−1

1

2

i

i

Links:

Nodes:

L

2

1

N

i+1

(b) (c)

2 4 6 8 10 12 14 16510

15−2−101234

ij

Hij

2 4 6 8 10 12 14 16510

15−5

−4

−3

−2

−1

0

ij

Hij

FIGURE 5.2.2 (a) Illustration of a one-dimensional periodic network consisting of nodes con-nected by links. Graph of (b) the Moore–Penrose Green’s function and (c) the normalizedMoore–Penrose Green’s function forN = 16.

Comparing the last expression in (5.2.31) with the cosine Fourier expansion(1.8.27), we obtain the complex Fourier coefficients c0 = 0 and

(5.2.32) cp =1

8N

1

sin2(12pk)

for p = 1, . . . ,N – 1. A graph of the periodic Moore–Penrose Green’s func-tion is shown in Figure 5.2.2(b). Because of translational invariance, all diagonalcomponents are equal.

The normalized Moore–Penrose Green’s function is given by

(5.2.33) Hij = –1

4N

N∑s= 2

1 – exp[–i (i – j)αs

]sin2

(12αs

) ,

where αs = 2π (s – 1)/N, which can be restated as

(5.2.34) Hi,j ≡ –k

N – 1∑p = 1

1 – cos[(i – j)pk

]sin2

(12pk) .

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Green’s Funct ions / / 171

A graph of the normalized Green’s function is shown in Figure 5.2.2(c). The apparentsymmetry is due to translational invariance along the periodic array for equal linkconductances in the absence of end effects.

5.2.6 Free-Space Green’s Function in One Dimension

In the limit N → ∞, the sum in (5.2.34) reduces into an integral and the right-handside provides us with the Green’s function of the one-dimensional infinite lattice,

(5.2.35) Lm ≡ Hj+m,j = –1

∫ 2π

0

1 – cos(mω)

sin2(12ω) dω

or

(5.2.36) Lm = –1

∫ 2π

0

1 – cos(mω)

1 – cosωdω,

where ω = pk. Performing the integration, we obtain

(5.2.37) Lm = –12 |m|.

This expression is the exact counterpart of the Green’s function of Laplace’s equationin one dimension,

(5.2.38)d2f

dx2+ δ1(x) = 0,

given by

(5.2.39) G = –12 |x|,

where δ1(x) is the one-dimensional Dirac delta function.

5.2.7 Complete Network

In the case of a complete network described by a complete graph, as illustratedin Figure 2.1.2, we use the eigenvalues and eigenvectors of the Laplacian given in(2.2.11) and (2.2.12) and obtain

(5.2.40) Hij =1

N2

N – 1∑q= 1

exp

(–i q p

N

)

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where p = i – j. A discrete Fourier orthogonality property states that

(5.2.41)N∑q= 1

exp

(–i qp

N

)=

{N if p = sN,0 otherwise,

where p and s are zero or arbitrary integers. Consequently,

(5.2.42) Hij =1

N2

{N – 1 if i = j,–1 if i = j

and

(5.2.43) H =1

N2L.

We have found that the Moore–Penrose Green’s function of a complete network isproportional to the corresponding Laplacian.

5.2.8 Discontiguous Networks

Suppose that an isolated network is unconnected, consisting of p isolated fragments.For example, when p = 2, an island consisting of N2 nodes may float in an ambientnetwork consisting of N1 nodes, where N = N1 + N2. The inverse of the Kirchhoffmatrix, K, does not exist due to the presence of p zero eigenvalues. The elements ofthe first eigenvector, ε(1), are equal to unity over the first nodal set and zero over thesecond nodal set, whereas the elements of the second eigenvector, ε(2), are equal tounity over the second nodal set and zero over the first nodal set.

A Moore–Penrose Green’s function, H, can be introduced, satisfying the equa-tion

(5.2.44) K · H = I –p∑

s= 1

1

Nsε(s) ⊗ ε(s),

where I is the N × N identity matrix. The elements of the N-dimensional vector ε(s)

are equal to unity over the sth nodal set and zero over the complement of the sthnodal set, so that

(5.2.45) K · ε(s) = 0.

Noting that ε(s) · ε(s) = Ns, we find that

(5.2.46) K · H · ε(s) = 0.

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Green’s Funct ions / / 173

In addition, we require that

(5.2.47) H · ε(s) = ε(s) · H = 0,

for s = 1, . . . , p. For any vector, c, orthogonal to the span of e(s), satisfying c ·e(s) = 0,we find that

(5.2.48) K · d = c,

where

(5.2.49) d = H · c,

and the vector d is also orthogonal to the span of e(s). In fact, H is a Moore–Penrosepseudoinverse satisfying the equation

(5.2.50) K · H · K = K.

The eigenvalues of H are s = 1/λs for s = 1, . . . ,N, except that s = 0 if λs = 0.The corresponding eigenvectors are the same as those of K.

Exercise

5.2.1 One-dimensional network

Confirm that (5.2.26) satisfies equation (5.2.5).

5.3 LATTICE GREEN’S FUNCTIONS

In Chapters 2 and 3, we discussed infinite lattices and studied the spectra of theirdoubly or triply periodic Laplacian. The results are useful in deriving specificexpressions for periodic and free-space Green’s functions.

5.3.1 Periodic Green’s Functions

An infinite two- or three-dimensional lattice admits periodic Green’s functions rep-resenting the nodal field due to a doubly or triply periodic array of point sources. Intwo dimensions, each node is parametrized by two indices, i1 and i2, and the periodicMoore–Penrose Green’s function is denoted by

(5.3.1) Hj1, j2i1, i2

.

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Physically, Hj1,j2i1, i2

represents the nodal value at the (i1, i2) vertex due to a sourceapplied at the (j1, j2) vertex and its periodic images. Three indices are employed inthree dimensions. The triply periodic Moore–Penrose Green’s function is denoted by

(5.3.2) Hj1, j2, j3i1,i2,i3

.

It is convenient to introduce two shifting indices in two dimensions, defined as

(5.3.3) m1 = i1 – j1, m2 = i2 – j2,

and write

(5.3.4) Hj1, j2i1,i2

= Hj1, j2j1+m1, j2+m2

≡ Mm1,m2 ,

where Mm1,m2 is the Moore–Penrose periodic lattice Green’s function. The nodalvalue at the point source corresponds to m1 = 0 and m2 = 0. Correspondingdefinitions are made in three dimensions.

Normalized Green’s FunctionsThe periodic Moore–Penrose Green’s function can be normalized so that it takes thereference value of zero at the source point and its images. In two dimensions, thenormalized Green’s function, indicated by a tilde, is defined as

(5.3.5) Hj1, j2i1, i2

≡ Hj1, j2i1,i2

– Hj1, j2j1, j2

.

The corresponding normalized periodic generalized lattice Green’s function is

(5.3.6) Mm1,m2 ≡ Mm1,m2 – M0,0.

By construction,

(5.3.7) Hj1, j2j1, j2

= 0, M0,0 = 0.

Three indices are employed in three dimensions.

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Green’s Funct ions / / 175

5.3.2 Free-Space Green’s Functions

The free-space lattice Green’s functions represents the nodal field due to a solitarypoint source. The free-space Green’s functions can be derived from the periodicGreen’s function by letting the size of the periodic patch tend to infinity, obtaining

(5.3.8) Gj1, j2i1,i2≡ lim

N1,N2→∞ Hj1, j2i1, i2

in two dimensions. Physically, Gj1, j2i1, i2represents the nodal value at the (i1, i2) node due

to a source applied at the (j1, j2) node. The corresponding lattice Green’s function is

(5.3.9) Lm1,m2 ≡ limN1,N2→∞ Mm1,m2 .

By definition, we have

(5.3.10) Gj1, j2j1+m1, j2+m2≡ Lm1,m2 ,

irrespective of the location of the nodal source determined by j1 and j2. Thenormalized lattice Green’s function, indicated by a tilde, is defined as

(5.3.11) Lm1,m2 ≡ Lm1,m2 – L0,0.

Analogous definitions are made in three dimensions.The nodal distribution induced by a point source diverges at a logarithmic rate

with respect to distance from the point source in two dimensions, and it decays likethe inverse of the distance in three dimensions. This behavior is consistent with thoseof the Green’s function of Laplace’s equation in an entire plane in two dimensions orspace in three dimensions, G, satisfying the forced Laplace equation

(5.3.12) ∇2G + δ(x – x0) = 0,

given by

(5.3.13) G = –1

2πlnr

a, G = –

1

4πr

in two or three dimensions, where r = |x – x0|, x is the position of a field point, x0is the position of the point source, a is an arbitrary length, and δ is the Dirac deltafunction in two or three dimensions.

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AB

B

B

C

C

C

C

C

C

FIGURE 5.3.1 Nearest neighbors (B) and sec-ond nearest neighbors (C) of a node (A) wherea source is applied on a honeycomb lattice.

Nearest NeighborsConsider an infinite honeycomb lattice with identical link conductances, and as-sume that a point source with strength s is applied at a node labeled A, as shownin Figure 5.3.1. Balancing the rates of transport of the entity associated with the cor-responding nodal potential, ψ , at that node and exploiting the inherent geometricalsymmetry of the honeycomb arrangement, we obtain

(5.3.14) c(ψB – ψA) + c(ψB – ψA) + c(ψB – ψA) + s = 0.

In terms of the normalized lattice Green’s function, Lm1,m2 , we find that

(5.3.15) ψB – ψA = s Lnn,

where the subscript nn indicates the nearest neighbor. Making substitutions, weobtain

(5.3.16) Lnn = –1

d,

where d = 3 is the lattice coordination number.In fact, expression (5.3.16) applies for any one-, two-, or three-dimensional sim-

ple lattice, provided that d is set equal to the lattice coordination number. We recallthat d = 2 for the one-dimensional lattice, d = 4 for the square lattice, and d = 6 forthe hexagonal (triangular) or simple cubic lattices.

Second Nearest Neighbor in the Honeycomb LatticeIn the particular case of the honeycomb lattice, but not more generally, we write abalance at a nearest neighbor of the node where the point source is applied and obtain

(5.3.17) 3 Lnn – 2Lsnn = 0,

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Green’s Funct ions / / 177

where the subscript snn indicates a second nearest neighbor, marked as node C inFigure 5.3.1. Accordingly,

(5.3.18) Lsnn = –12 .

This value will be confirmed by alternative methods in Section 5.7.

Exercise

5.3.1 Honeycomb lattice

Count the number of third and fourth nearest neighbors of a node on the honeycomblattice shown in Figure 5.3.1(b).

5.4 SQUARE LATTICE

Consider an infinite square lattice supporting a doubly periodic nodal field, as shownin Figure 5.4.1. Each periodic test section contains N1 square cells in the first direc-tion and N2 square cells in the second direction. The eigenvalues and eigenvectors ofthe doubly periodic Laplacian are given in (3.1.36) and (3.1.41).

5.4.1 Periodic Green’s Function

Substituting the eigenvalues and eigenvectors of the doubly periodic Laplacianinto the general expression (5.2.22), we derive the doubly periodic Moore–PenroseGreen’s function

(5.4.1) Hj1, j2i1,i2

=1

4N1N2

N1∑n1 = 1

N2∑n2 = 1

′ exp(–i[(i1 – j1)αn1 + (i2 – j2) βn2

])sin2

(12αn1

)+ sin2

(12βn2

) ,

1 N1

N2

2

2

1

i2

i1

FIGURE 5.4.1 Illustration of a periodic patch of a squarelattice consisting of N1 links in the first direction and N2

links in the second direction.

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where the prime indicates that the singular term, n1 = 1, n2 = 1, is excluded from thesum, and

(5.4.2) αn1 =n1 – 1

N12π , βn2 =

n2 – 1

N22π .

We recall that Hj1, j2i1,i2

is the field value at the (i1, i2) node due to a source applied atthe (j1, j2) node and its doubly periodic images. The first index parameterizes the firstdirection, and the second index parametrizes the second direction.

In terms of the shift indices, m1 ≡ i1 – j1 and m2 ≡ i2 – j2, we obtain the morecompact expression

(5.4.3) Mm1,m2 =1

4N1N2

N1∑n1 = 1

N2∑n2 = 1

′ exp(–i[m1 αn1 + m2 βn2

])sin2

(12αn1

)+ sin2

(12βn2

) ,where Mm1,m2 is the Moore–Penrose periodic lattice Green’s function defined in(5.3.4).

Fourier CoefficientsDefining p1 = n1 – 1 and p2 = n2 – 1, we obtain the equivalent representation

(5.4.4) Mm1,m2 =1

4N1N2

N1– 1∑p1= 0

N2– 1∑p2= 0

′ exp(–i[m1p1k1 + m2p2k2

])sin2

(12p1k1

)+ sin2

(12p2k2

) ,where the prime indicates that the singular term, p1 = 0, p2 = 0, is excluded fromthe sum, and

(5.4.5) k1 =2π

N1, k2 =

N2

are directional wave numbers. This expression reveals that the Fourier coefficients ofthe double Fourier series representing the Green’s function are

(5.4.6) cp1,p2 =1

8N1N2

1

sin2(12p1k1

)+ sin2

(12p2k2

) ,except that c0,0 = 0.

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Green’s Funct ions / / 179

Normalized Green’s FunctionNext, we normalize the Green’s function so that the point source generates a nodalfield that takes the reference value of zero at the application point. The normalizedGreen’s function, indicated by a tilde, is given by

(5.4.7) Mm1,m2 = –1

16π2k1k2

N1 – 1∑p1 = 0

N2 – 1∑p2 = 0

′ 1 – cos (m1p1k1 + m2p2k2)

sin2(12p1k1

)+ sin2

(12p2k2

)or

(5.4.8) Mm1,m2 = –1

8π2k1k2

N1 – 1∑p1 = 0

N2 – 1∑p2 = 0

′ 1 – cos (m1p1k1 + m2p2k2)

2 – cos (p1k1) – cos (p2k2),

where k1 and k2 are the directional wave numbers defined in (5.4.5).

5.4.2 Free-Space Green’s Function

In the limit N1 → ∞ and N2 → ∞, the double sum in (5.4.7) or (5.4.8) reduces intoa double integral, yielding the normalized free-space Green’s function of the infinitesquare lattice,

(5.4.9) Lm1,m2 = –1

16π2

∫ 2π

0

∫ 2π

0

1 – cos(m1ω1 + m2ω2)

sin2(12ω1

)+ sin2

(12ω2

) dω1 dω2or

(5.4.10) Lm1,m2 = –1

8π2

∫ 2π

0

∫ 2π

0

1 – cos(m1ω1 + m2ω2)

2 – cosω1 – cosω2dω1 dω2,

where ω1 and ω2 are auxiliary integration variables. By construction,

(5.4.11) L0,0 = 0.

By the fourfold symmetry of the square lattice, we have

(5.4.12) Lm,0 = L–m,0 = L0,m = L0,–m

for any positive or negative integer, m. By symmetry across a diagonal line, we have

(5.4.13) Lp,q = Lq,p

for any pair of integers, p and q.

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Alternative integral representations where the two terms in the argument of thecosine in the numerator of the integrand are separated are

(5.4.14) Lm1,m2 = –1

4π2

∫ π0

∫ π0

1 – cos(m1ω1) cos(m2ω2)

sin2(12ω1

)+ sin2

(12ω2

) dω1 dω2and

(5.4.15) Lm1,m2 = –1

2π2

∫ π0

∫ π0

1 – cos(m1ω1) cos(m2ω2)

2 – cosω1 – cosω2dω1 dω2.

Note that the limits of integration have been changed.Using (5.4.10), we find that

(5.4.16) L1,0 + L0,1 = –1

8π2

∫ 2π

0

∫ 2π

0dω1 dω2,

yielding

(5.4.17) L1,0 = L–1,0 = L0,1 = L0,–1 = –14 ,

in agreement with the general expression (5.3.16).

Horizontal and Vertical ProfilesSetting in (5.4.9) m2 = 0 or m1 = 0, we obtain a simplified expression for the nodalprofile in the first or second direction intercepting the nodal source:

(5.4.18) L±m,0 = L0,±m = –1

16π2

∫ 2π

0

∫ 2π

0

1 – cos(mt)

sin2(12 t)+ sin2

(12v) dv dt,

where v and t are two integration variables representing ω1 and ω2, or vice versa.Performing the integration with respect to v, we obtain

(5.4.19) L±m,0 = L0,±m = –1

∫ 2π

0

1 – cos(mt)

sin(12 t) [

1 + sin2(12 t) ]1/2 dt,

which can be restated as

(5.4.20) L±m,0 = L0,±m = –1

∫ π0

1 – cos(2mw)

sinw(1 + sin2 w)1/2dw,

where w = 12 t.

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Green’s Funct ions / / 181

For m = 1, we use a trigonometric identity and perform a straightforwardintegration to recover (5.4.17):

(5.4.21) L±1,0 = L0,±1 = –1

∫ π0

sinw

(1 + sin2 w)1/2dw = –

1

4.

For m = 2, we perform the integration to obtain

(5.4.22) L±2,0 = G0,±2 = –2

π

∫ π0

sinw cos2 w

(1 + sin2 w)1/2dw = –1 +

2

π.

Using (5.4.20), we find that

(5.4.23) Lm,0 – Lm–1,0 =1

∫ π0

cos(2mw) – cos[2 (m – 1)w]

sinw (1 + sin2 w)1/2wdw.

Simplifying the integrand, we obtain

(5.4.24) Lm,0 – Lm–1,0 = –1

∫ π0

sin[(2m – 1)w]

(1 + sin2 w)1/2dw.

Unfortunately, the definite integral can be found by analytical methods in terms ofelementary functions, and this prevents us from developing a recursion relation.

Diagonal ProfileA diagonal node corresponds tom1 = m2 = m, wherem is arbitrary. Applying the bal-ance equation defining the Green’s function at the node labeled m1 = 1 and m2 = 0,noting that, by symmetry, L1,1 = L1,–1 and rearranging, we extract the first diagonalvalue,

(5.4.25) L1,1 =1

2

(4 L1,0 – L2,0

)= –

1

π.

Note that this is higher in absolute value than the nearest-neighbor value, L±1,0 =L0,±1 = –1/4.

To obtain the diagonal profile of the Green’s function, we set in (5.4.9) m1 =m2 = m and derive the expression

(5.4.26) Lm,m = –1

16π2

∫ 2π

0

∫ 2π

0

1 – cos[m (ω1 + ω2)]

sin2(12ω1

)+ sin2

(12ω2

) dω1 dω2,which is equivalent to

(5.4.27) Lm,m = –1

8π2

∫ 2π

0

∫ 2π

0

1 – cos[m (ω1 + ω2)]

2 – cosω1 – cosω2dω1 dω2

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182 / / AN INTRODUCTION TO GRIDS, GRAPHS, AND NETWORKS

or

(5.4.28) Lm,m = –1

16π2

∫ π–π

∫ π–π

1 – cos(mω1) cos(mω2)

sin2(12ω1

)+ sin2

(12ω2

) dω1 dω2.It is helpful to introduce two new variables, w and v, such that

(5.4.29) ω1 = w + v, ω2 = w – v,

where

(5.4.30) w = 12 (ω1 + ω2), v = 1

2 (ω1 – ω2).

Substituting this transformation into (5.4.27), we obtain

(5.4.31) Lm,m = –1

16π2

∫ 2π

0

∫ 2π

0

1 – cos(2mw)

1 – cosw cos vdv dw.

Performing the inner integration with respect to v, we obtain a simple integralrepresentation:

(5.4.32) Lm,m = –1

∫ π0

1 – cos 2mw

sinwdw.

When m = 1, we use a trigonometric identity and compute the integral by elementarymethods to recover (5.4.25).

Using (5.4.32), we find that the diagonal profile satisfies a simple one-termrecursion relation,

(5.4.33) Lm,m – Lm – 1,m – 1 =1

∫ π0

cos 2mw – cos[2(m – 1)w]

sinwdw.

Using a trigonometric identity to simplify the integrand, we obtain

(5.4.34) Lm,m – Lm – 1,m – 1 = –1

∫ π0

sin[(2m – 1)w] dw

and then

(5.4.35) Lm,m = Lm – 1,m – 1 –1

π

1

2m – 1.

Accordingly, we have

(5.4.36) Lm,m = –1

π

m∑q= 1

1

2q – 1

for m ≥ 1, where L0,0 = 0.

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Green’s Funct ions / / 183

Subdiagonal ProfileFor convenience, we define the elements of the subdiagonal line,

(5.4.37) Km ≡ Lm + 1,m.

Writing a balance at a diagonal node and exploiting the symmetry across thediagonal, we obtain

(5.4.38) Km = 2 Lm,m – Km – 1,

where K0 = –1/4 is the value at the second nearest neighbor of the point source. Forexample,

(5.4.39) K1 = 2 L1,1 – K0 = –2

π+1

4.

Using (5.4.38), we obtain a recursion relation for the first subdiagonal array

(5.4.40) Km – Km – 2 = 2 (Lm,m – Lm – 1,m – 1) = –2

π

1

2m – 1

or

(5.4.41) Km = Km – 2 –2

π

1

2m – 1.

Thus,

(5.4.42) Km = –1

4–2

π

m/2∑q = 1

1

4q – 1

when m is even, and

(5.4.43) Km =1

4–2

π

(m–1)/2∑q= 0

1

6q – 1

when m is odd.

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Recursive RelationsThe Green’s function can be built using relations originating from nodal balancesbased on (a) the recursion relation (5.4.35) for the diagonal profile and (b) from therecursion relation (5.4.38) for the subdiagonal profile.

For points at the first axis, m2 = 0, we have

(5.4.44) Lm+ 1,0 = 4 Lm,0 – Lm – 1,0 – 2Lm,1.

For any other point, we have

(5.4.45) Lm1 + 1,m2 = 4 Lm1,m2 – Lm1–1,m2 – Lm1,m2 + 1 – Lm1,m2 – 1.

An algorithm for building the nodal field based on these recursive relations is imple-mented in the Fortran code shown in Table 5.4.1(a). Note that indices of arrays areallowed to take zero (or negative) values, assuming that these are declared at the be-ginning of the code. The output generated by the code is shown in Table 5.4.1(b), anda graph of the Green’s function is shown in Figure 5.4.2(a). Unfortunately, numericalinstability arises sufficiently far from the point source.

One-Dimensional Integral RepresentationA useful expression for the Green’s function arises by introducing a complexvariable, z, defined such that

(5.4.46) z ≡ exp(iω1), cosω1 =1

2

(z +

1

z

), dω1 = –i

dz

z,

and recasting the integral representation (5.4.10) into the form

(5.4.47) Lm1,m2 = –i

4π2

∫ 2π

0

∮1 – z|m1| exp(im2w)

z2 – 2 (2 – cosw)z + 1dz dw,

where the closed integration path is the unit circle in the z plane and we have setw = ω2 [1, 5, 47].

The roots of the denominator of the fraction inside the integral provide us withthe pole of the integrand with respect to z. Setting z = e–σ , we find that the realnumber σ satisfies the equation

(5.4.48) cosh σ = 2 – cosw.

For a pole to reside inside the unit circle, σ must be positive. A graph of σ as afunction of w is shown in Figure 5.4.3.

Page 200: An Introduction to Grids Graphs and Networks

TABLE 5.4.1 (a) Fortran Code for Computing the Normalized Green’s Function on an Infinite SquareLattice by Recursion and (b) Output of the Code

(a)

pi = 3.14159265358979D0

mmax = 8

L(0,0) = 0.0D0

L(1,0) = -0.25D0

L(2,0) = -1.00D0+2.0D0/pi

Do m=1,mmax ! diagonal elements

L(m,m) = L(m-1,m-1) -1.0D0/pi/(2.0D0*m-1)

End Do

Do m=1,mmax-1 ! first subdiagonal elements

L(m+1,m) = 2.0D0*L(m,m) - L(m,m-1)

End Do

Do m=1,mmax-1 ! rest of the elements

L(m+1,0) = 4.0D0*L(m,0) - L(m-1,0)- 2.0D0*L(m,1)

Do l=1,mmax-m-1

L(m+l+1,l) = 4.0D0*L(m+l,l)-L(m+l-1,l)-L(m+l,l+1)-L(m+l,l-1)

End Do

End do

Do j=1,mmax

Do i=0,j-1

L(i,j)=L(j,i)

End Do

End Do

(b )

0.000 -0.250 -0.363 -0.430 -0.477 -0.513 -0.542 -0.567 -0.588

-0.250 -0.318 -0.387 -0.440 -0.482 -0.516 -0.544 -0.568 -0.589

-0.363 -0.387 -0.424 -0.462 -0.496 -0.525 -0.551 -0.573 -0.593

-0.430 -0.440 -0.462 -0.488 -0.514 -0.538 -0.560 -0.580 -0.599

-0.477 -0.482 -0.496 -0.514 -0.534 -0.553 -0.572 -0.590 -0.606

-0.513 -0.516 -0.525 -0.538 -0.553 -0.569 -0.585 -0.600 -0.615

-0.542 -0.544 -0.551 -0.560 -0.572 -0.585 -0.598 -0.611 -0.624

-0.567 -0.568 -0.573 -0.580 -0.590 -0.600 -0.611 -0.622 -0.634

-0.588 -0.589 -0.593 -0.599 -0.606 -0.615 -0.624 -0.634 -0.644

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186 / / AN INTRODUCTION TO GRIDS, GRAPHS, AND NETWORKS

−20−10

010

20

−20−10

010

20−0.8

−0.6

−0.4

−0.2

0

m1m2

L

(b)

−50

0

50

−50

0

50−1

−0.8

−0.6

−0.4

−0.2

0

m1m2

L

(a)

FIGURE 5.4.2 (a) Nodal distribution of the normalizedGreen’s function over a square lattice computed by recur-sive relations. (b) The entire distribution can be computedby combining the recursive relations with the far-fieldasymptotics. Numerical instability arises far from the

point source.

The residue of the integrand at a pole, R, arises by evaluating the ratio of thenumerator and the derivative of the denominator with respect to z at a pole, finding

(5.4.49) R =1 – exp(–|m1|σ + im2w)

2 (e–σ – 2 + cosω)= –

1 – exp(–|m1|σ + im2w)

2 sinh σ.

Using the residue theorem, we find that

(5.4.50) Lm1,m2 = –i

4π2

∫ 2π

0(2π i)R dw,

yielding

(5.4.51) Lm1,m2 = –1

∫ π0

1 – exp(–|m1| σ + im2w)

sinh σdw,

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Green’s Funct ions / / 187

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

w/(2π)

σ

FIGURE 5.4.3 Graph of the pole location, σ , against the inte-gration variable, w , for computing the Green’s function on asquare lattice.

provided that σ > 0. By symmetry, we also have

(5.4.52) Lm1,m2 = –1

∫ π0

1 – exp(–|m2| σ + im1w)

sinh σdw.

Expressing the denominator of the integrand in (5.4.51) with respect to w using(5.4.48), we obtain

(5.4.53) Lm1,m2 = –1

∫ π0

1 – exp (–|m1|σ + im2 w)

sin(12 w) [

1 + sin2(12 w)]1/2 dw.

When m1 = 0, we recover precisely (5.4.20) for the horizontal or vertical profile.A Mathematica script that computes the Green’s function based on the integral

representation (5.4.51) adapted from Atkinson and van Steenwijk [1] is listed below:

sigma[omega_] := ArcCosh[2- Cos[omega]];

lgfs[m1_,m2_]:=Simplify[-1/(2*Pi)

*Integrate[(1-Exp[-Abs[m1]*sigma[omega]]*Cos[m2*omega])/Sinh[sigma[omega]]

,{omega,0,Pi}]];

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188 / / AN INTRODUCTION TO GRIDS, GRAPHS, AND NETWORKS

TABLE 5.4.2 Exact Values of the Normalized Green’s Function on a Square Lattice, Lm1,m2 , Basedon the Computation of a One-Dimensional Integral form1,m2 = 0, 2, . . . .

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 –14

–1 + 2π –17

4+ 12π –20 + 368

6π–401

4+ 1880

6π· · ·

· · · – 1π

14– 2π 2 – 23

3π494

– 40π 70 – 3223

15π· · ·

· · · · · · – 43π

–14– 2

3π–3 + 118

15π–97

4+ 1118

15π· · ·

· · · · · · · · · – 2315π

12– 12

5π4 – 499

35π· · ·

· · · · · · · · · · · · – 176105π

–12+ 20

21π· · ·

· · · · · · · · · · · · · · · – 563315π

· · ·

· · · · · · · · · · · · · · · · · · · · ·

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦Note: The lower triangular part of this symmetric matrix arises by reflection.

Results generated by this script are shown in Table 5.4.2. These numerical pre-dictions are consistent with those shown in Table 5.4.1 obtained by recursiverelations.

Far-Field AsymptoticsTo study the behavior far from the source point, we note that, for small w, the solutionof the algebraic equation (5.4.48) is

(5.4.54) σ � w + · · · ,where the three dots indicate higher-order terms. Accordingly, for large |m1|, therepresentation (5.4.51) yields

(5.4.55)Lm1,m2 � –

1

[ ∫ π0

1 – exp[(–|m1| + im2)w

wdw

+∫ π0

( 1

sinh σ–1

w

)dw

][5, 47]. Performing the integrations, we obtain the approximation

(5.4.56) Lm1,m2 � –1

(real [E(φ)] + 1

2ln 8 – lnπ

),

where

(5.4.57) φ = (|m1| – im2)π

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Green’s Funct ions / / 189

and

(5.4.58) E(z) ≡∫ z

0

1 – e–t

tdt

is the exponential integral. For large |φ|, we have

(5.4.59) real [E(φ)] � ln |φ| + E = 12 ln(m21 + m

22

)+ E + lnπ ,

where

(5.4.60) E = 0.577215665 · · ·

is the Euler constant. Substituting this expression into (5.4.56), we obtain

(5.4.61) Lm1,m2 � –1

(ln√m21 + m

22 + E +

1

2ln 8).

This asymptotic formula carries an error on the order of 10–4 around the edges of thesquare |m1| ≤ 16 and |m2| ≤ 16.

Like the Green’s function of Laplace’s equation in two dimensions, the cor-responding normalized lattice Green’s function diverges at a logarithmic rate.However, the lattice Green’s function is zero at the forced node, whereas the Green’sfunction of Laplace’s equation in two dimensions takes an infinite value at thepole.

Comparing the asymptotic expression (5.4.61) with (5.4.36), we derive theidentity

(5.4.62) limm→∞

⎛⎝ m∑q=1

2

2q – 1– lnm

⎞⎠ = E + 2 ln 2.

The sum on the left-hand side is an approximation to an integral computed by thetrapezoidal rule,

(5.4.63)m∑q= 1

2

2q – 1– 1 –

1

2m – 1�∫ m

1

2

2x – 1dx = ln(2m – 1).

The second and third terms on the left-hand side are included to render the weightsof the summed terms equal to 1/2 at the first and last points.

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5.4.3 Helmholtz Equation Green’s Function

Corresponding results can be derived for the generalized balance equation (4.6.13)originating from the Helmholtz equation, repeated below for convenience:

(5.4.64) t fi, j –1

2γ(fi+1, j + fi–1, j + fi, j–1 + fi, j+1

)= 0,

where t and γ are arbitrary coefficients (e.g., [31]).The counterparts of the integral representations (5.4.14) and (5.4.15) for the free-

space Green’s function are

(5.4.65) Lm1,m2 = –1

8γπ2

∫ π0

∫ π0

1 – cos(m1ω1) cos(m2ω2)t2γ – 1 + sin2

(12ω1

)+ sin2

(12ω2

) dω1 dω2and

(5.4.66) Lm1,m2 = –1

γπ2

∫ π0

∫ π0

1 – cos(m1ω1) cos(m2ω2)tγ– cosω1 – cosω2

dω1 dω2.

The Laplace Green’s functions (5.4.14) and (5.4.15) arise for t = 4 and γ = 2.Morita [31] developed a three-term recursive relation for the diagonal elements,

Lm ≡ Lm,m,

(5.4.67) Lm+1 =4m

2m + 1

(t2

2γ 2– 1

)Lm –

2m – 1

2m + 1Lm–1,

where L0 = 0 and L1 is available in terms of complete elliptic integrals. When t = 4and γ = 2, the term inside the parentheses is equal to unity. The nodal field can beproduced by recursion, as discussed in Section 5.4.2.

The asymptotic behavior of the Helmholtz lattice Green’s function far from thepoint source has been studied with reference to wave scattering (e.g., [29]).

5.4.4 Kirchhoff Green’s Function

The periodic of free-space Green’s function of the Laplace matrix corresponds tonetworks with uniform link conductances, c. In the case of a square network with ar-bitrary conductances in the directions of the indices i1 and i2, we use the eigenvaluesof the Kirchhoff matrix given in (4.5.3) or (4.5.4) and obtain the free-space Green’sfunction

(5.4.68) Lm1,m2 = –1

8π2

∫ 2π

0

∫ 2π

0

1 – cos(m1ω1 + m2ω2)

ς1 + ς2 – ς1 cosω1 – ς2 cosω2dω1 dω2,

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Green’s Funct ions / / 191

where ς1 and ς2 are dimensionless conductance coefficients. The double integral canbe evaluated by numerical methods.

Exercise

5.4.1 Integral representation of the free-space Green’s function

Derive the integral representation (5.4.53) from (5.4.51).

5.5 HEXAGONAL LATTICE

Consider a patch of a hexagonal lattice in its natural state supporting a doubly peri-odic nodal field, as shown in Figure 5.5.1. The base vectors of the underlying Bravaislattice are

(5.5.1) a1 = a (1, 0) , a2 = a 12

(–1,

√3),

where a is the distance between two nearest neighbors. Each periodic test sectioncontains N1 triangular cells in the first direction, and N2 triangular cells in the seconddirection. The eigenvalues and eigenvectors of the doubly periodic Laplacian aregiven in (3.3.36) and (3.3.21).

5.5.1 Periodic Green’s Function

Working as in Section 5.4.1 for the square lattice, we derive the normalized periodicGreen’s function

(5.5.2) Mm1,m2 = –k1k216π2

N1– 1∑p1= 0

N2– 1∑p2= 0

′ 1 – cos (m1p1k1 +m2p2k2)

sin2(12p1k1

)+ sin2

(12p2k2

)+ sin2

[12 (p1k1 + p2k2)

]

211

2

a1

N1

N2

i1x

a2

a

i2

y

FIGURE 5.5.1 Illustration of a Periodic Patch of a Hexagonal Lattice Con-sisting of N1 triangles in the first direction and N2 triangles in the seconddirection. The base vectors, a1 and a2, determined the node numbering

scheme.

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192 / / AN INTRODUCTION TO GRIDS, GRAPHS, AND NETWORKS

or

(5.5.3) Mm1,m2 = –k1k28π2

N1 – 1∑p1 = 0

N2 – 1∑p2 = 0

′ 1 – cos (m1p1k1 + m2p2k2)

3 – cos(p1k1) – cos(p2k2) – cos(p1k1 + p2k2),

where

(5.5.4) k1 =2π

N1, k2 =

N2

are directional wave numbers.

5.5.2 Free-Space Green’s Function

The normalized free-space Green’s function is given by the integral representation

(5.5.5) Lm1,m2 = –1

16π2

∫ 2π

0

∫ 2π

0

1 – cos(m1ω1 + m2ω2)

sin2(12ω1

)+ sin2

(12ω2

)+ sin2

[12 (ω1 + ω2)

] dω1 dω2or

(5.5.6) Lm1,m2 = –1

8π2

∫ 2π

0

∫ 2π

0

1 – cos(m1ω1 + m2ω2)

3 – cosω1 – cosω2 – cos(ω1 + ω2)dω1 dω2.

By symmetry, we have

(5.5.7) L±m,0 = L0,±m = Lm,m = L–m,–m

for any positive or negative integer, m. Using (5.5.6), we find that

(5.5.8) L1,0 + L0,1 + L1,1 = –1

8π2

∫ 2π

0

∫ 2π

0dω1 dω2,

yielding the nearest-neighbor value

(5.5.9) L±1,0 = L0,±1 = L1,1 = L–1,–1 = –1

6,

in agreement with the more general expression (5.3.16) for lattice coordinationnumber d = 6.

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Green’s Funct ions / / 193

One-Dimensional Integral RepresentationIt is convenient to introduce two new variables, w and v, such that

(5.5.10) ω1 = w + v, ω2 = w – v,

where

(5.5.11) w = 12 (ω1 + ω2), v = 1

2 (ω1 – ω2).

Substituting these transformations into (5.5.6), we obtain

(5.5.12) Lm1,m2 = –1

8π2

∫ 2π

0

∫ 2π

0

1 – exp[ i(m1 – m2)v] exp[ i(m1 + m2)w]

3 – cos(w + v) – cos(w – v) – cos 2wdv dw,

which can be rearranged into

(5.5.13) Lm1,m2 = –1

8π2

∫ 2π

0

∫ 2π

0

1 – exp[ i(m1 – m2)v] exp[ i |m1 + m2|w]

3 – 2 cosw cos v – cos 2wdv dw.

Next, we introduce a complex variable z, defined such that

(5.5.14) z ≡ exp( i v), cos v =1

2

(z +

1

z

), dv = –i

dz

z,

and find that

(5.5.15) Lm1,m2 = –i

8π2

∫ 2π

0

∮1 – z|m1–m2| exp[ i (m1 + m2)w]

z2 cosw – z (3 – cos 2w) + coswdz dw,

where the closed integration path is the unit circle centered at the origin of the z plane[5].

The roots of the denominator provide us with the poles of the integrand withrespect to z, satisfying the equation

(5.5.16)(z +

1

z

)cosw = 3 – cos 2w.

Setting z = e–σ , we find that the real number σ satisfies a nonlinear algebraicequation,

(5.5.17) cosh σ =3 – cos 2w

2 cosw=2 – cos2 w

cosw.

For a pole to reside inside the unit circle, σ must be positive. A graph of σ as afunction w is shown in Figure 5.5.2 in the interval [0, 12 π ]. A solution cannot be

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0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

w/(2π)

σ

FIGURE 5.5.2 Graph of the pole location, σ , against the inte-gration variable, w , for the hexagonal lattice. The dashed

line represents the linear dependence for small w .

found in the interval [ 12 π ,32 π ], and this means that the contour integral in (5.5.15)

is zero.The residue of the integrand, R, arises by evaluating the ratio of the numerator

and the derivative of the denominator with respect to z at a pole, finding

(5.5.18) R =1 – z|m1–m2| exp[ i (m1 + m2)w]

2 e–σ cosw – 3 + cos 2w.

Substituting 3 – cos 2w = 2 cosh σ cosw into the denominator and simplifying, weobtain

(5.5.19) R = –1

2

1 – z|m1–m2| exp[ i (m1 + m2)w]

cosw sinh σ.

Using the residue theorem, we find that

(5.5.20) Lm1,m2 = –i

8π2

∫ 2π

0(2π i)R dw

and then

(5.5.21) Lm1,m2 = –1

∫ π /20

1 – e–|m1–m2|σ cos[(m1 + m2)w]

cosw sinh σdw,

provided that σ > 0 [1]. Eliminating the dependent variable σ from the denominatorof the integrand, we obtain

(5.5.22) Lm1,m2 = –1

π

∫ π /20

1 – e–|m1 –m2|σ cos[(m1 + m2)w]

[ (3 – cos 2w)2 – 2 cos 2w – 2 ]1/2dw

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Green’s Funct ions / / 195

or

(5.5.23) Lm1,m2 = –1

π

∫ π /20

1 – e–|m1 –m2|σ cos[(m1 + m2)w]

(cos2 2w – 8 cos 2w + 7)1/2dw.

Note that the denominator becomes zero at w = 0.A radial profile arises by setting m1 = m2 = m, yielding

(5.5.24)

L±m,0 = L0,±m = Lm,m = L–m,–m

= –1

π

∫ π /20

1 – cos 2mw

( cos2 2w – 8 cos 2w + 7 )1/2dw

= –1

∫ π0

1 – cos(mv)

( cos2 v – 8 cos v + 7 )1/2dv,

where v = 2w. A Mathematica script that computes the radial profile based on(5.5.21) is presented next:

sigma[w_] := ArcCosh[2/Cos[w]- Cos[w]];lgft[m1_,m2_] := Simplify[ -1/(2*Pi) *Integrate[

(1-Exp[-Abs[m1-m2]*sigma[w]]*Cos[(m1+m2)*tau])/(Cos[w]*Sinh[sigma[w]]),{w,0,Pi/2}

]];

lgft[1,1]

Results for the five nearest neighbors of the forced node, marked as nodes A–E, areshown in Figure 5.5.3(a) [1].

Far-Field AsymptoticsTo study the behavior of the free-space Green’s function far from the point source,we note that, for small w, the solution of equation (5.5.17) is

(5.5.25) σ � √3w + · · ·,

represented by the dashed line in Figure 5.5.2, where the three dots indicate higher-order terms. For large |m1 – m2|, the integral representation (5.5.21) yields

(5.5.26)

Lm1,m2 � –

√3

⎡⎣∫ π /20

1 – exp[–|m1 – m2|

√3w + i (m1 + m2)w

]w

dw

+∫ π /20

( √3

cosw sinh σ–1

w

)dw

].

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196 / / AN INTRODUCTION TO GRIDS, GRAPHS, AND NETWORKS

Performing the integrations, we obtain

(5.5.27) Lm1,m2 � –

√3

6π( real [E (φ)] + 0.097723 · · · ),

where E it the exponential integral with complex argument defined in (5.4.58), and

(5.5.28) φ =[|m1 – m2|

√3 – i (m1 + m2)

] π2.

For large |φ|, we have

(5.5.29) real [E(φ)] � ln∣∣∣ [|m1 – m2|

√3 – i(m1 + m2)

] π2

∣∣∣ + E,

yielding

(5.5.30) real [E(φ)] � 12 ln

(m21 + m

22 – m1m2

)+ E + lnπ ,

where E = 0.577215665 · · · is the Euler constant. Substituting this expression into(5.5.27), we obtain

(5.5.31) Lm1,m2 � –

√3

(ln√m21 + m

22 – m1m2 + 1.81967 · · ·

).

Although this expression was derived under the assumption that |m1 – m2| is large, itdoes apply for arbitrarily large |m1| or |m2|. The asymptotic formula provides us withremarkably accurate results, as shown in the last column of Figure 5.5.3(a).

The entire nodal distribution of the Green’s function is shown in Figure 5.5.3(b).Like the Green’s function of the square lattice, the Green’s function for the hexagonallattice diverges at a logarithmic rate.

Exercises

5.5.1 Free-space Green’s function

Confirm by numerical integration that the predictions of formula (5.5.24) areconsistent with those shown in figure 5.5.3(a) for m = 1, 2, 3.

5.5.2 Far field

Derive the counterpart of (5.5.31) for the point indexing scheme shown inFigure 3.3.1.

5.6 MODIFIED UNION JACK LATTICE

Consider the modified Union Jack lattice shown in Figure 5.6.1, supporting a doublyperiodic nodal field. Each periodic test section contains N1 square cells in the firstdirection and N2 square cells in the second direction inside each period. The eigen-values and eigenvectors of the doubly periodic Laplacian were given in (3.4.13) and(3.3.21).

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Green’s Funct ions / / 197

(a)

A

A

A

A

A

A

B

B

B

B

B

B

C

C

C C

C

C

DD

D

D

D

D

D D

D

EE

E

D

D

D

E

E E

(b)

citotpmysAtcaxEedoN

A L±1,0 = L0,±1 = L1,1 = L−1,−1 − 16 = −0.1667 −0.1672

B L1,2 = L2,113 − 1

π

√3 = −0.2180 −0.2177

C L2,0 = L0,2 = L2,2 = L−2,−2 − 43 + 2

π

√3 = −0.2307 −0.2309

D L1,3 = L3,1 = L2,3 = L3,252 − 5

π

√3 = −0.2566 −0.2566

E L±3,0 = L0,±3 = L3,3 = L−3,−3 − 272 + 24

π

− 613 − 1

π

√3

− 43 + 2

π

√3

52 − 5

π

√3

27 + 24√

3 = −0.2681 −0.2682

FIGURE 5.5.3 (a) Normalized free-space Green’s function of the hexagonallattice at the five nearest neighbors of the point source, A–E. (b) Nodaldistribution of the normalized Green’s function over the hexagonal lattice.

5.6.1 Periodic Green’s Function

Working as in Section 5.4.1, we derive the normalized Moore–Penrose doublyperiodic Green’s function

(5.6.1)

Mm1,m2 = –k1k216π2

N1 – 1∑p1=0

N2 – 1∑p2=0

1 – cos (m1p1k1 +m2p2k2)

sin2(12p1k1

)+ sin2

(12p2k2

)+ sin2

[12 (p1k1 + p2k2)

]+ sin2

[12 (p1k1 – p2k2)

]

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198 / / AN INTRODUCTION TO GRIDS, GRAPHS, AND NETWORKS

1 N1

N2

2

21

i1

i2

FIGURE 5.6.1 Illustration of a periodic patch of a mod-ified Union Jack lattice consisting of N1 square cellsin the first direction and N2 square cells in the seconddirection.

or

(5.6.2)

Mm1,m2 = –k1k28π2

N1 – 1∑p1= 0

N2 – 1∑p2= 0

1 – cos (m1p1k1 + m2p2k2)

4 – cos(p1k1) – cos(p2k2) – cos(p1k1 + p2k2) – cos(p1k1 – p2k2),

where k1 = 2π /N1 and k2 = 2π /N2 are directional wave numbers, and the primeindicates that the singular term, p1 = 0 and p2 = 0, is excluded from the double sum.

5.6.2 Free-Space Green’s Function

The normalized free-space Green’s function is given by the integral representation

(5.6.3)

Lm1,m2 = –1

16π2

∫ 2π

0

∫ 2π

01 – cos(m1ω1 + m2ω2)

sin2(1

2ω1

)+ sin2

(1

2ω2

)+ sin2

[1

2(ω1 + ω2)

]+ sin2

[1

2(ω1 – ω2)

] dω1 dω2or

(5.6.4)Lm1,m2 = –

1

8π2

∫ 2π

0

∫ 2π

01 – cos(m1ω1 + m2ω2)

4 – cosω1 – cosω2 – cos(ω1 + ω2) – cos(ω1 – ω2)dω1 dω2.

By symmetry, we have

(5.6.5) L±m,0 = L0,±m, Lm,±m = L±m,m

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Green’s Funct ions / / 199

for any positive or negative integer, m. Using (5.6.4), we find that

L1,0 + L0,1 + L1,1 + L1,–1 = –1

8π2

∫ 2π

0

∫ 2π

0dω1 dω2,

yielding

(5.6.6) L1,0 + L1,1 = –14 .

However, it should be noted that L1,0 = L1,1.

One-Dimensional Integral RepresentationIt is convenient to write

(5.6.7) ω1 = w + v, ω2 = w – v,

where

(5.6.8) w = 12 (ω1 + ω2), v = 1

2 (ω1 – ω2)

are two new variables. Substituting these transformations into (5.6.4), we obtain

(5.6.9) Lm1,m2 = –1

2π2

∫ π0

∫ 2π

0

1 – exp[ i (m1 – m2)v] exp[ i (m1 + m2)w]

4 – 2 cosw cos v – cos 2w – cos 2vdv dw.

Next, we introduce a complex variable z such that

(5.6.10) z ≡ exp( iv), cos v =1

2

(z +

1

z

), cos 2v =

1

2

(z +

1

z

)2

– 1, dv = –idz

z

and find that

(5.6.11) Lm1,m2 = –i

2π2

∫ π0

∮1 – z|m1–m2| exp[ i (m1 + m2)w]

z(z2 + 1) cosw – (5 – cos 2w) z2 + 12 (z

2 + 1)2z dz dw,

where the closed integration path is the unit circle in the z plane.The roots of the denominator provide us with the pole of the integrand with re-

spect to z. Setting z = e–σ , we find that the real number κ ≡ cosh σ = (z2 + 1)/(2z)satisfies the equation

(5.6.12) 2 κ2 + 2 κ cosw – 5 + cos 2w = 0.

Solving this quadratic equation and retaining the positive root, we obtain

(5.6.13) κ ≡ cosh σ = –12 cosw +

(3 – 3

4 cos2 w)1/2

.

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For a pole to reside inside the unit circle, σ must be positive. The residue of theintegrand, R, arises by evaluating the ratio of the numerator and the derivative of thedenominator divided by z at a pole, finding

(5.6.14) R =1 – z|m1–m2| exp[ i (m1 + m2)w]

2 e–σ cosw – 5 + cos 2w + 2κ(κ – 2λ),

where λ ≡ sinh σ . Eliminating the expression 5 – cosw in favor of the rest of theterms in (5.6.12) and simplifying, we obtain

(5.6.15) R = –1

2

1 – z|m1–m2| exp[ i (m1 + m2)w]

cosw sinh σ + sinh 2σ.

It is instructive to compare this residue with that shown in (5.5.19) for the hexagonallattice. Now using the residue theorem, we find that

(5.6.16) Lm1,m2 = –i

2π2

∫ π0

(2π i)R dw

and thus

(5.6.17) Lm1,m2 = –1

∫ π0

1 – e–|m1–m2|σ cos[(m1 + m2)w]

sinh σ cosw + sinh 2σdw,

provided that σ > 0. Numerical computations show that

(5.6.18) L1,0 = –0.12101 . . . , L1,1 = –0.12899 . . . ,

in agreement with (5.6.6). The entire nodal distribution can be computed byconventional numerical integration.

Exercise

5.6.1 Far field

Derive an expression for the far-field behavior of the free-space Green’s function.

5.7 HONEYCOMB LATTICE

Consider a honeycomb lattice supporting a doubly periodic nodal field, as shownin Figure 5.7.1. Each periodic test section contains N1 hexagonal cells in the firstdirection and N2 hexagonal cells in the second direction. The nodes are distributedon two triangular Bravais lattices, denoted as A and B, as discussed in Section 3.5.Nodes on lattice A are shown as hollow circles connected by dashed lines, and nodeson lattice B are shown as filled circles connected by dotted lines in Figure 5.7.1.

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Green’s Funct ions / / 201

211

i

1 21Bi

A

N1

N1

A

1

Bi

N2 + 1

N1 + 1

N1 + 1

N2 + 1

N2

N2

2

i2a2

1

a1

b

x

y

a

FIGURE 5.7.1 Illustration of a periodic patch of a honeycomb lattice consisting of twohexagonal lattices, A and B, containing N1 cells in the first direction and N2 cellsin the second direction. For the configuration shown, N1 = 4 and N2 = 3. Nodes

on lattice A are shown as open circles connected by dashed lines, and nodes on

lattice B are shown as filled circles connected by dotted lines.

Without loss of generality, we assume that the point source associated with theGreen’s function is applied at a node on lattice A. The eigenvalues of the doublyperiodic Laplacian were given in (3.5.63) as

(5.7.1) λ±n1,n2 = 3 ± [ 3 + 2 cosαn1 + 2 cos βn2 + 2 cos(αn1 + βn2)

]1/2,

where

(5.7.2) αn1 =n1 – 1

N12π , βn2 =

n2 – 1

N22π

for n1 = 1, . . . ,N1 and n2 = 1, . . . ,N2. The corresponding eigenvectors were givenin (3.5.38) and (3.5.65).

5.7.1 Periodic Green’s Function

The normalized Moore–Penrose doubly periodic Green’s function on the constituentlattice A is given by

(5.7.3) (Mm1,m2 )A = –

1

2N1N2

N1∑n1 =1

N2∑n2 =1

′( 1

λ+n1,n2+

1

λ–n1,n2

) (1 – exp[–i(m1 αn1 + m2 βn2 ) ]

),

where the prime indicates that the singular term, n1 = 1 and n2 = 1, is excluded fromthe sum. Consolidating the two terms inside the sum, we obtain

(5.7.4) (Mm1,m2)A = –

1

2N1N2

N1∑n1 =1

N2∑n2 =1

′λ+n1,n2 + λ

–n1,n2

λ+n1,n2λ–n1,n2

(1 – exp[–i(m1 αn1 + m2 βn2) ]

).

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202 / / AN INTRODUCTION TO GRIDS, GRAPHS, AND NETWORKS

Substituting expression (5.7.1) for the eigenvalues and simplifying, we obtain

(5.7.5) (Mm1,m2)A = –

3

2N1N2

N1∑n1 =1

N2∑n2 =1

′ 1 – exp[–i(m1αn1 + m2βn2

)]

3 – cos αn1 – cosβn2 – cos(αn1 + βn2)

or

(5.7.6)

(Mm1,m2)A = –

3

4N1N2

N1∑n1 =1

N2∑n2 =1

1 – exp[–i(m1 αn1 + m2 βn2) ]

sin2(1

2αn1

)+ sin2

(1

2βn2

)+ sin2

[1

2(αn1 + βn2)

] .Physically, this expression provides us with the nodal field generated at the point( j1 + m1, i2 + m2) of the constituent Bravais lattice A when a source is applied at thepoint ( j1, j2) of the same lattice. By construction, we obtain (M0,0)A = 0. Expression(5.7.6) shows that the nodal values of the Green’s function on lattice A are three timesthose on a hexagonal lattice.

Lattice BWorking in a similar fashion for the nodes of the second constituent lattice B, weobtain the periodic lattice Green’s function

(5.7.7)(Mm1,m2)

B =1

2N1N2

N1∑n1 =1

N2∑n2 =1

′(

1

λ+n1,n2

1

3 – λ+n1,n2+

1

λ–n1,n2

1

3 – λ–n1,n2

)× exp[–i (m1 αn1 + m2 βn2) ] ( 1 + e

iαn1 + e–iβn2 ).

Physically, this expression provides us with the nodal field generated at the point( j1 + m1, i2 + m2) of the constituent Bravais lattice B when a source is applied at thepoint ( j1, j2) of lattice A. The second and fourth fractions inside the tall parenthesesoriginate from the eigenvectors of the doubly periodic Laplacian on lattice B.

We find that

(5.7.8)

1

λ+n1,n2

1

3 – λ+n1,n2

1

λ–n1,n2

1

3 – λ–n1,n2=

1√D

(–

1

3 +√

D+

1

3 –√

D

)=

2

9 – D =1

3 – cos αn1 – cosβn2 – cos(αn1 + βn2),

where

(5.7.9) D = 3 + 2 cosαn1 + 2 cosβn2 + 2 cos(αn1 + βn2 ).

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Green’s Funct ions / / 203

Substituting this expression into (5.7.7), we obtain

(5.7.10)

(Mm1,m2)B =

1

2N1N2

N1∑n1 =1

N2∑n2 =1

exp[–i (m1 αn1 + m2 βn2) ]

3 – cosαn1 – cosβn2 – cos(αn1 + βn2)

(1 + eiαn1 + e–iβn2

).

The normalized Green function, indicated by a tilde, is given by

(5.7.11) (Mm1,m2)B = –

1

2N1N2

N1∑n1=1

N2∑n2=1

′ Am1,m2

3 – cos αn1 – cosβn2 – cos(αn1 + βn2),

where

(5.7.12)Am1,m2 = 3 – cos(m1 αn1 + m2 βn2) – cos[(m1 – 1)αn1 + m2 βn2)]

– cos[m1 αn1 + (m2 + 1)βn2 )].

We may confirm that

(5.7.13) (Mm1,m2)B =

1

3

[(Mm1,m2)

A + (Mm1–1,m2)A + (Mm1,m2+1)

A],

which is consistent with the linear equation defining the Green’s function at the(m1,m2) node of lattice B.

5.7.2 Free-Space Green’s Function

The normalized free-space Green’s function on lattice A is given by the integralrepresentation

(5.7.14) (Lm1,m2)A = –

3

16π2

∫ 2π

0

∫ 2π

0

1 – cos(m1ω1 + m2ω2)

sin2(12ω1

)+ sin2

(12ω2

)+ sin2

[12 (ω1 + ω2)

] dω1dω2or

(5.7.15) (Lm1,m2)A = –

3

8π2

∫ 2π

0

∫ 2π

0

1 – cos(m1ω1 + m2ω2)

3 – cosω1 – cosω2 – cos(ω1 + ω2)dω1 dω2.

By definition, we have (L0,0)A = 0. Comparing the representations (5.7.14) and(5.7.15) with those derived in Section 5.5, we find that the nodal values of the Green’sfunction on lattice A are three times those on a triangular lattice.

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204 / / AN INTRODUCTION TO GRIDS, GRAPHS, AND NETWORKS

By symmetry, we have

(5.7.16) (L±m,0)A = (L0,±m)A = (Lm,–m)A = (L–m,m)

A

for any positive or negative integer, m. Using (5.7.15), we obtain

(5.7.17) (L1,0 + L0,1 + L1,1)A = –

3

8π2

∫ 2π

0

∫ 2π

0dω1 dω2,

yielding

(5.7.18) (L±1,0)A = (L0,±1)

A = (L1,–1)A = (L–1,1)

A = –12 ,

which is the value at the second nearest neighbor derived earlier in (5.3.18).

Lattice BWriting a balance equation at an arbitrary node of the constituent lattice B, we obtain

(5.7.19) (Lm1,m2)B = 1

3

[(Lm1,m2)

A + (Lm1–1,m2)A + (Lm1,m2+1)

A].

This equation allows us to generate the nodal field on lattice B in terms of the nodalfield on lattice A. For m1 = 0 and m2 = 0, we obtain

(5.7.20) (L0,0)B = 1

3

[(L–1,0)

A + (L0,+1)A]= –1

3 .

By symmetry, we have

(5.7.21) (L0,0)B = (L1,0)

B = (L0,–1)B = –

1

3,

which is the value at the nearest neighbor.The Green’s function on lattice B is given by the integral representation

(5.7.22)

(Lm1,m2

)B= –

1

16π2

∫ 2π

0

∫ 2π

0 Am1,m2(ω1,ω2)

sin2(1

2ω1

)+ sin2

(1

2ω2

)+ sin2

[1

2(ω1 + ω2)

] dω1 dω2or

(5.7.23)(Lm1,m2

)B= –

1

8π2

∫ 2π

0

∫ 2π

0

Am1,m2(ω1,ω2)

3 – cosω1 – cosω2 – cos(ω1 + ω2)dω1 dω2,

where

(5.7.24)Am1,m2(ω1,ω2) = 3 – cos(m1ω1 + m2ω2)

– cos[(m1 – 1)ω1 + m2 ω2)] – cos[m1 ω1 + (m2 + 1)ω2)].

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Green’s Funct ions / / 205

SummaryThe first several nearest neighbors of a node on a honeycomb lattice are identifiedin Figure 5.2.2(a). Nodes A, B, and C fall on lattice A involving the source point,and nodes A′, B′, and C′ fall on lattice B. The corresponding values of the Green’sfunction are given in a table under the illustration in Figure 5.7.2(a). The entiredistribution of the Green’s function is plotted in Figure 5.7.2(b).

(a)

(b)

B

B

A

A’

A’

A

A

A

A

A

C

CC

C

C C

B’

B’ B’

A’

B

B

B

B

C’

C’C’

C’

C’ C’

eulaVedoN

A’ (L0,0)B − 13 = −0.3333

A (L1,0)A − 12 = −0.5

B’ (L1,1)B = 13 2 LA + LB − 1

π

√3 = −0.5513

C’ (L2,0)B = 13 LA + LB + LC − 7

6 + 1π

√3 = −0.6153

B (L2,1)A 1 − 3π

√3 = −0.6540

C (L2,0)A −4 + 6π

√3 = −0.6920

− 3 = −0.3− 1

2 = −0.513 2 LA + LB − 1

π

√3 = −

13 LA + LB + LC − 7

6 + 1π

√3

1 − 3π

√3 =

−4 + 6√

3

FIGURE 5.7.2 (a) Normalized free-space Green’s function ofthe honeycomb lattice at the six nearest neighbors of a pointsource applied at the central node. (b) Nodal distribution ofthe normalized Green’s function over the hexagonal lattice.

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206 / / AN INTRODUCTION TO GRIDS, GRAPHS, AND NETWORKS

Exercises

5.7.1 Lattice B

Confirm by numerical integration that the integral representation (5.7.22) reproduces(5.7.21).

5.7.2 Alternative node indexing

Derive expressions for the free-space Green’s function for the point indexing schemeshown in Figure 3.5.1.

5.8 SIMPLE CUBIC LATTICE

Consider a slab of a simple cubic network containing N1 links in the first direction,N2 links in the second direction, and N3 links in the third direction, as shown inFigure 5.8.1. The lattice is parametrized by three indices, i1, i2, and i3 running inthree perpendicular directions.

5.8.1 Periodic Green’s Function

The triply periodic normalized Moore–Penrose periodic Green’s is given by

(5.8.1)

Mm1,m2,m3 = –k1k2k332π3

N1– 1∑p1 = 0

N2– 1∑p2 = 0

N3– 1∑p3 = 0

1 – cos(m1p1k1 + m2p2k2 + m3p3k3)

sin2(1

2p1k1

)+ sin2

(1

2p2k2

)+ sin2

(1

2p3k3

)

i1

i2

i3

FIGURE 5.8.1 Illustration of a rectangularslab of a simple cubic network containingN1 links in the first direction, N2 links in thesecond direction, andN3 links in the third di-rection. The conductances of all links are

assumed to be the same. In the configura-

tion shown, N1 = 2, N2 = 2, and N3 = 1.

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Green’s Funct ions / / 207

or

(5.8.2) Mm1,m2,m3 = –k1k2k316π3

N1–1∑p1= 0

N2–1∑p2= 0

N3–1∑p3= 0

′ 1 – cos(m1p1k1 + m2p2k2 + m3p3k3)

3 – cos(p1k1) – cos(p2k2) – cos(p3k3),

where

(5.8.3) k1 =2π

N1, k2 =

N2, k3 =

N3

are directional wave numbers, and the prime after the summation symbol indicatesthat the troublesome term ( p1 = 0, p2 = 0, and p3 = 0) is excluded from the sum.

5.8.2 Free-Space Green’s Function

The free-space Green’s function is given by the triple integral representation

(5.8.4)

Lm1,m2,m3 = –1

32π3

∫ 2π

0

∫ 2π

0

∫ 2π

0

1 – cos(m1 ω1 + m2ω2 + m3ω3)

sin2(1

2ω1

)+ sin2

(1

2ω2

)+ sin2

(1

2ω3

) dω1 dω2 dω3

or

(5.8.5) Lm1,m2,m3 = –1

16π3

∫ 2π

0

∫ 2π

0

∫ 2π

0

1 – cos(m1 ω1 + m2ω2 + m3ω3)

3 – cosω1 – cosω2 – cosω3dω1 dω2 dω3.

Alternative integral representations where the three terms in the argument of thecosine in the numerator of the integrand are separated are

(5.8.6)

Lm1,m2,m3 = –1

4π3

∫ π0

∫ π0

∫ π0

1 – cos(m1ω1) cos(m2ω2) cos(m3ω3)

sin2(12 ω1

)+ sin2

(12 ω2

)+ sin2

(12 ω3

) dω1 dω2 dω3and

(5.8.7)

Lm1,m2,m3 = –1

2π3

∫ π0

∫ π0

∫ π0

1 – cos(m1ω1) cos(m2ω2) cos(m2ω3)

3 – cosω1 – cosω2 – cosω3dω1 dω2 dω3.

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208 / / AN INTRODUCTION TO GRIDS, GRAPHS, AND NETWORKS

Note the new limits of integration.By symmetry, we have

(5.8.8) L±m,0,0 = L0,±m,0 = L0,0,±m

for any positive or negative integer, m.Using the integral representation (5.8.5), we find that the sum of the three nearest-

neighbor values is

(5.8.9) L1,0,0 + L0,1,0 + L0,0,1 = –1

16π3

∫ 2π

0

∫ 2π

0

∫ 2π

0dω1 dω2 dω3 = –

1

2,

yielding

(5.8.10) L±1,0,0 = L0,±1,0 = L0,0,±1 = –1

6,

in agreement with the general expression (5.3.16).It is interesting to consider the nodal distribution in the (i1, i2) plane correspond-

ing to a fixed value of i3. Setting in (5.8.5) m3 = 0, we obtain

(5.8.11) Lm1,m2,0 =1

∫ 2π

0Fm1,m2(δ) dω3,

where

(5.8.12) Fm1,m2(δ) ≡ –1

8π2

∫ 2π

0

∫ 2π

0

1 – cos(m1 ω1 + m2ω2)

δ – cosω1 – cosω2dω1 dω2

and δ ≡ 3 – cosω3. The integral representation (5.8.12) arises from the Green’sfunction for the square lattice given in (5.4.10) by replacing the 2 in the denominatorof the fraction of the integrand with δ.

Alternative representations of the Green’s function have been developed in termsof complete elliptic integrals and products [21–23]. Unfortunately, an efficientmethod for computing the Green’s function is not available.

Far-Field AsymptoticsFar from the point source, in the limit as m1 or m2 or m3 tends to infinity, thenormalized Green’s function tends to the asymptotic value

(5.8.13) L∞ = –1

16π3

∫ π0

∫ π0

∫ π0

dω1 dω2 dω33 – cosω1 – cosω2 – cosω3

.

A detailed analysis shows that

(5.8.14) L∞ = –2

π2

(18 + 12

√2 – 10

√3 – 7

√6)K2(α)

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Green’s Funct ions / / 209

or

(5.8.15) L∞ = –1

π3

√3 – 1

192�2(

1

24

)�2(11

24

)= –0.25273 . . . ,

where α = (2 –√3)(

√3 –

√2), K is the complete elliptic integral of the first kind,

(5.8.16) K(α) ≡∫ π /20

dw√1 – α2 sin2 w

,

and � is the Gamma function [14, 20, 21, 50]. In contrast with the logarithmic growthof the Green’s function of the square lattice, the normalized Green’s function of thecubic lattice tends to a constant value far from the point source.

Exercise

5.8.1 Numerical evaluation of a triple integral

Write a code that computes the triple integral in (5.8.6) using the trapezoidal rule andconfirm the value given in (5.8.10) (e.g., [35]).

5.9 BODY-CENTERED CUBIC (BCC) LATTICE

Using the results of Section 3.8, we find that the normalized free-space Green’s func-tion of the body-centered cubic network shown in Figure 5.9.1 is given by the integralrepresentation

(5.9.1)

Lm1,m2,m3 = –1

32π3×∫ 2π

0

∫ 2π

0

∫ 2π

0

1 – cos(m1 ω1 + m2ω2 + m3ω3)

sin2(1

2ω1

)+ sin2

(1

2ω2

)+ sin2

(1

2ω3

)+ sin2

[1

2(ω1 + ω2 + ω3)

] dω1 dω2 dω3or

(5.9.2)Lm1,m2,m3 = –

1

16π3×∫ 2π

0

∫ 2π

0

∫ 2π

0

1 – cos(m1 ω1 + m2ω2 + m3ω3)

4 – cosω1 – cosω2 – cosω3 – cos(ω1 + ω2 + ω3)dω1 dω2 dω3,

where the relative node indices, m1, m2, and m3, correspond to the base vectors, a1,a2, and a3, as shown in (3.8.1).

By symmetry,

(5.9.3) L±m,0,0 = L0,±m,0 = L0,0,±m

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210 / / AN INTRODUCTION TO GRIDS, GRAPHS, AND NETWORKS

i1

i2

i3

x

z

y

a

FIGURE 5.9.1 Illustration of the body-centered cubic(bcc) lattice. The nodes are parametrized by three in-

dices, i1,i2, and i3, corresponding to the base vectors,

a1, a2, and a3, shown in (3.8.1).

for any positive or negative integer, m. Following a procedure similar to thatdescribed in previous sections, we find that we have

(5.9.4) L±1,0,0 = L0,±1,0 = L0,0,±1 = –1

8,

in agreement with the general expression (5.3.16).The nodes can be identified by an alternative trio of primed indices,

(5.9.5) i′1 = –i1 + i2 + i3, i′2 = i1 – i2 + i3, i′3 = i1 + i2 – i3,

corresponding to a Cartesian base, as discussed in Section 3.8, The correspondinglattice Green’s function is

(5.9.6) Lm′1,m

′2,m

′3= –

1

64π3

∫ 2π

0

∫ 2π

0

∫ 2π

0

1 – cos(m′1 ω1 + m

′2ω2 + m

′3ω3)

1 – cosω1 cosω2 cosω3dω1 dω2 dω3,

where

m′1 = –m1 + m2 + m3, m′

2 = m1 – m2 + m3,

m′3 = m1 + m2 – m3.(5.9.7)

An alternative representation is

(5.9.8) Lm′1,m

′2,m

′3=–

1

8π3

∫ π0

∫ π0

∫ π0

1 – cos(m′1 ω1) cos(m

′2 ω2) cos(m

′3 ω3)

1 – cosω1 cosω2 cosω3dω1dω2dω3.

Exercise

5.9.1 Numerical evaluation of a triple integral

Write a code that computes the triple integral in (5.9.1) using the trapezoidal rule andconfirm the value given in (5.9.4) (e.g., [35]).

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Green’s Funct ions / / 211

5.10 FACE-CENTERED CUBIC (FCC) LATTICE

Using the results of Section 3.9, we find that the normalized free-space Green’sfunction of the body-centered cubic network shown in Figure 5.10.1 is given by theintegral representation

(5.10.1) Lm1,m2,m3 = –1

32π3

∫ 2π

0

∫ 2π

0

∫ 2π

0

1 – cos(m1 ω1 + m2ω2 + m3ω3)

D dω1dω2dω3,

where

(5.10.2)D = sin2

(12 ω1

)+ sin2

(12 ω2

)+ sin2

(12 ω3

)+ sin2

[12 (ω1 – ω2)

]+ sin2

[12 (ω2 – ω3)

]+ sin2

[12 (ω3 – ω1)

],

or

(5.10.3) Lm1,m2,m3 = –1

16π3

∫ 2π

0

∫ 2π

0

∫ 2π

0

1 – cos(m1 ω1 + m2ω2 + m3ω3)

E dω1dω2dω3,

where

(5.10.4)E = 6 – cosω1 – cosω2 – cosω3

– cos(ω1 – ω2) – cos(ω2 – ω3) – cos(ω3 – ω1).

The node indices, m1, m2, and m3 (corresponding to i1, i2, and i3), are associatedwith the base vectors a1, a2, and a3, as shown in (3.9.1).

By symmetry, we have

(5.10.5) L±m,0,0 = L0,±m,0 = L0,0,±m

i3

i2

i1

a

x

y

z

FIGURE 5.10.1 Illustration of the face-centered cubic (fcc) lattice.The nodes are parametrized by three indices, i1, i2, and i3,corresponding to the base vectors, a1, a2, and a3 shown in (3.9.1).

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212 / / AN INTRODUCTION TO GRIDS, GRAPHS, AND NETWORKS

for any positive or negative integer, m. Following a procedure similar to thatdescribed in previous sections, we find that

(5.10.6) L±1,0,0 = L0,±1,0 = L0,0,±1 = –1

12,

in agreement with the general expression (5.3.16).The nodes can be identified by an alternative trio of indices, i′1, i

′2, and i′3

(corresponding to a different base), defined as

(5.10.7) i′1 = i2 + i3, i′2 = i3 + i1, i′3 = i1 + i2,

as discussed in Section 3.9. The corresponding normalized lattice Green’s functionis

(5.10.8)

Lm′1,m

′2,m

′3= –

3

128π3

∫ 2π

0

∫ 2π

0

∫ 2π

0

1 – cos(m′1 ω1 + m

′2ω2 + m

′3ω3)

3 – cosω1 cosω2 – cosω2 cosω3 – cosω3 cosω1dω1 dω2 dω3,

where

(5.10.9) m′1 = m2 + m3, m′

2 = m3 + m1, m′3 = m1 + m2.

An alternative representation is

(5.10.10)

Lm1,m2,m3 = –3

16π3

∫ π0

∫ π0

∫ π0

1 – cos(m1 ω1) cos(m2 ω2) cos(m3 ω3)

3 – cosω1 cosω2 – cosω2 cosω3 – cosω3 cosω1dω1 dω2 dω3.

Exercise

5.10.1 Numerical evaluation of a triple integral

Write a code that computes the triple integral in (5.10.2) using the trapezoidal ruleand confirm the value given in (5.10.6) (e.g., [35]).

5.11 FREE-SPACE LATTICE GREEN’S FUNCTIONS

In Section 5.2.6, we found that the normalized free-space Green’s function associatedwith the Laplacian matrix of an infinite lattice in one dimension is given by theintegral representation

(5.11.1) Lm = –1

d

1

∫ 2π

0

1 – cos(mω)

1 –�(ω)dω,

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Green’s Funct ions / / 213

where d = 2 is the lattice coordination number and �(ω) = cosω is a structurefunction arising from the eigenvalues of the periodic Laplacian.

The normalized free-space Green’s function of a Bravais lattice in two dimen-sions admits the unified integral representation

(5.11.2) Lm1,m2 = –1

d

1

4π2

∫ 2π

0

∫ 2π

0

1 – cos(m1ω1 + m2ω2)

1 –�(ω1,ω2)dω1 dω2,

where d is the lattice coordination number and �(ω1,ω2) is a structure functionarising from the eigenvalues of the doubly periodic Laplacian.

The normalized free-space Green’s function of a Bravais lattice in three dimen-sions admits the unified integral representation

(5.11.3) Lm1,m2,m3 = –1

d

1

8π3

∫ 2π

0

∫ 2π

0

∫ 2π

0

1 – cos(m1ω1 + m2ω2 + m3ω3)

1 –�(ω1,ω2,ω3)dω1dω2dω3,

where d is the lattice coordination number and �(ω1,ω2,ω3) is a structure functionarising from the eigenvalues of the triply periodic Laplacian.

The structure function, �, is tabulated in Table 5.11.1 for several Bravais latticesalong with the lattice coordination number, d. It should be noted that the form ofthe structure function is not unique for each lattice type, but depends on the nodeindexing scheme associated with the choice of base vectors.

The normalized free-space Green’s function of a composite lattice consisting oftwo or a higher number of Bravais lattices admits a more involved representation. Forthe honeycomb lattice consisting of two interwoven hexagonal lattices, we found that

(5.11.4) Lm1,m2 = –1

8π2

∫ 2π

0

∫ 2π

0

1 – cos(m1ω1 + m2ω2)

1 –�(ω1,ω2)dω1 dω2

on the lattice hosting the nodal source, where

(5.11.5) �(ω1,ω2) = 13 [cosω1 + cosω2 + cos(ω1 ± ω2) ] .

5.11.1 Probability Lattice Green’s Function

Suppose that a random walker wanders over the nodes of a Bravais lattice, startingat a node labeled m = 0 in one dimension, m1 = 0 and m2 = 0 in two dimensions, orm1 = 0, m2 = 0, and m3 = 0 in three dimensions. The walker stays in its current posi-tion with probability 1 – z, and jumps indiscriminantly to one of its nearest neighborswith probability z.

In one dimension, we introduce the the probability lattice Green’s function

(5.11.6) Pm(z) =1

∫ 2π

0

cos(mω)

1 – z�(ω)dω =

1√1 – z2

.

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214 / / AN INTRODUCTION TO GRIDS, GRAPHS, AND NETWORKS

In two dimensions, we introduce the probability lattice Green’s function

(5.11.7) Pm1,m2(z) =1

4π2

∫ 2π

0

∫ 2π

0

cos(m1ω1 + m2ω2)

1 – z�(ω1,ω2)dω1 dω2.

In three dimensions, we introduce the probability lattice Green’s function

(5.11.8) Pm1,m2,m3(z) =1

8π3

∫ 2π

0

∫ 2π

0

∫ 2π

0

cos(m1ω1 + m2ω2 + m3ω3)

1 – z�(ω1,ω2,ω3)dω1 dω2 dω3

(e.g., [16]). These representations are strikingly similar to those of the free-spaceGreen’s function discussed earlier in this section.

The probability that the walker returns to the origin after any number of steps is

(5.11.9) �0(z) = 1 –1

P0(z),

where

(5.11.10) P0(z) =1

4π2

∫ 2π

0

∫ 2π

0

dω1 dω21 – z�(ω1,ω2)

TABLE 5.11.1 Tabulation of the Structure Function, �, of Several Lattices with Coordination Num-ber d .

Lattice d �

One-dimensional 2 cosω

Square 4 12(cosω1 + cosω2)

Hexagonal 6 13

[cosω1 + cosω2 + cos(ω1 ± ω2)

]ModiÞed Union Jack 8 1

4

[cosω1 + cosω2 + cos(ω1 + ω2)

+ cos(ω1 – ω2)]

Simple cubic 6 13

(cosω1 + cosω2 + cosω3

)Body-centered cubic (bcc) 8 1

4

[cosω1 + cosω2 + cosω3

+ cos(ω1 + ω2 + ω3)]

Body-centered cubic (bcc) 8 cosω1 cosω2 cosω3

Face-centered cubic (fcc) 12 16

[cosω1 + cosω2 + cosω3

+ cos(ω1 – ω2) + cos(ω2 – ω3)

+ cos(ω3 – ω1)]

Face-centered cubic 12 13

[cosω1 cosω2 + cosω2 cosω3

+ cosω3 cosω1]

Note: The minus or plus sign in the argument of the cosine for the hexagonal and honeycomb

lattices apply when the base vectors in the natural state form a 60◦ or 120◦ angle.

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Green’s Funct ions / / 215

in two dimensions. Similar equations can be written in two, three, and higherdimensions. For example, in the case of the square lattice, we have

(5.11.11) P0(z) =2

πK(z),

where K is the complete elliptic integral of the first kind. Since P0(1) is infinite intwo dimensions, �0(1) = 1, which shows that the walker is certain to return to theorigin after an unspecified number of steps.

Of particular interest is the Taylor series expansion of the lattice Green’s functionwith respect to z. In two dimensions, we have

(5.11.12) Pm1,m2(z) = a(0)m1,m2

+ a(1)m1,m2z + a(2)m1,m2

z2 + · · ·.

The coefficients, a(n)m1,m2 , are the probabilities that a walker starting at (0, 0) is locatedat (m1,m2) after n steps. The structure function is given in terms of the single-stepprobabilities by the Fourier expansion

(5.11.13) �(ω1,ω2) =∑m1

∑m2

a(1)m1,m2exp [ i(m1 ω1 + m2 ω2) ],

where i is the imaginary unit. Similar interpretations apply in one and threedimensions.

For example, in the case of the square lattice, all a(1)m1,m2 are zero, except that

(5.11.14) a(1)±1,0 =14 , a(1)0,±1 =

14 ,

expressing equal probabilities in four directions. Applying (5.11.13) reproduces thestructure function shown in the second entry of Table 5.11.1.

In one dimension, we find that

(5.11.15) a(n)0 =1

2n

(nn/2

)=

1

2nn!

[(n/2)!]2

if n is even, and an = 0 if n is odd, where the large parentheses denote thecombinatorial and the exclamation mark denotes the factorial.

Exercise

5.11.1 Eigenvalues of the Laplacian

Confirm by numerical computation the coefficients given in (5.11.15).

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216 / / AN INTRODUCTION TO GRIDS, GRAPHS, AND NETWORKS

5.12 FINITE DIFFERENCE SOLUTION IN TERMS OF GREEN’S FUNCTIONS

In Section 4.5, we saw that the Laplacian matrix arises from the finite differencediscretization of the Laplacian of an unknown function of two variables in twodimensions, f (x, y), or three variables in three dimensions, f (x, y, z), on a uniformCartesian finite difference grid. The Kirchhoff matrix arises from correspondingdiscretizations on a nonuniform or non-Cartesian grid.

As an example, we consider a uniform Nx × Ny Cartesian grid described by twoindices, i and j, covering a rectangular area in the xy plane, as shown in Figure 5.12.1.Our objective is to compute a numerical solution of the Poisson equation,

(5.12.1) ∇2f + g(x, y) = 0,

where g(x, y) is a given distributed source. Using the five-point formula to approxi-mate the Laplacian at the (i, j) node, we obtain the finite difference equation

(5.12.2) 2 (1 + β)fi, j – fi+1, j – fi–1, j – β (fi, j–1 + fi, j+1) = �x2gi, j,

where β = (�y/�x)2 (e.g., [35]). In the case of a square grid,�x = �y, we set β = 1.The finite difference solution at the (i, j) node can be expressed as a linear su-

perposition of the fields due to (a) boundary nodal sources with a priori unknownstrength sij and (b) interior nodal sources with strength �x2gi,j. In the case of La-place’s equation, g = 0, only boundary nodal sources are employed. Introducing thenormalized free-space Green’s function of the infinite square lattice correspondingto the prevailing value of β, Lm1,m2 , we write

(5.12.3) fij = f(1)ij + f (2)ij + f (3)ij + f (4)ij + f (5)ij ,

where

(5.12.4) f (1)ij =Nx∑p=1

sp,1 Li – p,j – 1

11

Nx

Ny

ix

j

Δx

Δy

y

FIGURE 5.12.1 A Cartesian finite difference grid used tosolve the Poisson equation in two dimensions.

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Green’s Funct ions / / 217

is a nodal distribution at the bottom,

(5.12.5) f (2)ij =

Ny∑q =1

sNx+1,q Li –Nx – 1,j – q

is a nodal distribution at the right,

(5.12.6) f (3)ij =Nx∑p=1

sp,Ny + 1 Li – p,j –Ny – 1

is a nodal distribution at the top,

(5.12.7) f (4)ij =

Ny∑q=1

s1,q Li – 1,j – q

is a nodal distribution at the left, and

(5.12.8) f (5)ij = �x2Nx∑p =2

Ny∑q=2

gp,q Li – p,j – q

is a nodal distribution in the interior of the solution domain. Thanks to the linearity ofthe governing equations, expression (5.12.3) satisfies the difference equation (5.12.3)for any boundary and interior source terms.

The representation in terms of the lattice Green’s function is inspired by theintegral representation of the solution of Laplace’s equation in terms of the corre-sponding Green’s function. The boundary source terms, sij, must be computed tosatisfy the boundary conditions around the four edges of the rectangle.

In most applications, we impose the Dirichlet boundary condition, specifying theboundary values of f , or the Neumann boundary condition, specifying the normalderivative. After the solution has been found, the nodal field can be reconstructedfrom the point source distribution. The efficiency of this approach hinges on theavailability of the lattice Green’s function.

In the presence of a source term, the computational cost for assembling the linearsystem is proportional to the product NxNy, and the computational cost for solvingthe linear system is proportional to the sum Nx +Ny. In the absence of a source term,both costs are proportional to Nx + Ny.

Page 233: An Introduction to Grids Graphs and Networks

(a)

20 40 60 80 100 120−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

n

20 40 60 80 100 120

n

20 40 60 80 100 120

n

s ns n

s n

5 10 15 20 25 3010

2030

−1

−0.5

0

0.5

1

i2 i1

ψ

i2 i1

ψ

5 10 15 20 25 3010

2030

−1

−0.5

0

0.5

1

i2i1

ψ

−12

−10

−8

−6

−4

−2

0

2x 10−3

5 10 15 20 25 3010

2030

0

0.02

0.04

0.06

0.08

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

(b)

(c)

FIGURE 5.12.2 Finite-difference solution of the Laplace or Poisson equation computed in termsof the square lattice Green’s function. The strength of nodal sources around the four edges

of a square domain starting from the southwestern point and moving along the bottom,

right, top, and left is shown in the left column, where n is a node count. The finite difference

solution of (a) Laplace’s or (b) Poisson’s equation is shown in the right column. (c) Same as

(a) but with the zero-flux condition along the bottom.

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Green’s Funct ions / / 219

Dirichlet Boundary ConditionAssume the the boundary values of f are prescribed as a Dirichlet boundary con-dition. Applying (5.12.3) at the boundary nodes and rearranging the differenceequations, we formulated a system of linear equations with a unique solution for the2 (Nx + Ny) unknown strengths of the boundary point sources sij. The linear systemcan be solved by a direct or iterative method.

A solution of Laplace’s equation, gp,q = 0, on a 32 × 32 grid is shown inFigure 5.12.2(a). In this case, the boundary conditions specify half a sinusoidal wavealong the bottom and top sides and a full sinusoidal wave along the left and rightsides with equal amplitude.

A solution of Poisson’s equation with uniform source term gp,q = 1/N2x and

the homogeneous Dirichlet boundary condition on a 32 × 32 grid is shown inFigure 5.12.2(b). Physically, the nodal distribution describes the velocity profile ofPoiseuille flow inside a square duct or the deformed shape of an elastic membraneattached to a square frame.

Neumann Boundary ConditionTo implement the Neumann boundary condition specifying the normal derivative,we approximate the normal derivative with a one-side finite difference with a desireddegree of accuracy (e.g., [35]). The emerging algebraic equation is then used to com-pute the boundary nodal sources, as in the case of the Dirichlet boundary conditiondiscussed in the previous section. A pertinent finite difference solution is shown inFigure 5.2.2(c). Other types of boundary conditions can be handled in similar ways.

Exercise

5.12.1 Constant field

Compute a finite difference solution with the Dirichlet boundary condition spec-ifying the same boundary values around the four edges of a square. Discuss thecomputed boundary source distribution.

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/ / / 6 / / / NETWORK PERFORMANCE

The efficiency and performance of a conductive or convectivenetwork depends on the node connectivity and conductance of the individual links.Of particular interest is the pairwise resistance determining the rate of transport ofa suitable entity associated with a nodal potential across an arbitrary pair of nodes.The pairwise resistance also admits a probabilistic interpretation in the context ofrandom walks. The sum of all pairwise resistances over all possible sets of nodesprovides us with an overall measure of the network performance. Link disruption orclipping weakens a network, whereas link addition improves the performance of anetwork. Pertinent concepts and quantitative measures are discussed in this chapterfor isolated and embedded networks.

6.1 PAIRWISE RESISTANCE

Suppose that a transported entity associated with a scalar nodal potential, ψ , is sup-plied at a rate s at the ith node and withdrawn at the same rate from the jth node of anetwork, as illustrated in Figure 6.1.1. The induced difference in the potential across

+s

−s

i

j

FIGURE 6.1.1 A transported entity, such asheat, is supplied at a rate s at the ith nodeand withdrawn at the same rate from the jthnode of an embedded or isolated network.The dashed lines connect selected nodes to

external Dirichlet nodes.

220

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Network Performance / / 221

this pair of nodes can be used to define a corresponding pairwise resistance, in that,the more pathways connecting the two nodes, the lower the associated pairwise re-sistance. Nodes belonging to disconnected parts of an unconnected network registeran infinite pairwise resistance.

In the case of heat or mass transfer through a conductive network of rods orconduits, ψ is the temperature or species concentration and s is the rate of heat ormass transport. In the case of fluid flow through a capillary tube network, ψ is thepressure and s is the volumetric or mass fluid rate. In the case of electricity transport,ψ is the electrical voltage and s is an electrical current. Physically, the pairwiseresistance arises when the ith node is connected to a positive battery pole or Ohmmeter, while the jth node is connected to the negative battery pole or Ohm meter,or vice versa. The Ohm meter will register a resistance that depends on the overallstructure of the network.

6.1.1 Embedded Networks

Consider an embedded network where the Dirichlet nodes are grounded to zeroslpotential. Using (5.1.8), we express the nodal field induced by a source appliedthe ith node and a sink applied at the j node in terms of the Green’s function matrix,G, as

(6.1.1) ψ = rsG ·(e(i) – e(j)

),

where e(i) and e(j) are unit vectors, and

(6.1.2) r ≡ 1

c

is a reference resistance associated with a reference conductance, c. The differencein the induced nodal field at the source and sink is

(6.1.3) ψi – ψj = ψ ·(e(i) – e(j)

),

yielding

(6.1.4) ψi – ψj = rs(e(i) – e(j)

)· G ·

(e(i) – e(j)

).

This equation motivates defining the dimensionless pairwise resistance

(6.1.5) Rij ≡ ψi – ψjrs

=(e(i) – e(j)

)· G ·

(e(i) – e(j)

).

Carrying out the multiplications, we obtain

(6.1.6) Rij = e(i) · G · e(i) + e(j) · G · e(j) – e(i) · G · e(j) – e(j) · G · e(i),

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yielding

(6.1.7) Rij = Gii + Gjj – Gij – Gji.

Taking into consideration the symmetry of the Green’s function, Gij = Gji, we obtain

(6.1.8) Rij = Gii + Gjj – 2Gij

for i = j. In vector notation, the N × N pairwise resistance matrix is given by

(6.1.9) R = (ε ⊗ ε) · G′ + G ′ · (ε ⊗ ε) – 2G,

where the N-dimensional vector ε and N × N matrix ε × ε are filled with ones, ⊗denotes the tensor product, and G′ is the diagonal part of G.

IdentitiesPremultiplying (6.1.9) by the augmented Kirchhoff matrix, K, and recalling that, bydefinition, K · G = I and also K · ε = τ , as discussed in Section 4.4, we obtain

(6.1.10) K · R = (τ ⊗ ε) · G ′ + K · G′ · (ε ⊗ ε) – 2 I,

where I is the identity matrix. Postmultiplying this equation by K, we obtain

(6.1.11) K · R · K = (τ ⊗ ε) · G′ · K + K · G ′ · (ε ⊗ τ ) – 2K.

We recall that the vector τ contains the scaled conductances of links connectingnetwork nodes to external Dirichlet nodes.

Spectral ExpansionSubstituting into (6.1.8) the spectral expansion of the Green’s function given in(5.1.13), we obtain

(6.1.12) Rij =N∑n=1

1

λn

(u(n)i – u(n)j

) (u(n)i – u(n)j

)∗,

where an asterisk denotes the complex conjugate [56].

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Representation in Terms of the Normalized Green’s FunctionExpression (6.1.8) can be rearranged as

(6.1.13) Rij = –(Gij – Gii) – (Gij – Gjj),

yielding

(6.1.14) Rij = – Gij – Gji,

where Gij is the normalized Green’s function, defined such that Gii = 0, wheresummation is not implied over the repeated index, i. We recall that the normalizedGreen’s function is not necessarily symmetric, that is, Gij is not necessarily equal toGji. In contrast, Gij is always equal to Gji.

6.1.2 Isolated Networks

A network is isolated in the absence of Dirichlet nodes, T = 0 andK = K, whereK isthe Kirchhoff matrix. Introducing the Moore–Penrose Green’s function, H, workingas in Section 6.1.1 for an embedded network, and using (5.2.16), we obtain

(6.1.15) Rij = (e(i) – e(j)) · H · (e(i) – e(j)),

yielding

(6.1.16) Rij = Hii + Hjj – 2Hij.

In vector notation, we have

(6.1.17) R = (ε ⊗ ε) · H′ + H′ · (ε ⊗ ε) – 2H,

where the N-dimensional vector ε and the N×N matrix ε⊗ ε are filled with ones, ⊗denotes the tensor product, and H′ is the diagonal part of H. Simplifications occurin the case of an infinite regular lattice where the diagonal components of H are allequal.

IdentitiesUseful identities can be derived from (6.1.17). Premultiplying (6.1.17) by theKirchhoff matrix, K, and recalling that K · ε = 0 and K · H = I , we obtain

(6.1.18) K · R = K · H′ · (ε ⊗ ε) – 2I ,

where the matrix

(6.1.19) I ≡ I –1

Nε ⊗ ε

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was introduced in (5.2.6), and I is the N×N identity matrix. Equation (6.1.18) is thecounterpart of (6.1.10).

Postmultiplying (6.1.18) by I , recalling that I · ε = 0, and noting that I2 = I ,we obtain

(6.1.20) K · R · I = –2I .

Because all matrices involved in this equation are symmetric, we can write

(6.1.21) trace(K · R · I) = trace

(K · I · R) = –trace

(K · R),

yielding

(6.1.22) trace(K · R) = –2 trace(I) = –2 (N – 1).

We conclude that [55]

(6.1.23) trace(K · R) = K : R = 2 (N – 1)

where the colon denotes the double dot product, that is, the sum of the products ofcorresponding elements of the two matrices on either side.

Postmultiplying equation (6.1.18) by K, we obtain

(6.1.24) K · R · K = –2K.

Postmultiplying this equation by an arbitrary symmetric matrix, S, we obtain

(6.1.25) K · R · K · S = –2K · S,which is the counterpart of (6.1.11). Because all matrices involved in this equationare symmetric, we can write

(6.1.26)trace

(K · R · K · S) = trace

(K · S · K · R)

= (K · S · K) : R = –2 trace(K · S).A chain of identities can be derived by setting S = Kn, where n is a positive integer.

Spectral ExpansionSubstituting into (6.1.16) the spectral expansion of the Green’s function given in(5.2.22), we obtain

(6.1.27) Rij =N∑s=2

1

λs

(u(s)i – u(s)j

) (u(s)i – u(s)j

)∗,

where an asterisk denotes the complex conjugate. Note that summation begins ats = 2 to skip the zero eigenvalue, λ1 = 0.

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Representation in Terms of the Normalized Green’s FunctionExpression (6.1.16) can be rearranged into

(6.1.28) Rij = –(Hij – Hii) – (Hij – Hjj),

yielding

(6.1.29) Rij = –Hij – Hji,

where Hij is the normalized Green’s function defined such that Hii = 0, wheresummation is not implied over the repeated index, i. We recall that the normalizedGreen’s function is not necessarily symmetric, that is, Hij is not necessarily equalto Hji.

Complete NetworkIn the case of a complete network with identical link conductances, c, we use theMoore–Penrose Green’s function given in (5.2.42) and find that

(6.1.30) Rij =2

N

for any nodal pair, i and j. This is the minimum possible pairwise resistance for anyuniform network.

6.1.3 One-Dimensional Network

Substituting into the general expression (6.1.16) the Moore–Penrose Green’s func-tion for a one-dimensional isolated network with uniform conductances, given in(5.2.26), we obtain the pairwise resistance

(6.1.31) Rij =1

2N

N∑s=2

⎛⎝cos[ (i – 1

2

)αs

]– cos

[ (j – 1

2

)αs

]sin(12 αs

)⎞⎠2

,

where αs = (s – 1)π /N. It can be shown by algebraic manipulation, or else confirmedby numerical computation, that

(6.1.32) Rij = |i – j|,

in agreement with physical intuition [56]. The same result can be obtained by substi-tuting into (6.1.29) the corresponding normalized Moore–Penrose Green’s functiongiven in (5.2.28).

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6.1.4 One-Dimensional Periodic Network

Substituting into the general expression (6.1.16) the Moore–Penrose Green’s func-tion for a one-dimensional periodic network with uniform conductances, given in(5.2.30), we obtain the corresponding pairwise resistance

(6.1.33) Rij =1

4N

N∑s=2

∣∣∣ exp(–i iαs) – exp(–i jαs)sin(12 αs

) ∣∣∣2,where αs = 2(s–1)π /N. It can be shown by algebraic manipulation, or else confirmedby numerical computation, that

(6.1.34) Rij =1

N|i – j|

(N – |i – j|

),

in agreement with physical intuition [56]. The same results is obtained by substi-tuting into (6.1.29) the corresponding normalized periodic Moore–Penrose Green’sfunction given in (5.2.34).

6.1.5 Infinite Lattices

In the case of an infinite regular lattice with a uniform coordination number d in one,two, or three dimensions, the diagonal components of the Green’s function matrixare equal. Expression (6.1.8) for the pairwise resistance simplifies to

(6.1.35) Rij = 2(Gii – Gij

).

This relation applies for the square, hexagonal, modified Union Jack, honeycomb,cubic, or any other appropriate lattice.

In terms of the normalized Green’s function, denoted by a tilde, we obtain thesimplified expression

(6.1.36) Rij = –2 Gij.

Since all pairwise resistances are positive, every component of the normalized latticeGreen’s function must be negative.

Using expression (5.3.16) for the nearest-neighbor Green’s function, we obtainthe nearest-neighbor pairwise resistance

(6.1.37) Rnn =2

d,

where d is the lattice coordination number. For example, d = 2 for the one-dimensional lattice, d = 4 for the square lattice, and d = 6 for the hexagonal(triangular) or simple cubic lattice.

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6.1.6 Triangle Inequality

The pairwise resistance obeys a triangle inequality stating that

(6.1.38) Rij ≤ Rik + Rkj,

where i, j, and k is an arbitrary triplet of nodes [8, 26]. In the case of an embeddednetwork, the inequality implies that

(6.1.39) Gij + Gji ≥ Gik + Gki + Gkj + Gjk.

In the case of an isolated network, the inequality implies that

(6.1.40) Hij + Hji ≥ Hik + Hki + Hkj + Hjk.

For a uniform infinite lattice, we have

(6.1.41) Gij ≥ Gik + Gkj.

6.1.7 RandomWalks

The pairwise resistance of an isolated network admits a physical interpretation in thecontext of random walks. With reference to the Kirchhoff matrix, K, we define theprobability that a compulsory random walker jumps from the ith to the jth node,

(6.1.42) pi, j ≡ –Ki,jKi,i

,

provided that pi, i = 0. By construction,

(6.1.43)N∑j=1

pi, j = 1,

as required. Note that pi, j is not necessarily equal to pj, i.The first passage probability, Pα,β is defined as the probability that the random

walker starts at node α and reaches node β before returning to α. It can shown that

(6.1.44) Pα,β =1

Kα,αRα,β

(e.g., [38]). In the case of uniform conductances, Kα,α is the degree of the α node.

Exercise

6.1.1 One-dimensional networks

(a) Confirm (6.1.32) by numerical computation. (b) Repeat for (6.1.34).

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6.2 MEAN PAIRWISE RESISTANCE

To assess the overall transport efficiency of a network, we require a global measureof the node pairwise resistance. One such measure is the mean resistance, defined asthe scaled sum of all elements of the pairwise resistance matrix, R,

(6.2.1) Rmean ≡ 1

2N

N∑i=1

⎛⎝ N∑j=1

Rij

⎞⎠ =1

N

N–1∑i=1

⎛⎝ N∑j=i+1

Rij

⎞⎠.The product NRmean is sometimes called the effective network resistance or theresistance distance (e.g., [9, 26]).

The mean pairwise resistance is a mathematically sound and physically intuitivemeasure of the overall network robustness (e.g., [8, 9]). Complete networks are themost robust and tree networks are the least robust connected networks, in agreementwith physical intuition.

Substituting into (6.2.1) expression (6.1.8) for an embedded network or expres-sion (6.1.16) for an isolated network, and noting that the sum of elements in eachrow or column of G or H is zero, we obtain

(6.2.2) Rmean = trace(G)

for an embedded network or

(6.2.3) Rmean = trace(H)

for an isolated network.If a network is unconnected, containing fragments or secluded islands of nodes,

the pairwise resistance of nodes residing in two different fragments or inside andoutside an island is infinite, and the mean resistance is not defined.

6.2.1 Spectral Representation

Since the trace of a matrix is equal to the sum of its eigenvalues, we have

(6.2.4) Rmean =N∑s=1

1

λs

for an embedded network, where λs are the eigenvalues of the modified Kirchhoffmatrix defined in (4.4.6). For an isolated network,

(6.2.5) Rmean =N∑s=2

1

λs,

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Network Performance / / 229

where λs are the eigenvalues of the Kirchhoff matrix. Note that the zero eigenvalueis excluded from the sum in (6.2.5).

Based in (6.2.5), we derive the inequality

(6.2.6)1

λ2< Rmean ≤ N – 1

λ2.

As expected, when λ2 = 0, the mean pairwise resistance is infinite. Other tighterbounds of the mean pairwise resistance are available (e.g., [49]).

Substituting into (6.2.1) formula (6.1.12) for an embedded network or formula(6.1.27) for an isolated network, along with comparing the resulting expression with(6.2.4) or (6.2.5), we find that

(6.2.7)N∑i=1

N∑j=i+1

∣∣u(s)i – u(s)j∣∣2 = N,

where s = 1, . . . ,N for an embedded network or s = 2, . . . ,N for an isolated network.

6.2.2 Complete Network

In the case of a complete network with uniform link conductances, c, we substituteinto (6.2.1) the pairwise resistances given in (6.1.30) and obtain

(6.2.8) Rmean =N – 1

N< 1.

Precisely the same result is obtained by substituting into (6.2.5) the eigenvalues ofthe graph Laplacian matrix given in (2.2.11). The mean resistance of a completenetwork is lower than that of any other network with the same number of nodes.

6.2.3 One-Dimensional Isolated Network

In the case of a one-dimensional isolated network with uniform conductances, wesubstitute into (6.2.1) the pairwise resistances given in (6.1.32) and obtain

(6.2.9) Rmean =1

N

N–1∑i=1

⎛⎝ N∑j=i+1

(j – i)

⎞⎠ =1

2N

N–1∑i=1

(N – i + 1)(N – i)

or

(6.2.10) Rmean =1

2N

N–1∑p=1

p (p + 1) =1

6(N2 – 1).

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We observe that the mean resistance increases as the square of the number of nodes,N. In the case of two nodes, N = 2, connected by one link, Rmean = 1/2.

Substituting into (6.2.5) the eigenvalues of the Laplace matrix given in (1.7.2),we obtain

(6.2.11) Rmean =1

4

N∑s=2

1

sin2(s–12N π

) .Comparing (6.2.10) with (6.2.11), we derive the identity

(6.2.12)N–1∑m=1

1

sin2( m2N π

) = 2

3(N2 – 1),

which can be recast into the form

(6.2.13)N∑m=1

1

sin2( m2N π

) = 1

3(2N2 + 1).

6.2.4 One-Dimensional Periodic Network

In the case of a one-dimensional periodic network with uniform link conductances,we substitute into (6.2.1) the pairwise resistances given in (6.1.34) and obtain

(6.2.14) Rmean =1

N

N–1∑i=1

⎛⎝ N∑j=i+1

(j – i)

(1 –

j – i

N

)⎞⎠.Computing the inner sum, we find that

(6.2.15) Rmean =1

6N2

N–1∑i=1

(N – i + 1)(N – i)(N + 2i – 1),

which can be summed to

(6.2.16) Rmean =1

12(N2 – 1).

The mean resistance of a one-dimensional periodic network is half that of a one-dimensional isolated network.

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Network Performance / / 231

Substituting into (6.2.5) the eigenvalues of the Laplacian matrix given in (1.8.2),we obtain

(6.2.17) Rmean =1

4

N∑s=2

1

sin2(s–1N π

) .Comparing (6.2.16) with (6.2.17) we derive the identity

(6.2.18)N–1∑m=1

1

sin2(mN π) = 1

3(N2 – 1).

6.2.5 Periodic Lattice Patches

In Chapter 3, we derived expressions for the eigenvalues of the Laplacian of severallattice patches in isolated or periodic configurations. For any two-dimensional peri-odic lattice whose eigenvalues, λn1, n2 , are parametrized by two indices, n1 and n2,the mean resistance is

(6.2.19)(Rmean

)2D =

N1∑n1=1

N2∑n2=1

′ 1

λn1, n2,

where the integers N1 and N2 determine the size of the periodic patch and the primeafter the summation symbol indicates that the zero eigenvalue, n1 = 1 and n2 = 1, isexcluded from the sum. For a three-dimensional lattice, we obtain the correspondingexpression

(6.2.20)(Rmean

)3D =

N1∑n1=1

N2∑n1=1

N3∑n3=1

′ 1

λn1, n2, n3,

where the prime has a similar meaning. Expressions for the eigenvalues are shownin Table 6.2.1 for several lattices, where

(6.2.21) αn1 =n1 – 1

N12π , βn2 =

n2 – 1

N22π , γn3 =

n3 – 1

N32π .

Composite lattices consisting of dual or multiple Bravais lattices are treated inspecial ways. In the case of the honeycomb lattice discussed in Section 3.5, we have

(6.2.22)(Rmean

)2D =

N1∑n1=1

N2∑n2=1

(1

λ–n1,n2+

1

λ+n1,n2

),

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TABLE 6.2.1 Eigenvalues of Periodic Patches of Several Lattices with Coordination Number d

Lattice d λn1,n2 or λn1,n2,n3Square 4 4 – 2 cosαn1 – 2 cosβn2

Hexagonal 6 6 – 2 cos αn1 – 2 cosβn2 – 2 cos(αn1 ± βn2 )

ModiÞed Union Jack 8 8 – 2 cosαn1 – 2 cosβn2 – 4 cosαn1 cosβn2

Honeycomb 3 1 – 13

[cosαn1 + cosβn2 + cos(αn1 ± βn2 )

]Kagomé 4

18 – 6 cos αn1 – 6 cosβn2 – 6 cos(αn1 ± βn2 )21 – cosαn1 – cosβn2 – cos(αn1 ± βn2 )

Simple cubic 6 6 – 2 cosαn1 – 2 cosβn2 – 2 cos γn3

bcc 8 8 – 2 cos αn1 – 2 cosβn2 – 2 cos γn3

–2 cos(αn1 + βn2 + γn3 )

fcc 12 12 – 2 cosαn1 – 2 cosβn2 – 2 cos γn3

–2 cos(αn1 – βn2 ) – 2 cos(βn2 – γn3 )–2 cos(γn3 – αn1 )

Note: The plus or minus sign applies for different node indexing schemes associated with a

different set of base vectors.

which can be reduced into (6.2.19) with

(6.2.23) λn1,n2 = 1 – 13

[cosαn1 + cosβn2 + cos(αn1 ± βn2)

].

The plus or minus sign correspond to different node numbering schemes associatedwith different base vectors.

In the case of the kagomé lattice discussed in Section 3.6, we have

(6.2.24)(Rmean

)2D =

N1∑n1=1

N2∑n2=1

(1

λ◦n1,n2

+1

λ–n1,n2+

1

λ+n1,n2

),

which can be reduced into (6.2.19) with

(6.2.25) λn1,n2 =18 – 6 cosαn1 – 6 cosβn2 – 6 cos(αn1 ± βn2)21 – cosαn1 – cosβn2 – cos(αn1 ± βn2)

.

The plus or minus sign correspond to different node numbering schemes associatedwith different base vectors.

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Graphs of the mean resistance scaled by the number of nodes inside each period,N, are shown in Figure 6.2.1(a) for two-dimensional lattices with N1 = N2 on alinear-logarithmic scale. As the size of the periodic unit increases, N → ∞, thescaled mean resistance tends to a well-defined limit,

(6.2.26) Rmean ∼ αN,

where the coefficient α depends on the lattice type. For simple Bravais lattices, thecoefficient α decreases as the lattice coordination number becomes higher due to the

(a)

(b)

0 0.5 1 1.5 2 2.50.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22

log N1

log N1

Rm

ean/

NR

mea

n/N

square (d=4)hexagonal (d=6)mod Union Jack (d=8)honeycomb (3)kagome(4)

0 0.5 1 1.5 2 2.50.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4simple cubic (d=6)bcc (d=8)fcc (d=12)

FIGURE 6.2.1 (a) Dependence of the scaled mean resistanceof periodic two-dimensional lattices with dimensions N1 =

N2 and (b) periodic three-dimensional lattices with dimensionsN1 = N2 = N3.

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availability of a higher number of conductive pathways. The dual honeycomb andkajomé lattices exhibit a higher scaled effective resistance.

Graphs of the scaled mean resistance for three-dimensional lattices with N1 =N2 = N3, shown in Figure 6.2.1(b), exhibit a similar behavior. We may conclude thatthe coefficient α is a sensible index of the efficiency of lattice transport.

Exercises

6.2.1 Trigonometric identity

Confirm identity (6.2.13) by direct numerical evaluation.

6.2.2 Tree network

Compute the mean pairwise resistance of a tree network with N = 4 nodes.

6.3 DAMAGED NETWORKS

Consider an arbitrary network involving L links with arbitrary conductances, andassume that the conductances of M ≤ L links, numbered ms for s = 1, . . . ,M,are perturbed from the unperturbed value, cms , to a perturbed value indicated by aprime, c′ms .

For example, the network shown in Figure 6.3.1 has L = 19 total links, M = 10damaged links drawn with thin lines labeled

(6.3.1) m1 = 3, m2 = 18, m3 = 5, . . . , m10 = 8,

and L –M = 9 intact lines drawn with heavy lines.Our goal is to assess the effect of these perturbations on the overall performance

of the network. In Section 6.4, we will consider the complementary problem of linkaddition.

1

3

4

2

6

1011

13

1214

17

1915

9

5 7

1618

8

FIGURE 6.3.1 Illustration of a networkwith L = 19 total links,M = 10 damagedlinks (thin lines), and L – M = 9 intactlines (heavy lines.)

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Network Performance / / 235

6.3.1 Damaged Kirchhoff Matrix

The Kirchhoff matrix of an isolated network after the M links have been altered isgiven by

(6.3.2) K = K0 +M∑s=1

ζms υ(ms) ⊗ υ(ms),

where a superscript 0 indicates the unperturbed state

(6.3.3) ζj ≡ 1

c(c′j – cj) = σj (ξj – 1)

are dimensionless coefficients, c is a reference conductance, ξj ≡ c′j/cj = 1 is theratio of the perturbed to the unperturbed conductance of the perturbed link labeledj, σj ≡ cj/c, and υ(j) is the jth column of the pristine oriented incidence matrix, R0,before link removal. Specifically, the N-dimensional vector υ(j) is null, except that

(6.3.4) υ(j)kj = –1, υ(j)lj

= 1,

where kj is the label of the first end node and lj is the label of the second end node ofthe jth link.

Unperturbed links make trivial contributions to the right-hand side of (6.3.2). Ifthree links labeled 7, 9, and 14 are removed, then we have M = 3, m1 = 3, m2 = 9,and m3 = 14.

It is useful to introduce a rectangular N × M matrix holding in its columns thevectors υ corresponding to the perturbed links,

(6.3.5) V =

⎡⎢⎣ ↑ ↑ ↑ ↑ ↑υ(m1)

... υ(ms)... υ(mM)

↓ ↓ ↓ ↓ ↓

⎤⎥⎦.The N × L matrix V encompassing all links, M = L, is the oriented incidence matrixof the network, R. In the case of selected damaged links, we obtain a reduced N×Mincidence matrix referring to the set of defective links.

Moreover, it is useful to introduce anM ×M diagonal matrix,

(6.3.6) Z ≡

⎡⎢⎢⎢⎢⎢⎣ζm1 0 · · · 0 00 ζm2 · · · 0 0...

.... . .

......

0 0 · · · ζmM–1 00 0 · · · 0 ζmM

⎤⎥⎥⎥⎥⎥⎦,

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under the stipulation that ζi = 1 so that the matrix Z is nonsingular. This means thatundamaged links are not allowed into the matrices V and Z.

The matrices V and Z are defined such that the sum on the right-hand side of(6.3.2) is given by the matrix product V · Z · VT , so that

(6.3.7) K = K0 + V · Z · VT .

Neither the unperturbed nor the perturbed Kirchhoff matrix is invertible.

6.3.2 Embedded Networks

An expression analogous to (6.3.7) can be written for the modified Kirchhoff matrixof an embedded network,

(6.3.8) K = K0 + V · Z · VT ,

provided that links connecting network nodes to Dirichlet nodes are not disrupted.The inverse of the unperturbed modified Kirchhoff matrix, K, is the correspondingGreen’s function, G,

(6.3.9) K · G = I,

where I is the N × N identity matrix.Using the generalized Woodbury formula discussed in Appendix B, we find that

the Green’s function after perturbation is given by

(6.3.10) G = G0 · [ I – V · (Z–1 +�)–1 · VT · G0 ],where I is the N × N identity matrix, � is anM ×M matrix with elements

(6.3.11) �pq = υ(mp) · w(mq)

for p, q = 1, . . . ,M, and the vector w(mq) satisfies the linear system

(6.3.12) K0 · w(mq) = υ(mq).

We can write

(6.3.13) � = VT · W = VT · K0 · V,

where

(6.3.14) W =

⎡⎢⎣ ↑ ↑ ↑ ↑ ↑w(m1)

... w(ms)... w(mM)

↓ ↓ ↓ ↓ ↓

⎤⎥⎦ = G0 · V.

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By construction, the matrixW is symmetric for any network topology due to the sym-metry of the lattice Green’s function. Physically, the pq component of � expressesthe difference in the nodal values across the mp damaged link due to a point-sourcedipole applied across the mq damaged link. In terms of the unperturbed Green’sfunction, we have

(6.3.15) �pq = G0lp,lq + G0

kp,kq – G0lp,kq – G0

kp,lq ,

where kp and lp are the end nodes of the pth link and kq and lq are the end nodes ofthe qth link. The diagonal components,

(6.3.16) �pp = G0lp,lp + G0

kp,kp – 2G0lp,kp ,

express the difference in the nodal values across a damaged link due to a point-sourcedipole applied across the same link. In terms of the normalized Green’s function,

(6.3.17) �pp = –G0lp,kp – G0

kp,lp .

We recall that in the case of an infinite regular lattice, but not more generally, wehave �pp = 2/d, where d is the lattice coordination number.

Subject to the preceding definitions, we have

(6.3.18) G = G0 –W · (Z–1 +�)–1 · WT .

It is interesting that nodal field differences corresponding to damaged links, but notintact links, appear in the final expressions for the Green’s function in the perturbedstate, G.

Perturbation Nodal FieldIt is useful to introduce the matrix

(6.3.19) P ≡ –W · (�–1 +�)–1 · VT

and obtain

(6.3.20) G = (I + P) · G0.

The perturbation nodal field, denoted by a prime, is given by

(6.3.21) ψ ′ = P · ψ0,

where the superscript 0 denotes the unperturbed field corresponding to the pristinenetwork.

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6.3.3 One Damaged Link

In the case of one damaged link,M = 1, connecting nodes k and l in the pristine state,the matrices Z and � are scalars, yielding

(6.3.22) G = G0 –σ (ξ – 1)

1 + σ (ξ – 1)φw ⊗ w

and

(6.3.23) P = –σ (ξ – 1)

1 + σ (ξ – 1)φw ⊗ v,

where the coefficients σ and ξ and the vectors υ and w are associated with theperturbed link,

(6.3.24) w = G0 · υ,

and

(6.3.25) φ ≡ υ · w = wl – wk = υ · G0 · υ = R0kl.

Physically, the scalar φ represents the nodal difference across the link when electricalcurrent is supplied at the first node of the link and withdrawn from the second nodeof the link in the pristine state. In terms of the unperturbed Green’s function,

(6.3.26) wi = G0i,l – G0

i,k

and

(6.3.27) φ = G0l,l + G0

k,k – 2G0l,k = –G0

l,k – G0k,l,

where k and l are the end points of the perturbed link.Using expression (6.1.8), we find that the pairwise resistance matrix in the

perturbed state is given by

(6.3.28) Rij = R0ij –

σ (ξ – 1)

1 + σ (ξ – 1)φ(w2

i + w2j – 2wiwj)

or

(6.3.29) Rij = R0ij –

σ (ξ – 1)

1 + σ (ξ – 1)φ(wi – wj)

2.

The second term on the right-hand side expresses the effect of the perturbation. Forthe resistance to increase when ξ = 0, the denominator must be positive, and this

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Network Performance / / 239

requires that φ ≥ 1. The equality applies in the case of a one-dimensional network.Applying (6.3.29) for i = k and j = l, setting φ = R0

kl, and rearranging, we obtain

(6.3.30) Rkl =R0kl

1 + σ (ξ – 1)R0kl

.

As the conductance of the altered ring increases, ξ → ∞, the effective resistancetends to zero.

An Infinite Regular Lattice with One Damaged LinkIn the case of an infinite homogeneous regular lattice, σ = 1, we obtain φ =2/d, where d is the lattice coordination number. Consequently, the altered Green’sfunction is

(6.3.31) G = G0 –ξ – 1

1 + (ξ – 1)2

d

w ⊗ w,

the altered projection matrix is

(6.3.32) P = –ξ – 1

1 + (ξ – 1)2

d

w ⊗ v,

and the altered pairwise resistance matrix is

(6.3.33) Rij = R0ij –

ξ – 1

1 + (ξ – 1)2

d

(wi – wj)2.

Using the altered projection matrix, we obtain

(6.3.34) v · ψ ′ = –ξ – 1

1 +2

d(ξ – 1)

(v · w) (v · ψ0).

Substituting once again v · w = 2/d and simplifying, we obtain

(6.3.35) v · ψ ′ = 1 – ξ1

2d + ξ – 1

(v · ψ0).

Physically, v ·ψ ′ is the difference in the perturbation field and v ·ψ0 is the differencein the unperturbed field across the defective link.

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6.3.4 Clipped Links

In the case of a network whose links have the same conductance in the pristine state,σi = 1 for i = 1, . . . ,N, and M clipped links with zero conductance in the perturbedstate, ξms = 0 for s = 1, . . . ,M, we find that Z = –IM , where IM is theM×M identitymatrix. Accordingly, we have

(6.3.36) Q ≡ �–1 +� = � – IM .

When the matrix Q is singular, the projection matrix P does not exist and the dis-turbance nodal field is not defined. Physically, isolated nodes or clusters of nodesunconnected to their neighbors are encountered inside the network. The number ofthese isolated groups is equal to the number of zero eigenvalues of the matrix Q.Eigenvalues equal to –1 correspond to isolated nodes or clusters of nodes attached tothe Dirichlet nodes.

6.3.5 Isolated Networks

In the case of isolated networks, we use (5.2.14) and compute the Moore–PenroseGreen’s function of the perturbed network

(6.3.37) H = H0 –W · (�–1 +�)–1 · WT ,

where

(6.3.38) W = H0 · V, � = VT · H0 · Y,

subject to the preceding definitions for embedded networks.

Exercise

6.3.1 Perturbed network

Derive the matrix V corresponding to the damaged network shown in Figure 6.3.1.

6.4 REINFORCED NETWORKS

The analysis of Section 6.3 can be adapted to address the effect of link addition,intended to strengthen or reinforce a network.

Consider the addition of one link labeled L + 1 with conductance cL+1 = σcanchored at nodes labeled k and l of an embedded network, as shown in Figure 6.4.1.The Green’s function matrix after link addition is given by

(6.4.1) G = G0 –σ

1 + σφw ⊗ w,

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k

l

M + 1

FIGURE 6.4.1 Illustration of a reinforced embedded networkwith one added link anchored at the k th and lth nodes, drawnas a heavy line.

the nodal projection matrix providing us with the perturbation field due to linkaddition is given by

(6.4.2) P = –σ

1 + σ φw ⊗ v,

and the pairwise resistance matrix is given by

(6.4.3) Rij = R0ij –

σ

1 + σ φ(wi – wj)

2,

where i, j = 1, . . . ,N. The vector υ is null, except that the lth entry is equal to 1 andthe kth entry is equal to –1. The vector w = G0 · υ and scalar φ are given in (6.3.26)and (6.3.27) in terms of the unperturbed Green’s function. Physically, the scalar

(6.4.4) φ = υ · w = wl – wk ≡ R0kl

represents the difference in the induced potential across the added link when currentis supplied at the first node of the link and withdrawn from the second node of thelink in the pristine state. Applying (6.4.3) for i = k and j = l, we obtain [8]

(6.4.5) Rkl =R0kl

1 + σ R0kl

.

To address the general case of L′ added links, we introduce a rectangular N × L′matrix holding in its columns the vectors υ corresponding to the added links,

(6.4.6) V =

⎡⎢⎣ ↑ ↑ ↑ ↑ ↑υ(L+1)

... υ(i)... υ(L+L

′)

↓ ↓ ↓ ↓ ↓

⎤⎥⎦,

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and the L′ × L′ diagonal matrix

(6.4.7) Z ≡

⎡⎢⎢⎢⎢⎢⎣ζL+1 0 · · · 0 00 ζL+2 · · · 0 0...

.... . .

......

0 0 · · · ζL+L′–1 00 0 · · · 0 ζL+L′

⎤⎥⎥⎥⎥⎥⎦,

under the stipulation that ζi = 1 so that the matrix Z is nonsingular.The matrices V and Z are defined such that the Kirchhoff matrix after reinforce-

ment is

(6.4.8) K = K0 + V · Z · VT .

Neither the original nor the reinforced Kirchhoff matrix is invertible. However, thecorresponding modified Kirchhoff matrices, K and K0, are invertible. The conceptsand formulas discussed in Section 6.3 for link damage also apply to link additionwith sensible modifications.

Exercise

6.4.1 Reinforced lattices

Explain how a square network (d = 4) can be transformed into a hexagonal network(d = 6) with systematic link addition.

6.5 DAMAGED LATTICES

In Section 6.3, we discussed the performance of arbitrary damaged networks andderived general expressions for the Green’s function and pairwise resistance. In thissection, we consider the particular case of networks configured as infinite regularlattices.

6.5.1 One Damaged Link

Consider an infinite square lattice where all links have the same conductance, c, ex-cept that one defective link extending between nodes labeled A and B has a differentconductance, c′, as shown in Figure 6.5.1. We are interested in assessing the effectof the defect on the nodal distribution of a potential, ψ , associated with a transportedentity.

A balance of the transported entity at node labeled A requires that

(6.5.1) c′ (ψB – ψA) + c (ψC – ψA) + c (ψD – ψA) + c (ψE – ψA) = 0.

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c’C E

F

G

H B

AD

FIGURE 6.5.1 Illustration of scalar transportthrough an infinite square network of resis-tors arranged on a square lattice. The con-

ductance of one link is different than that of

all other links.

Rearranging, we obtain

(6.5.2) c(ψB – ψA) + c(ψC – ψA) + c(ψD – ψA) + c(ψE – ψA) + s = 0,

where the term

(6.5.3) s ≡ (c′ – c) (ψB – ψA)

is regarded as an a priori unknown nodal source applied in a pristine network withuniform conductance, c, at node numbered A. A similar balance at node labeled Brequires that

(6.5.4) c′ (ψA – ψB) + c (ψF – ψB) + c (ψG – ψB) + c (ψH – ψB) = 0.

Rearranging, we obtain

(6.5.5) c(ψA – ψB) + c(ψF – ψB) + c(ψG – ψB) + c(ψH – ψB) – s = 0.

The solution of the linear system that arises by writing balance equations at allnodes can be decomposed into a homogeneous solution, ψ0, a particular solution,ψ (1), due to the source (sink) in equation (6.5.2), and another particular solution,ψ (2), due to the sink (source) in equation (6.5.5). The nodal value at an arbitrarynode X is

(6.5.6) ψX = ψ0X + ψ (1)

X + ψ (2)X .

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Considering the nodal field ψ (1), we note that, by symmetry, the flow rate of thetransported field is divided into d = 4 equal flow rates upon entering node A, whered is the lattice coordination number. Consequently,

(6.5.7) s = dc(ψ

(1)A – ψ (1)

B

).

A similar conclusion can be reached regarding the field ψ (2), yielding

(6.5.8) s = dc(ψ

(2)A – ψ (1)

B

).

Now using equation (6.5.6), we obtain

(6.5.9) ψA – ψB =(ψ0A – ψ0

B

)+(ψ

(1)A – ψ (1)

B

)+(ψ

(2)A + ψ (2)

B

).

Substituting the preceding expressions for the particular solutions and rearranging,we find that

(6.5.10) δψ ′AB ≡ (ψA – ψB) –

(ψ0A – ψ0

B

)=2

d

s

c=2

d

c′ – cc

(ψA – ψB

).

Solving for the difference across the defective link, we obtain

(6.5.11) δψAB ≡ ψA – ψB =1

1 +2

d

c′ – cc

(ψ0A – ψ0

B

).

Consequently,

(6.5.12)δψ ′

AB

δψ0AB

≡ =1 – ξ

α + ξ,

where ξ ≡ c′/c and

(6.5.13) α = 12 d – 1,

which is positive since d = 4. Expression (6.5.12) is consistent with the more generalresult stated in (6.3.35).

In fact, expressions (6.5.12) and (6.5.13) apply for any one-, two-, or three-dimensional regular network consisting of links with equal conductances, providedthat the coefficient d is set equal to the lattice coordination number [24, 25]. In thecase of a one-dimensional lattice, d = 2, in the case of a honeycomb lattice, d = 3,in the case of a square lattice, d = 4, and in the case of a hexagonal (triangular) orsimple cubic lattice, d = 6.

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Network Performance / / 245

6.5.2 Effective-Medium Theory

Assume that a defective link with conductance c′ occurs with probability densityfunction φ(c′). The expected value of the coefficient defined in (6.5.12) is

(6.5.14) < >=∫ ∞

0 (c′)φ(c′) dc′.

To be consistent with the imposed boundary conditions far from the defective link,we require that < >= 0 and invoke the definition of ξ to obtain an algebraicequation for c,

(6.5.15)∫ ∞

0

c – c′

α c + c′φ(c′) dc′ = 0.

In the case of a binary distribution with two possible link conductances c′ = c0and βc0, we set

(6.5.16) φ(c′) = (1 – q) δ(c′ – c0) + q δ(c′ – βc0),

where β is a specified positive coefficient, q the number density of links withconductance βc0, and δ is the Dirac delta function. Also setting c = ζc0, we obtain

(6.5.17)ζ – 1

α ζ + 1(1 – q) +

ζ – β

α ζ + βq = 0,

which can be rearranged into a quadratic equation for the dimensionless coefficient ζ ,

(6.5.18) α ζ 2 –[(α + 1)(1 – q) – 1 + β

((α + 1) q – 1

) ]ζ – β = 0.

Substituting the value of α from (6.5.13), we find that

(6.5.19)(12 d – 1

)ζ 2 –

[ 12 d (1 – q) – 1 + β

( 12 d q – 1

) ]ζ – β = 0.

The positive root of this quadratic equation provides us with a rational estimate forthe effective conductivity of the network.

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6.5.3 Percolation Threshold

When β = 0, corresponding to disrupted links, and lattice coordination numberd > 2, equation (6.5.19) has the uninteresting root ζ = 0 and the interesting root

(6.5.20) ζ = 1 –α + 1

αq

or

(6.5.21) ζ =12 d (1 – q) – 1

12 d – 1

,

which is plotted in Figure 6.5.2 for several lattices. We find that ζ = 0 at theapproximate percolation threshold

(6.5.22) pc = 1 – qc � 2

d.

The fraction on the left-hand side is the ratio of number of nodes to the numberof links, N/L, according to (2.1.6). Considering the heuristic nature of the effective

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

q

ζ

FIGURE 6.5.2 Coefficient ζ determining the effective conduc-tivity of a network with a binary distribution of conductances.The solid line is for the honeycomb lattice (d = 3), the dashed

line is for the square lattice (d = 4), and the dotted dashed

line is for the hexagonal lattice (d = 6). The symbols on the

q axis represent percolation thresholds, qc = 2 sin(π /18) �0.3473 for the honeycomb lattice (circle), qc = 0.5 for the

square lattice (square), qc = 1–sin(π /18) � 0.6527 for the hex-

agonal (diamond), and qc = 0.7512 · · · for the simple cubic

lattice (×) [28, 44, 52].

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Network Performance / / 247

medium theory, the predictions of the critical threshold for complete disruption areremarkably accurate.

Exercise

6.5.1 Effective medium theory

Derive the counterpart of (6.5.18) for three types of links with conductances c0, β1c0,and β2c0, occurring with probabilities 1 – q1 – q2, q1, and q2.

6.6 DAMAGED SQUARE LATTICE

Consider transport through an infinite square lattice whose nodes are parametrized bytwo indices, i1 and i2, as shown in Figure 6.6.1. All links have the same conductance,c, except for two unrelated defective links that have different conductances, c′ andc′′. Our objective is to assess the effect of the defects on the nodal distribution of atransported field, ψ . For simplicity, we assign the labels A–D to the end points of thedefective links, as shown in Figure 6.6.1.

Without loss of generality, we may assume that the first defective link withconductance c′ is horizontal, extending between two nodes labeled (n1, n2) and(n1 + 1, n2). When the defective links are parallel, the second defective link withconductance c′′ extends between nodes (m1,m2) and (m1 + 1,m2), as shown inFigure 6.6.1(a). When the defective links are perpendicular, the second defective linkwith conductance c′′ is subtended between nodes (m1,m2) and (m1,m2 +1), as shownin Figure 6.1.1(b).

For any relative defective link orientations, a balance of the transported entityassociated with the potential ψ at each end node of the first defective link requiresthat

(6.6.1)c′ (ψB – ψA) + c (ψA1 – ψA) + c (ψA2 – ψA) + c (ψA3 – ψA) = 0,

c′ (ψA – ψB) + c (ψB1 – ψB) + c (ψB2 – ψB) + c (ψB3 – ψB) = 0.

(a) (b)

1m

c’’m2

2i

D

1m

m2

2i

C

c’

1n

n2A BA

c’

n1

nBA2

A3 B1

B2

A1 B3

2

c’’

C

DC2

D3

C1

C2

C3

D1

D2

C3 D1

C1

D2

A1

A2

A3 B1

B2

B3

D3

i i

FIGURE 6.6.1 Illustration of transport through an infinite square network withtwo parallel or perpendicular defective links.

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Rearranging, we obtain

(6.6.2)c (ψB – ψA) + c (ψA1 – ψA) + c (ψA2 – ψA) + c (ψA3 – ψA) + s

′ = 0,

c (ψA – ψB) + c (ψB1 – ψB) + c (ψB2 – ψB) + c (ψB3 – ψB) – s′ = 0,

where the term

(6.6.3) s′ ≡ (c′ – c) (ψB – ψA)

is regarded as an a priori unknown nodal source applied to a pristine network withuniform conductance, c.

Working similarly with the second defective link, we derive correspondingequations involving a nodal source with strength

(6.6.4) s′′ ≡ (c′′ – c) (ψD – ψC)

at the point C, and a nodal sink with opposite strength at the point D.The solution of the linear system that arises by writing balance equations at all

nodes can be decomposed into a homogeneous solution, ψ0, a particular solution dueto a source (sink) at node A accompanied by a sink (source) at node B, denoted asψAB, and another particular solution due to a source (sink) at node C accompaniedby a sink (source) at node D, denoted as ψCD. The value at an arbitrary node X is

(6.6.5) ψX = ψ0X + ψAB

X + ψCDX .

In terms of the lattice Green’s function, GXY,

(6.6.6) ψX = ψ0X +

s′

c

(GXA – GXB) +

s′′

c

(GXC – GXD).

Physically, GXY is the potential induced at node X by a point source of unit strengthapplied at point Y.

To compute the strengths of the fictitious sources, s′ and s′′, we apply equation(6.6.6) at the end points of the defective links, obtaining

(6.6.7)

ψA = ψ0A +

s′

c

(GAA – GAB) +

s′′

c

(GAC – GAD),

ψB = ψ0B +

s′

c

(GBA – GBB) +

s′′

c

(GBC – GBD),

ψC = ψ0C +

s′

c

(GCA – GCB) +

s′′

c

(GCC – GCD),

ψD = ψ0D +

s′

c

(GDA – GDB) +

s′′

c

(GDC – GDD).

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Next, we subtract the second from the first equation and the third from the secondequation, and obtain

(6.6.8)

[1 –(GBA + GAB) c

′–cc

](ψB – ψA)

–c′′ – cc

(GBC + GAD – GAC – GBD

)(ψD – ψC) = ψ0

B – ψ0A

and

(6.6.9)

–c′ – cc

(GDA + GCB – GCA – GDB

)(ψB – ψA)[

1 – (GDC + GCA)c′′ – cc

](ψD – ψC) = ψ

0D – ψ0

C,

where

(6.6.10) GXY ≡ GXY – GYY

is the normalized Green’s function defined such that GXX = 0. Solving this linearsystem provides for the nodal differences ψA – ψB and ψD – ψC and thereby allowsus to compute the strengths of the sources, s′ and s′′.

Parallel and Adjacent Defective LinksWhen the two defective links are parallel and adjacent, m1 = n1 + 1 and m2 = n2,nodes B and C coincide. Referring to Section 5.4, we find that

(6.6.11) GAB = GBA = GBD = GDB = –1

4, GAD = GDA = –1 +

2

π.

Substituting these values into equations(6.6.8) and (6.6.9), we obtain

(1 +

1

2

c′ – cc

)(ψB – ψA) –

(2

π–1

2

)c′′ – cc

(ψD – ψB) = ψ0B – ψ0

A,

(2

π–1

2

)c′ – cc

(ψB – ψA) +

(1 +

1

2

c′′ – cc

)(ψD – ψB) = ψ

0D – ψ0

B.

(6.6.12)

When c′ = c′′, these equations can be added and rearranged to yield

(6.6.13) ψD – ψA =1

1 +

(1 –

2

π

)c′ – cc

(ψ0D – ψ0

A

).

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The corresponding perturbation difference is

(6.6.14) δψ ′DA ≡ (ψD – ψA) – (ψ

0D – ψ0

A).

Making substitutions, we obtain

(6.6.15)δψ ′

DA

δψ0DA

≡ =1 – ξ

α + ξ

where δψ0DA ≡ ψ0

D – ψ0A, ξ ≡ c′/c, and

(6.6.16) α =2

π – 2.

Node Damage and Effective Medium TheoryIf a node of a square lattice is damaged, the conductances of the four links sharingthe node are altered, as shown in Figure 6.6.2. In one special configuration, the con-ductance of all damaged links is the same, c′. In the case of unidirectional transportin the first direction, corresponding to the index i1, the nodal values of the unper-turbed potential, ψ0, are independent of the second index, i2. An effective mediumtheory may then be developed following the analysis of Section 6.4.2. The analysisculminates in equation (6.5.18) for the effective conductance coefficient, ζ , where qis the fraction of damaged links and the coefficient α is given in (6.6.16).

In the case of clipped links, β = 0, we obtain (6.5.20) and substitute the value ofα from (6.6.16) to obtain

(6.6.17) ζ = 1 –1

2π q

i2

i1

c’

n1

n2

FIGURE 6.6.2 Illustration of scalar transportthrough an infinite square network of resistorswith one damaged node disrupting the operationof four links.

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[51]. The percolation threshold corresponding to ζ = 0 is predicted to be

(6.6.18) qc � 2

π= 0.637, pc = 1 – qc = 1 –

2

π= 0.363.

Using (2.7.2), we set pc = pnode2

c and obtain

(6.6.19) pnodec �(1 –

2

π

)1/2= 0.60281,

which compares favorably with the known value for the square lattice, pnodec =0.59275, as discussed in Section 2.7.

Exercises

6.6.1 Perpendicular adjacent links

Derive the counterpart of system (6.6.12) for two adjacent perpendicular links, asshown in Figure 6.6.1(b).

6.6.2 Simple cubic lattice

Derive an estimate for the node percolation threshold of the simple cubic lattice basedon the effective medium theory.

6.7 DAMAGED HONEYCOMB LATTICE

The analysis of Section 6.6 for the square lattice can be extended to the honeycomblattice. Consider transport through a honeycomb lattice, as shown in Figure 6.7.1.All links have the same conductance, c, except for three adjoining defective linksthat have different conductances, c′, c′′, and c′′′. Our objective is to assess the effect

A

B

C

E

F

G

H

I

c′′′c″

c′

D

J

FIGURE 6.7.1 Illustration of transportthrough an infinite honeycomb latticewith three adjoining defective links.

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of the defects on the nodal distribution of a potential,ψ , associated with a transportedentity. We will see that this calculation will allow us to obtain an accurate estimatefor the node percolation threshold.

Ten nodes of interest, labeled A–J, are shown in Figure 6.7.1. Balancing the ratesof transport at nodes A–D provides us with four equations:

(6.7.1)

c′ (ψB – ψA) + c′′ (ψC – ψA) + c′′′ (ψD – ψA) = 0,c′ (ψA – ψB) + c (ψE – ψB) + c (ψF – ψB) = 0,c′′ (ψA – ψC) + c (ψG – ψC) + c (ψH – ψC) = 0,c′′′ (ψA – ψD) + c(ψI – ψD) + c (ψJ – ψD) = 0.

Rearranging, we obtain an identical set of equations:

(6.7.2)

c (ψB – ψA) + c (ψC – ψA) + c (ψD – ψA) + s′ + s′′ + s′′′ = 0,c (ψA – ψB) + c (ψE – ψB) + c (ψF – ψB) – s′ = 0,c (ψA – ψC) + c (ψG – ψC) + c (ψH – ψC) – s′′ = 0,c (ψA – ψD) + c (ψI – ψD) + c (ψj – ψD) – s′′′ = 0,

where

(6.7.3)s′ = (c′ – c) (ψB – ψA), s′′ = (c′′ – c) (ψC – ψA),

s′′′ = (c′′′ – c) (ψD – ψA)

are fictitious sources applied at the four nodes.The solution of the linear system that arises by writing balance equations at all

nodes can be decomposed into a homogeneous solution, ψ0, a particular solution dueto the source (sink) at node A accompanied by a sink (source) at point B, denoted byψAB, another particular solution due to the source (sink) at node A accompanied bya sink (source) at point C, denoted by ψAC, and a third particular solution due to thesource (sink) at node A accompanied by a sink (source) at point D, denoted by ψAD.The nodal value at an arbitrary node, X, is

(6.7.4) ψX = ψ0X + ψAB

X + ψACX + ψAD

X .

In terms of the lattice Green’s function, GXY, we obtain the representation

(6.7.5) ψX = ψ0X +

s′

c(GXA – GXB) +

s′′

c(GXA – GXC) +

s′′′

c(GXA – GXD).

Physically, GXY is the potential induced at node X by a point source of unit strengthapplied at point Y.

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To compute the strengths of the sources, we apply the representation (6.7.5) atnodes A–D, obtaining

(6.7.6)

ψA = ψ0A +

s′

c(GAA – GAB) +

s′′

c(GAA – GAC), +

s′′′

c(GAA – GAD),

ψB = ψ0B +

s′

c(GBA – GBB) +

s′′

c(GBA – GBC) +

s′′′

c(GBA – GBD),

ψC = ψ0C +

s′

c(GCA – GCB) +

s′′

c(GCA – GCC) +

s′′′

c(GCA – GCD),

ψD = ψ0D +

s′

c(GDA – GDB) +

s′′

c(GDA – GDC) +

s′′′

c(GDA – GDD).

Subtracting the first from the second, third, and fourth equations, substituting ex-pressions (6.7.3) for the fictitious sources, and rearranging, we obtain a system ofthree linear equations for the differences ψB – ψA, ψC – ψA, and ψD – ψA. The firstequation reads

(6.7.7)

[1 –

c′ – cc

(GBA + GAB)

](ψB – ψA) –

c′′ – cc

(GBA – GBC + GAC) (ψC – ψA)

–c′′′ – cc

(GBA – GBD + GAD) (ψD – ψA) = ψ0B – ψ0

A,

the second equation reads

(6.7.8)

–c′ – cc

(GCA – GCB + GAB) (ψB – ψA) +

[1 –

c′′ – cc

(GCA + GAC)

](ψC – ψA)

–c′′′ – cc

(GCA – GCD + GAD) (ψD – ψA) = ψ0C – ψ0

A,

and the third equation reads

(6.7.9)

–c′ – cc

(GDA – GDB + GAB) (ψB – ψA) –c′′ – cc

(GDA – GDC + GAC) (ψC – ψA)

+

[1 –

c′′′ – cc

(GDA + GAD)

](ψD – ψA) = ψ0

D – ψ0A,

where

(6.7.10) GXY ≡ GXY – GYY

is the normalized Green’s function defined such that GXX = 0.Using the results of Section 5.6, we find that

(6.7.11)GAB = GBA = GAC = GCA = GAD = GDA = –1

3 ,

GBC = GCB = GBD = GDB = GCD = GDC = –12 .

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Substituting these values into equations (6.7.7)–(6.7.9), we obtain

(6.7.12)

(1 +

2

3

c′ – cc

)(ψB – ψA) +

1

6

c′′ – cc

(ψC – ψA) +1

6

c′′′ – cc

(ψD – ψA)

= ψ0B – ψ0

A,

(6.7.13)

1

6

c′ – cc

(ψB – ψA) +

(1 +

2

3

c′′ – cc

)(ψC – ψA) +

1

6

c′′′ – cc

(ψD – ψA)

= ψ0C – ψ0

A,

and

(6.7.14)

1

6

c′ – cc

(ψB – ψA) +1

6

c′′ – cc

(ψC – ψA) +

(1 +

2

3

c′′′ – cc

)(ψD – ψA)

= ψ0D – ψ0

A.

Node Damage and Effective Medium TheoryIf node A is damaged, the conductances of the three links sharing this node aremodified. Assume that the conductances of the three affected links is the same, givenas

(6.7.15) c′ = c′′ = c′′′ = ξc,

where ξ is an arbitrary positive or zero coefficient. In the case of vertical unperturbedtransport, the nodal values of the unperturbed potential are independent of horizontalposition,

(6.7.16) δψ0BA ≡ ψ0

B – ψ0A = ψ0

A – ψ0C, ψ0

D – ψ0A = 0.

By symmetry, the perturbed nodal field satisfies the same equations. Equation(6.7.14) is trivially satisfied and equation (6.7.12) or (6.7.13) yields

(6.7.17) δψBA ≡ ψB – ψA = ψA – ψC =2

1 + ξ(ψ0

B – ψ0A).

The corresponding perturbation difference is

(6.7.18) δψ ′BA ≡ (ψB – ψA) – (ψ

0B – ψ0

A).

Making substitutions, we obtain

(6.7.19)δψ ′

BA

δψ0BA

≡ =1 – ξ

1 + ξ.

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An effective medium theory can be developed following the analysis of Sec-tion 6.4.2 for one defective link, culminating in equation (6.5.18) for the effectiveconductance coefficient, ζ , where q is the fraction of damaged links and α = 1. Inthe case of clipped links, β = 0, we substitute α = 1 into equation (6.5.20) and obtain

(6.7.20) ζ = 1 – 2 q

[19]. The percolation threshold corresponding to ζ = 0 is predicted to be

(6.7.21) qc � 0.5, pc = 1 – qc � 0.5.

Using (2.7.2), we set pc = pnode2

c and obtain

(6.7.22) pnodec � 1√2= 0.707,

which is in surprisingly good agreement with the exact value for the honeycomblattice, pnodec = 0.69704, as discussed in Section 2.7.

Exercise

6.7.1 Effective conductance and node percolation threshold

Derive the effective conductance and estimate the node percolation threshold for thecase of horizontal unperturbed transport.

6.8 DAMAGED HEXAGONAL LATTICE

Consider transport through a hexagonal lattice, as shown in Figure 6.8.1. To study theperformance of the network, we consider separately the case of longitudinal transportwhere the unperturbed potential varies along horizontal links, and the case of lateraltransport where the unperturbed potential is constant along horizontal links.

6.8.1 Longitudinal Transport

To study the case of longitudinal unperturbed transport, we refer to Figure 6.8.1(a)and assume links have the same conductance, c, except for two adjoining defec-tive links that have different conductances, c′ and c′′. Repeating the analysis ofSections 6.5 and 6.6, we derive the balance equations

(6.8.1)

[1 –

c′ – cc

(GBA + GAB)

](ψB – ψA)

–c′′ – cc

(GBA – GBC + GAC) (ψC – ψA) = ψ0B – ψ0

A

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(a) (b)

c′ c″CB A

B

C

D

E

F

GA

FIGURE 6.8.1 Illustration of (a) longitudinal and (b) lateral transport through an infinitehexagonal network.

and

(6.8.2)–c′ – cc

(GCA – GCB + GAB) (ψB – ψA)

+

[1 –

c′′ – cc

(GCA + GAC)

](ψC – ψA) = ψ

0C – ψ0

A.

Using the results of Section 5.5, we find that

(6.8.3)GAB = GBA = GAC = GCA = –

1

6,

GBC = GCB = –4

3+2

π

√3.

Equations (6.8.1) and (6.8.2) then become

(6.8.4)(1 +

1

3

c′ – cc

)(ψB – ψA) +

(2

π

√3 – 1

)c′′ – cc

(ψC – ψA) = ψ0B – ψ0

A

and

(6.8.5)(2

π

√3 – 1

)c′ – cc

(ψB – ψA) +

(1 +

1

3

c′′ – cc

)(ψC – ψA) = ψ

0C – ψ0

A.

Assume that the conductances of the links is the same, c′ = c′′ = ξc, where ξis an arbitrary positive or zero coefficient. In the case of longitudinal unperturbedtransport, the nodal values of the unperturbed potential are independent of lateralposition and

(6.8.6) δψ0BA ≡ ψ0

B – ψ0A = ψ0

A – ψ0C.

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By symmetry, the perturbed nodal field satisfies the same equations. Equation (6.8.4)or (6.8.5) yields

(6.8.7) ψB – ψA = ψA – ψC =1

1 + η(ξ – 1)(ψ0

B – ψ0A)

and

(6.8.8)δψ ′

BA

δψ0BA

≡ =1 – ξ

α + ξ,

where

(6.8.9) η =4

3–2

π

√3, α =

1 – η

η.

The effective medium theory culminates in equation (6.5.20), yielding

(6.8.10) ζ = 1 –1

1 – ηq.

The percolation threshold corresponding to ζ = 0 is predicted to be

(6.8.11) qc � 1 – η, pc = 1 – qc � η.

Using (2.7.2), we set pc = pnode2

c and obtain

(6.8.12) pnodec � √η = 0.480,

which is in surprisingly good agreement with the exact value for the hexagonallattice, pnodec = 0.5, as discussed in Section 2.7.

6.8.2 Lateral Transport

To study the case of lateral transport, we consider a more general configuration wheresix links originating from a node labeled A and ending at nodes B–G are damaged,as shown in Figure 6.8.1(b). All links have the same conductance, c, except for thesix defective links that have a different conductance, c′ = ξc. The nodal value at anarbitrary node, X, can be expressed in terms of the lattice Green’s function, GXY, as

(6.8.13)

ψX = ψ0X + (ξ – 1)

[(ψB – ψA)(GXA – GXB) + (ψC – ψA)(GXA – GXC)

+(ψD – ψA)(GXA – GXD) + (ψE – ψA)(GXA – GXE)

+(ψF – ψA)(GXA – GXF) + (ψG – ψA)(GXA – GXG)].

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Subtracting the equation at node X = A from that at node X = B, and rearranging,we obtain

(6.8.14)

(ψB –ψA)[1 – (ξ – 1)(GBA + GAB)

]–(ξ – 1)

[(ψC – ψA)(GBA – GBC + GAC)

–(ψD – ψA)(GBA – GBD + GAD) – (ψE – ψA)(GBA – GBE + GAE)

–(ψF – ψA)(GBA – GBF + GAF) – (ψG – ψA)(GBA – GBG + GAG)]

= ψ0B – ψ0

A,

where GXY is the normalized Green’s function defined so that GXX = 0. Similarequations can be written for the other nodes.

Now we consider the field induced by a vertical potential gradient. By symmetry,we have

(6.8.15) ψ0B – ψ0

A = ψ0A – ψ0

E = ψ0G – ψ0

A = ψ0A – ψ0

D,

and

(6.8.16) ψ0F = ψ0

A, ψ0A = ψ0

C.

A set of identical equations can be written for the perturbed potential, ψ . Equation(6.8.14) simplifies into

(6.8.17) (ψB – ψA)[1 – (ξ – 1)(GBD + GBE – GBA)

]= ψ0

B – ψ0A.

Using the results of Section 5.6, we find that

(6.8.18) GBA = –1

6, GBD =

1

3–1

π

√3, GBE = –

4

3+2

π

√3.

Substituting these values into (6.8.17), we obtain expressions (6.8.7) and (6.8.8),where

(6.8.19) η =5

6–1

π

√3, α =

1 – η

η.

The effective medium theory yielding the node percolation threshold

(6.8.20) pnodec � √η = 0.531,

which is in surprisingly good agreement with the exact value for the hexagonallattice, pnodec = 0.5 as discussed in Section 2.7 [18].

Exercise

6.8.1 Effective medium

Derive the values of η and α stated in (6.8.19).

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APPENDIX A

EIGENVALUES OF MATRICES

A brief account of eigenvalues and eigenvectors of matrices is given in this appendix.Further information is available in texts on linear algebra and numerical methods(e.g., [35]).

A.1 EIGENVALUES AND EIGENVECTORS

An eigenvector, u, of an N×N square matrix, A, and the corresponding eigenvalues,λ, satisfy the equation

(A.1.1) A · u = λ u,

with the understanding that the eigenvector, u, is not null, where a centered dotindicates the regular matrix product. An equivalent statement is

(A.1.2) (A – λ I) · u = 0,

where I is the N × N identity matrix. Requiring that this homogeneous equation hasa nontrivial solution for u, we find that the matrix

(A.1.3) A – λI =

⎡⎢⎢⎢⎢⎣A1,1 – λ A1,2 · · · A1,N–1 A1,NA2,1 A2,2 – λ · · · A2,N–1 A2,N· · · · · · · · · · · · · · ·AN–1,1 AN–1,2 · · · AN–1,N–1 – λ AN–1,NAN,1 AN,2 · · · AN,N–1 AN,N – λ

⎤⎥⎥⎥⎥⎦must be singular, that is, its determinant must be zero. Conversely, the eigenvaluesof a matrix, A, render the diagonally shifted matrix A –λI singular. By definition, aneigenvector belongs to the null space of the matrix A – λI.

If u is an eigenvector corresponding to a certain eigenvalue, then au is also an ei-genvector corresponding to the same eigenvalue, for any real or complex constant a.

259

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However, eigenvectors that arise from one another by multiplication with a nonzeroscalar constant are not distinct.

A real or complex matrix may have real or complex eigenvalues and associatedeigenvectors. A real matrix has either real eigenvalues or pairs of complex conjugateeigenvalues. A real and symmetric matrix, and more generally a Hermitian complexmatrix, has only real eigenvalues. If a matrix is real, an eigenvector correspondingto a real eigenvalue must be real if the eigenvalue is not repeated or complex if theeigenvalue is repeated, whereas an eigenvector corresponding to complex eigenval-ues is necessarily complex. If a matrix is complex, an eigenvector corresponding toa real eigenvalue is necessarily complex.

A.2 THE CHARACTERISTIC POLYNOMIAL

Expressing the determinant of the shifted N×N matrix,A–λ I, in terms of the cofac-tors, we obtain an Nth-degree polynomial with respect to λ, called the characteristicpolynomial of the matrix A:

(A.2.1) PN(λ) = det(A – λ I),

where det denotes the determinant.Monitoring the first three highest powers of λ in the Laplace expansion of the

determinant of the matrix A–λ I, and noting that the constant term is the determinantof A, we find that the characteristic polynomial takes the form

(A.2.2) PN(λ) = (–λ)N + c1 (–λ)N–1 + · · · + cm (–λ)N–m + · · · + cN ,

where

(A.2.3)

c1 = trace(A) ≡ A1,1 + A2,2 + · · · + AN,N ,

c2 =N∑i=1

i–1∑j=1

(Ai,i Aj,j – Ai,j Aj,i),

cN = det(A).

When N = 2, we have c3 = det(A) and the characteristic polynomial is

(A.2.4) P2(λ) = λ2 – (A1,1 + A2,2)λ + (A1,1A2,2 – A1,2A2,1).

When N = 3, we have c4 = det(A) and the characteristic polynomial is

P3(λ) = – λ3 + (A1,1 + A2,2 + A3,3) λ2

–[(A1,1A2,2 – A1,2A2,1) + (A2,2A3,3 – A2,3A3,2) + (A3,3A1,1 – A3,1A1,3)

+ A1,1 (A2,2A3,3 – A2,3A3,2) – A2,1 (A1,2A3,3 – A1,3A3,2)

+ A3,1 (A1,2A2,3 – A1,3A2,2).(A.2.5)

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Eigenvalues of Matr ices / / 261

Algorithms for the programmable computation of the coefficients, ci, for arbitrarypolynomials are available.

We have demonstrated that computing the eigenvalues of a matrix is equivalentto finding the roots of its characteristic polynomial satisfying

(A.2.6) PN(λ) = 0.

Since an Nth-degree polynomial has precisely N roots, an N×N matrix is guaranteedto have exactly N real or complex eigenvalues, λ1,λ2, . . . ,λN . If an eigenvalue, λi,is repeated m times, its algebraic multiplicity is m, meaning that

PN(λi) = 0, P′N(λi) = 0, . . . ,

(A.2.7) P(m–1)N (λi) = 0, P(m)N (λi) = 0,

where P(k)N denotes the kth derivative. Since the coefficients of the characteristic pol-ynomial associated with a real matrix are real, the eigenvalues must be real or appearin pairs of complex conjugates.

Spectrum and Spectral RadiusThe set of all eigenvalues of a matrix is the spectrum of eigenvalues of the matrix.The maximum of the norm of all real and complex eigenvalues is the spectral radiusof the matrix,

(A.2.8) ρ ≡ maxi

|λi|.

The spectral radius of a matrix is an important diagnostic of certain importantproperties of the matrix regarded as a engine that drives a linear map.

Diagonal and Triangular MatricesThe characteristic polynomial of a diagonal or triangular matrix, A, takes the form

(A.2.9) PN (λ) = (A1,1 – λ)(A2,2 – λ) · · · (AN,N – λ),

which shows that the eigenvalues are equal to the diagonal elements. A repeateddiagonal element reveals a multiple eigenvalue. For example, the N × N identitymatrix has a single eigenvalue equal to unity with algebraic multiplicity m = N.

A.2.1 Eigenvalues, Trace, and the Determinant

The characteristic polynomial can be expressed in an alternative form in terms of itsroots,

(A.2.10) PN(λ) = (λ1 – λ) (λ2 – λ) · · · (λN–1 – λ) (λN – λ).

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In the case of a diagonal or triangular matrix, this expression is equivalent to thatshown in (A.2.9). Expanding the product on the right-hand side of (A.2.10), we findthat

(A.2.11)PN(λ) = (–λ)N + (–λ)N–1 (λ1 + λ2 + · · · + λN) + · · · + cm (–λ)N–m

+ · · · + λ1 λ2 · · ·λN–1 λN .

Comparing the right-hand side of this equation with the right-hand side of (A.2.11),we derive expressions for the trace and determinant in terms of the eigenvalues:

(A.2.12) trace(A) = λ1 + λ2 + · · · + λN , det(A) = λ1 λ2 · · · λN .

Thus, if one eigenvalue is zero, the determinant is also zero and the matrix is singular.

A.2.2 Powers, Inverse, and Functions of a Matrix

Multiplying both sides of the definition A · u = λu by A, we find that

(A.2.13) A2 · u = A · (A · u) = λ (A · u) = λ2u,

which shows that λ2 is an eigenvalue of the matrix A2 with corresponding eigenvec-tor u. Working in a similar fashion, we find that λk is an eigenvalue of the matrix Ak

with corresponding eigenvector u, for any positive integer exponent, k.Multiplying both sides of the definition A · u = λu by the inverse matrix A–1, we

find that u = λ (A–1 · u), and then

(A.2.14) A–1 · u =1

λu.

Thus, 1/λ is an eigenvalue of the inverse matrix, A–1, with corresponding eigenvec-tor u.

Working in a similar fashion, we find that, if Q(x) is an arbitrary polynomial,then Q(λ) is an eigenvalue of the matrix Q(A) with corresponding eigenvector u. IfQ(x) and R(x) are two arbitrary polynomials, then Q(λ)/R(λ) is an eigenvalue of thematrix R–1(A) · Q(A) with corresponding eigenvector u. To show this, we observethat R(λ)Q(A) · u = Q(λ)R(A) · u.

A.2.3 Hermitian Matrices

By definition, a Hermitian matrix is equal to the complex conjugate of its transpose.Hermitian matrices, and their inclusive real and symmetric matrices, have real eigen-values. To show this, we take the complex conjugate of the definition A · u = λu,finding that A∗

ij u∗j = λ

∗u∗i , where summation is implied over the repeated index j, and

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Eigenvalues of Matr ices / / 263

an asterisk denotes the complex conjugate. Because the matrix A is assumed to beHermitian, A∗

ij = Aji and thus Aji u∗j = λ∗ u∗

i . Taking the inner product of both sideswith u, we find that

(A.2.15) ui Aji u∗j = λ

∗ui u∗i , or λ uj u

∗j = λ

∗ui u∗i ,

where summation is implied over the repeated index i. The last equation requires thatλ = λ∗, which guarantees that λ is real.

Consider anN×N Hermitian matrix,A. If the scalar x∗i Aijxj is real and positive forany N-dimensional vector x, then the matrix A is called positive definite. Identifyingx with an eigenvector, we find that u∗

i Aijuj = λ u∗i ui > 0. Since u∗ · u is real and

positive, the eigenvalue, λ, must also be real and positive. We conclude that a positivedefinite Hermitian matrix has real and positive eigenvalues.

A.2.4 Diagonal Matrix of Eigenvalues

It is useful to introduce a diagonal matrix, �, whose diagonal entries are the Neigenvalues of a matrix, A:

(A.2.16) � =

⎡⎢⎢⎢⎢⎢⎣λ1 0 · · · 0 00 λ2 · · · 0 0...

.... . .

......

0 0 · · · λN–1 00 0 · · · 0 λN

⎤⎥⎥⎥⎥⎥⎦ .

Note that some or all of the eigenvalues may be the same.Next, we consider the characteristic polynomial of the matrixA, given in (A.2.1),

replace λ with � and unity with the N × N identity matrix, I, and obtain the matrixpolynomial

(A.2.17) PN(�) = (λ1I –�) · (λ2I –�) · · · (λNI –�).We note that the ith column of the matrix enclosed by the ith set of parentheses onthe right-hand side is zero for i = 1, . . . ,N, and we carry out the multiplications toobtain

(A.2.18) PN(�) = 0,

which shows that the diagonal matrix of eigenvalues is a root of the characteristicpolynomial.

A.3 EIGENVECTORS AND PRINCIPAL VECTORS

If the eigenvalues of a matrix are available, the eigenvectors can be found by solvingthe homogeneous linear system (A.1.2). For each eigenvalue, the linear system has

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multiple solutions reflecting the arbitrary length of the corresponding eigenvector.However, this degree of freedom can be removed by imposing a suitable constraint.For example, we may specify the value of one eigenvector component, solve for therest of the components, and then scale the eigenvector so that the magnitude of onechosen component or the length of the eigenvector is equal to unity.

A.3.1 Properties of Eigenvectors

Eigenvectors corresponding to distinct eigenvalues are linearly independent. To showthis, we express one eigenvector as a linear combination of all other eigenvectors,multiply the linear expansion by the matrix A, use the definition of the eigenvectors,compare the resulting equation with the original expansion of the eigenvector, andfind that the eigenvector must be the null vector, which is a contradiction.

If a matrix has N distinct eigenvalues, it is guaranteed to have N linearly inde-pendent eigenvectors that form a base of the N-dimensional space. Any vector canbe expressed as a linear combination of the eigenvectors.

If one or more eigenvalues appear multiple times, we may not be able to find Nlinearly independent eigenvectors. The number of eigenvectors, k, corresponding to aparticular eigenvalue of algebraic multiplicity m, is the geometric multiplicity of theeigenvalue. Since equation (A.1.2) has at least one family of solutions, the geometricmultiplicity satisfies the inequality 1 ≤ k ≤ m.

Hermitian matrices are guaranteed to have N linearly independent and mutuallyorthogonal eigenvectors, even in the case of multiple eigenvalues. The proof relieson the existence of the Schur normal (e.g. [35]).

Two different matrices may have the same set of linearly independent eigen-vectors. For example, two Hermitian matrices that commute with respect tomultiplication share eigenvectors but not necessarily eigenvalues.

A.3.2 Left Eigenvectors

The determinant, and therefore the characteristic polynomial and eigenvalues of amatrix, A, are the same as those of its transpose, AT . However, unless the matrix issymmetric, the eigenvectors are different. The eigenvectors of the transpose, AT, arealso called the left eigenvectors of A.

An eigenvector of AT corresponding to an eigenvalue λ1 is orthogonal to aneigenvector of A corresponding to a different eigenvalue, λ2. To show this, we for-mulate the inner product of both sides of the definition, A · u = λ2u, with the lefteigenvector, v, and find that viAijuj = λ2viui, where summation is implied over therepeated indices i and j. Substituting the definition viAij = λ1vj and rearranging, weobtain (λ1 – λ2) u · v = 0, which shows that u · v = 0.

The number of linearly independent eigenvectors of a matrix and its transposecorresponding to a particular multiple eigenvalue is the same. A matrix and its

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Eigenvalues of Matr ices / / 265

transpose have identical eigenvalues and the same number of linearly independenteigenvectors.

A.3.3 Matrix of Eigenvectors

If an N×N matrix, A, has N eigenvectors, u(i), its transpose also has N eigenvectors,v(i). It is useful to arrange the first set of eigenvectors at the columns of the matrix

(A.3.1) U =

⎡⎢⎣ ↑ ↑ ↑ ↑ ↑u(1) u(2)

... u(N–1) u(N)

↓ ↓ ↓ ↓ ↓

⎤⎥⎦ ,

and the second set of eigenvectors at the columns of the matrix

(A.3.2) V =

⎡⎢⎣ ↑ ↑ ↑ ↑ ↑v(1) v(2)

... v(N–1) v(N)

↓ ↓ ↓ ↓ ↓

⎤⎥⎦ .

Next, we normalize the eigenvectors so that corresponding pairs satisfy the condition

(A.3.3) v(i) · u(i) = 1.

Subject to these definitions,

(A.3.4) VT · U = I, UT · V = I,

which shows that

(A.3.5) U–1 = VT , V–1 = UT .

The collection, u(i), and the collection, v(i), provide us with two mutually orthogonal(biorthonormal) sets.

Symmetric MatricesSince the eigenvalues and eigenvectors of a symmetric matrix and its transposeare identical, two eigenvectors corresponding to two different eigenvalues areorthogonal. Consequently,

(A.3.6) U = V, U–1 = UT ,

which demonstrates that the matrix of eigenvectors, U, is orthogonal. An N ×N realsymmetric matrix has N real eigenvalues and N real and orthogonal eigenvectors.

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A.3.4 Eigenvalues and Eigenvectors of the Adjoint

The determinant of a matrix, A, is equal to the complex conjugate of the determinantof its adjoint, denoted by the superscript A:

(A.3.7) AA ≡ A∗T .

Thus, the characteristic polynomial, and therefore the eigenvalues, are complex con-jugates of those of the adjoint. However, the associated eigenvectors are not generallyassociated by a simple relationship.

An eigenvector of AA corresponding to an eigenvalue, μ1, call it w, is orthogonalto the complex conjugate of an eigenvector of A corresponding to an eigenvalueλ2, call it u∗, where μ1 = λ∗

2, that is, w · u∗ = 0. This property follows from thebiorthogonality of the eigenvectors of a matrix and its transpose discussed earlier inthis section.

Let us assume that AA has N eigenvectors, w(i), arranged at the columns of amatrix,W. Moreover, let us assume that the two sets of eigenvectors w(i) and u(i) arenormalized so that

(A.3.8) w(i) · u(i)∗ = 1.

By construction, we have

(A.3.9) U–1 = WA, W–1 = UA,

in agreement with (A.3.5). If the matrix A is Hermitian, thenW = U and U–1 = UA.

A.3.5 Eigenvalues of Positive Definite Hermitian Matrices

We have seen that a positive definite Hermitian matrix, A, has real and positiveeigenvalues. Conversely, if all eigenvalues of a Hermitian matrix are positive, thematrix is positive definite. To show this, we express an arbitrary vector, x, as a linearcombination of the eigenvectors:

(A.3.10) x = c1u(1) + · · · + cNu(N).

Multiplying both sides by A, we find that

(A.3.11) A · x = c1λ1 u(1) + · · · + cN λNu(N).

Next, we compute the scalar

(A.3.12) x∗ · A · x =(c∗1 u(1)

∗+ · · · + c∗N u(N)

∗) ·(c1 λ1 u(1) + · · · + cN λN u(N)

).

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Eigenvalues of Matr ices / / 267

Using the orthogonality property, u(i)∗ · u(j) = 0 for i = j, we obtain

(A.3.13) x∗ · A · x = c∗1 c1 λ1 u(1)∗ · u(1) + · · · + c∗N cN λN u(N)

∗ · u(N)

or

(A.3.14) x∗ · A · x = |c1|2 λ1 |u(1)|2 + · · · + |cN |

2 λN |u(N)|2,

which shows that, if all eigenvalues are positive, x∗ · A · x is guaranteed to also bepositive.

A.4 CIRCULANT MATRICES

Each row of a circulant matrix derives from the previous row by shifting each elementto the right by one place, and then returning the last element to the first place. Byconstruction, all elements along any super- or subdiagonal line of a circulant matrixare the same.

A 2 × 2 circulant matrix, a 3 × 3 circulant matrix, and a 4 × 4 circulant matrixare shown below,

(A.4.1) A =

[a bb a

], A =

⎡⎣ a b cc a bb c a

⎤⎦ , A =

⎡⎢⎢⎣a b c dd a b cc d a bb c d a

⎤⎥⎥⎦ ,

where a, b, c, and d are arbitrary elements.Circulant matrices arise in the mathematical modeling of problems involving

temporal or spatial periodicity. It is remarkable that the eigenvalues and eigenvectorsof an arbitrary circulant matrix can be found explicitly in closed form.

Let A be an N × N circulant matrix and qm be an Nth complex root of unity,satisfying qNm = 1, given by

(A.4.2) qm = exp[ (m – 1)k i ],

for i = 1, . . . ,N, where k = 2π /N and i is the imaginary unit, i2 = –1. Directsubstitution shows that the eigenvalues of A are given by

(A.4.3) λm = A1,1 + A1,2 qm + A1,3 q2m + · · · + A1,N qN–1m ,

and the corresponding eigenvectors are

(A.4.4) u(m) =[1, qm, q

2m, . . . , q

N–1m

]T ,

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where m = 1, . . . ,N. The first eigenvalue, λ1, is equal to the sum of the elements ineach row. The corresponding eigenvector is filled with ones:

(A.4.5) u(1) = [ 1, 1 , . . . , 1 ]T .

When N = 2, we obtain q1 = 1 and q2 = –1, yielding

(A.4.6) λ1 = a + b, λ2 = a – b,

which are the eigenvalues of the first matrix in (A.4.1).

A.5 BLOCK CIRCULANT MATRICES

A block circulant matrix consists of N repeated matrix blocks in the arrangement ofcirculant matrices:

(A.5.1) A =

⎡⎢⎢⎢⎢⎢⎣A(1) A(2) · · · A(N–1) A(N)

A(N) A(1) · · · A(N–2) A(N–1)

......

. . ....

...A(3) A(4) · · · A(N–1) A(1)

A(2) A(3) · · · A(N–1) A(1)

⎤⎥⎥⎥⎥⎥⎦ ,

where A(i) are square matrices with the same dimensions, M ×M. Each row of thismatrix derives from the previous row by shifting each block to the right by one placeand then bringing the last block to the first place.

Consider the following square M × M matrices defined in terms of the blockmatrices, A(i):

(A.5.2) B(m) = A(1) + qm A(2) + q2mA(2) + · · · + qN–1m A(N)

for m = 1, . . . ,N. It can be shown that the determinant of the matrix A is the productof the determinants of the matrices B(m), the characteristic polynomial of A is theproduct of the characteristic polynomials of B(m), and the spectrum of eigenvalues ofA is the union of the spectra of eigenvalues of B(m) [12].

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APPENDIX B

THE SHERMAN–MORRISONANDWOODBURY FORMULAS

TheWoodbury and Sherman–Morrison formulas allow us to compute the inverse of amatrix that is perturbed with respect to a reference matrix whose inverse is availablein an explicit or readily computable form (e.g., [17], p. 123).

B.1 THE WOODBURY FORMULA

Woodbury’s formula relates the inverse of a perturbed N×N matrix, B, to the inverseof an unperturbed N × N matrix, A. The two matrices are related by

(B.1.1) A = B + U · VT ,

where U and V are two N × K matrices, K ≥ 1 is an arbitrary dimension, the su-perscript T denotes the matrix transpose, and a centered dot denotes the usual matrixproduct. The inverse of the perturbed matrix is

(B.1.2) A–1 = B–1 · [ I – U · (IK +G)–1 · VT · B–1 ],where I is the N × N identity matrix, IK is the K × K identity matrix, and

(B.1.3) G ≡ VT · B–1 · U

is a K × K matrix.

Direct ProofWoodbury’s formula can be proved by direct substitution, invoking the definition ofthe matrix inverse. Using (B.1.1) and (B.1.2), we compute

(B.1.4) A · A–1 = (B + U · VT ) · B–1 · [ I – U · (IK +G)–1 · VT · B–1 ].269

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Carrying out the multiplications and invoking the definition of the matrix G, weobtain

(B.1.5) A · A–1 = I + U · VT · B–1 – U · (IK +G) · (IK +G)–1 · VT · B–1,

where the first two terms on the right-hand side correspond to the matrix I inside thesquare brackets on the right-hand side of (B.1.4). Carrying out the multiplications,we obtain

(B.1.6) A · A–1 = I,

as required. Other proofs based on block Gauss elimination of LU decomposition areavailable.

Proof by Gauss EliminationBy definition, we have

(B.1.7) (B + U · VT ) · A–1 = I

and thus

(B.1.8) B · A–1 + U · D = I,

where

(B.1.9) D ≡ VT · A–1

is an intermediate K × N matrix. The last two equations can be collected into theblock linear system

(B.1.10)[

B U–VT IK

]·[

A–1

D

]=

[I0

].

Solving the first equation for A–1, we obtain

(B.1.11) A–1 = –B–1 · U · D + B–1.

Substituting this expression into the second equation of (B.1.10), we obtain thereduced system

(B.1.12)[

B U0 IK +G

]·[

A–1

D

]=

[I

VT · B–1

].

From the second equation, we find that

(B.1.13) D =(IK +G

)–1 · VT · B–1.

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Sherman–Morr ison and Woodbury Formulas / / 271

Substituting this expression into the first equation yields the Woodbury formula.The procedure described is the counterpart of the method of Gauss elimination

for solving systems of linear equations. The counterpart of the LU decomposition isthe block decomposition

(B.1.14)[

B U–VT IK

]=

[I 0

–VT · B–1 IK

]·[

B U0 IK +G

].

The determinant of the matrix on the left-hand side is equal to the determinant of thematrixA = B +U·VT . Since the first matrix on the right-hand side is lower triangularwith ones along the diagonal, its determinant is equal to unity. The determinant ofthe second matrix on the right-hand side is equal to the product of the determinantsof the two square matrices along the diagonal, B and IK +G. Taking the determinantof both sides of (B.1.13), recalling that the determinant of the product of two squarematrix is the product of the determinants, and rearranging, we obtain

(B.1.15)det(A)det(B)

= det(IK +G

).

When B is the N × N identity matrix, I, we find that

(B.1.16) det(I + U · VT) = det

(IK + VT · U),

expressing Sylvester’s determinant theorem.

Alternative ProofA third way of proving the Woodbury formula proceeds by applying the generalidentity

(B.1.17) (B – C)–1 = B–1 ·[I +

∞∑n=1

(C · B–1)n]

for the square matrix C = –U · VT , obtaining

(B.1.18) (B + U · VT )–1 = B–1 ·[I +

∞∑n=1

(–U · VT · B–1)n].

Rearranging the sum, we find that

(B.1.19) A–1 = B–1 ·[I – U ·

(IK +

∞∑m=1

(–G)m)

· VT · B–1],

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where m = n – 1. Also applying the general identity

(B.1.20) (IK – C)–1 = IK +∞∑n=1

Cn

for the K × K matrix C = –G, we obtain

(B.1.21) (IK +G)–1 = IK +∞∑n=1

(–G)n,

which completes the proof.

Generalized Woodbury FormulasA generalization of the Woodbury formula (B.1.2) provides us with the inverse ofthe N × N perturbed matrix

(B.1.22) A = B + U ·� · VT ,

where U and V are two N × K matrices and � is an arbitrary nonsingular K × Kmatrix with K ≥ 1. The inverse of the perturbed matrix is

(B.1.23) A–1 = B–1 · [ I – U · (�–1 +G)–1 · VT · B–1 ].Formula (B.1.23) arises from (B.1.2) by replacing UwithU·�. When� is the K×Kidentity matrix we recover the standard Woodbury formula.

Further GeneralizationA further generalization incorporates M deviations of a matrix of interest, A, froman unperturbed matrix, A,

(B.1.24)A = B + U(1) ·�(1) · V(1)T + · · · + U(q) ·�(q) · V(q)T

+ · · · + U(M) ·�(M) · V(M)T ,

where U(q) and V(q) are collections of N × Kq matrices and �(q) are Kq × Kq squarematrices for q = 1, . . . ,M. Let

(B.1.25) K ≡M∑q=1

Kq.

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Sherman–Morr ison and Woodbury Formulas / / 273

The inverse of the perturbed matrix is given in (B.1.23), where I is the N×N identitymatrix,

(B.1.26) U ≡⎡⎢⎣ ↑ ↑ ↑ ↑ ↑

U(1) ... U(q) ... U(M)

↓ ↓ ↓ ↓ ↓

⎤⎥⎦is a N × K matrix,

(B.1.27) VT ≡

⎡⎢⎢⎢⎢⎢⎣← V(1)T →← · · · →← V(q)T →← · · · →← V(M)T →

⎤⎥⎥⎥⎥⎥⎦is a K × N matrix,

(B.1.28) � ≡

⎡⎢⎢⎢⎢⎢⎣�(1) 0 · · · 0 00 �(2) · · · 0 0...

.... . .

......

0 · · · 0 �(M–1) 00 · · · 0 0 �(M)

⎤⎥⎥⎥⎥⎥⎦is a square block-diagonal K × K matrix, and

(B.1.29) G =

⎡⎢⎢⎢⎢⎣V(1)T · B–1 · U(1) V(1)T · B–1 · U(2) · · · V(1)T · B–1 · U(p)

V(2)T · B–1 · U(1) V(2)T · B–1 · U(2) · · · V(2)T · B–1 · U(p)

......

......

V(p)T · B–1 · U(1) V(p)T · B–1 · U(2) · · · V(p)T · B–1 · U(p)

⎤⎥⎥⎥⎥⎦is a K × K matrix [2]. Formula (B.1.23) corresponds to M = 1. To prove thegeneralized formula, we simply observe that

(B.1.30) U(1) ·�(1) · V(1)T + · · · + U(M) ·�(p) · V(M)T = U ·� · VT .

B.2 THE SHERMAN–MORRISON FORMULA

In the particular case where K = 1, the otherwise arbitrary matrices U and V reduceinto N-dimensional column vectors, u and v, and

(B.2.1) A = B + u · vT .

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The matrix G ≡ IK + vT · B–1 · u is scalar and the Woodbury formula reduces to theSherman–Morrison formula,

(B.2.2) A–1 = B–1 · ( I – 1

1 + su · vT · B–1 ),

where

(B.2.3) s ≡ vT · B–1 · uis a scalar (e.g., [4], p. 39). Equation (B.2.4) yields

(B.2.4)det(A)det(B)

= 1 + s.

It will be noted that the Sherman–Morrison formula fails when s = –1, in which caseB–1·u is an eigenvector of the perturbed matrixA corresponding to a zero eigenvalue.

The following Matlab script uses the internal Matlab function inv to verify theSherman–Morrison formula:

B = [1 2 3; 2 3 4; 1 4 5];u = [3 4 9]’;v = [2 3 7]’;A = B+u*v’;invA = inv(A)invB = inv(B);s = v’*invB*u;invA1 = invB-invB*u*v’*invB/(1+s)

A prime denotes the vector or matrix transpose. The output of the code is

invA =7.0000 -1.0000 -2.0000

-18.0000 5.0000 4.00006.2500 -2.0000 -1.2500

invA1 =7.0000 -1.0000 -2.0000

-18.0000 5.0000 4.00006.2500 -2.0000 -1.2500

We observe that the matrix inverses computed directly or by using the Sherman–Morrison formula are identical.

If all matrices involved are scalars, N = 1 and K = 1, the Sherman–Morrisonformula provides us with the identity

(B.2.5)1

b + uv=1

b

(1 –

uv

b + uv

),

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Sherman–Morr ison and Woodbury Formulas / / 275

whose veracity can be readily confirmed.An alternative form of (B.2.2) is

(B.2.6) A–1 =

(I –

1

1 + s�

)· B–1,

where

(B.2.7) � = w · vT , w = B–1 · u, s ≡ wT · v.

The N × N matrix � satisfies the property

(B.2.8) �n = sn�

for any integer, n.If A is the identity matrix, I, we obtain

(B.2.9) (I + u · vT )–1 = I –1

1 + su · vT ,

where s ≡ uT · v.

Generalized Sherman–Morrison FormulaConsider two collections of N-dimensional column vectors, u(q) and v(q) for q =1, . . . ,M, and formulate the perturbed N × N matrix

(B.2.10) A = B + ζ1u(1) · v(1)T + · · · + ζqu(q) · v(q)T + · · · + ζMu(M) · v(M)T ,

where ζq are arbitrary constants for q = 1, . . . ,M. In compact notation, we have

(B.2.11) A = B + U · Z · VT ,

where

(B.2.12) U ≡⎡⎢⎣ ↑ ↑ ↑ ↑ ↑ ↑

u(1)... u(p)

... u(M)

↓ ↓ ↓ ↓ ↓

⎤⎥⎦is an N ×M matrix,

(B.2.13) V ≡⎡⎢⎣ ↑ ↑ ↑ ↑ ↑

v(1)... v(p)

... v(M)

↓ ↓ ↓ ↓ ↓

⎤⎥⎦

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276 / / AN INTRODUCTION TO GRIDS, GRAPHS, AND NETWORKS

is another N ×M matrix, and

(B.2.14) Z ≡

⎡⎢⎢⎢⎢⎢⎢⎢⎣

ζ1 0 · · · 0 0

0 ζ2 · · · 0 0...

.... . .

......

0 · · · 0 ζM–1 0

0 · · · 0 0 ζM

⎤⎥⎥⎥⎥⎥⎥⎥⎦is anM ×M diagonal matrix.

Applying (B.1.23) with � = Z and K = M, we find that the inverse of theperturbed matrix A is given by

(B.2.15) A–1 = B–1 · [ I – U · (Z–1 +G)–1 · VT · B–1 ],

where

(B.2.16) G ≡ VT · B–1 · U

is anM ×M matrix with components

(B.2.17) Gij = v(i)l B

–1lmu

(j)m = B–1

lm :(v(i) · u(j)T

)= v(i)

T · w(j),

summation is implied over the repeated indices l and m, and the vector w(j) satisfiesthe linear system

(B.2.18) B · w(j) = u(j)

for j = 1, . . . ,M. We may set B–1 · U = W and obtain G ≡ VT · W, where

(B.2.19) W =

⎡⎢⎢⎣↑ ↑ ↑ ↑ ↑

w(1) ... w(p) ... w(M)

↓ ↓ ↓ ↓ ↓

⎤⎥⎥⎦is an N ×M matrix.

The following Matlab script confirms the generalized Sherman–Morrison for-mula for N = 3 andM = 2:

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Sherman–Morr ison and Woodbury Formulas / / 277

B = [1 2 3;2 3 4;1 4 5];

z1 = 1.4;u1 = [3 4 9]’;v1 = [2 3 7]’;

z2 = 3.4;u2 = [1 2 3]’;v2 = [6 5 4]’;

U(:,1) = u1;U(:,2) = u2;

V(:,1) = v1;V(:,2) = v2;

w1 = u1’/B’;w2 = u2’/B’;

G(1,1) = v1’*w1’; G(1,2) = v1’*w2’;G(2,1) = v2’*w1’; G(2,2) = v2’*w2’;

Z(1,1) = c1; Z(1,2) = 0.0;Z(2,1) = 0.0; Z(2,2) = c2;

A = B + z1*u1*v1’ + z2*u2*v2’;invA = inv(A)invB = inv(B);invA1 = invB - invB*U*inv(inv(Z)+G)*V’*invB

The output of the code is

invA =4.8017 0.2443 -1.7767-7.6920 -0.1932 2.74202.1952 -0.0256 -0.7327

invA1 =4.8017 0.2443 -1.7767-7.6920 -0.1932 2.74202.1952 -0.0256 -0.7327

The matrix inverse computed directly is the same as that computed by the Sherman–Morrison formula.

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INDEX

addition of a link, 49adjacency matrix, 18, 26periodic, 21weighed, 136

admittance matrix, 138Archimedean lattice, 53

bathroom tile lattice, 55bcc lattice, 59, 124Green’s function, 209

biharmonic operator, 155biorthonormal sets, 265bounce lattice, 56boundary conditionDirichlet, 3Neumann, 6periodic, 13

bow-tie lattice, 57Bravais lattice, 50, 86bridge lattice, 56Brillouin zone, 51

Cartesian grid, 153characteristic polynomial, 260Cheeger’s constant, 39circulant matrix, 21, 267block, 268

clique, 29complement of a graph, 29Laplacian of, 38

complete graph, 29, 34, 171conductance, 130arbitrary, 135matrix, 136scaled, 136

connected graph, 30connectivityalgebraic, 34

list, 19, 30coordination number, 28cross lattice, 55cubic lattice, 58bcc, 59, 124Green’s function, 209

fcc, 59, 126Green’s function, 211

simple, 59, 122Green’s function, 206

degree of a node, 18, 28delta function, 171determinant, 259–261Sylvester theorem, 271

diagonal matrix, 261of eigenvalues, 263

diameter of a graph, 30differential equation, 1partial, 153

digraph, 30directed graph, 30Dirichletboundary condition, 3node, 130

discontiguous network, 172dual lattice, 56

edgelist, 30weight, 136

effective medium theory, 250, 254eigenvalue, 259algebraic multiplicity of, 261

eigenvector, 259, 263left, 264

elliptic integral, 209embedded network, 130, 134, 142, 161

281

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embedding matrix, 131weighed, 142

Euler constant, 189, 196exponential integral, 189

fcc lattice, 59, 126Green’s function, 211

finitedifference method, 2, 153, 216element method, 156

Fourier expansionin one dimension, 22in two dimensions, 76

Gamma function, 209gradient, 3graph, 26complement, 29, 38complete, 29connected, 30diameter, 30directed, 30Laplacian, 17one-dimensional, 16order, 26periodic, 20random, 31size, 26unconnected, 30undirected, 30

Green’s functionbcc lattice, 209fcc lattice, 211free-space, 175, 212hexagonal lattice, 191honeycomb lattice, 200in one dimension, 171in probability theory, 213lattice, 173Moore–Penrose, 164normalized, 163, 167periodic, 173simple cubic lattice, 206square lattice, 177Union Jack modified lattice, 196

Green’s functions, 161grid, 1finite difference, 153finite element, 156

Heaviside function, 133Helmholtz equation, 1, 156Green’s function, 190

Hermitian matrix eigenvalues, 262, 266hexagonalgrid, 155lattice, 86, 150damaged, 255Green’s function, 191

hexagonal lattice, 54honeycombgrid, 155lattice, 98, 176damaged, 251Green’s function, 200

honeycomb lattice, 55

incidence matrixnormalized, 140weighed, 139

innerdisplacement, 98, 110product, 3

isolated network, 130, 134, 142, 164, 223

kagomé lattice, 55, 111Kirchhoffmatrix, 138damaged, 235Green’s function, 190modified, 142normalized, 140properties, 139

spanning-tree theorem, 36kisquadrille lattice, 57Klein bottle, 84Kronecker’s delta, 51

Laplace equation, 1, 156Laplacianfactorization, 2matrix, 32in one dimension, 17modified, 134

normalized, 38operator, 3

lattice, 56Archimedean, 53bathroom tile, 55bcc, 59, 124Green’s function, 209

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INDEX // 283

bounce, 56bow-tie, 57bridge, 56coordination number, 28, 50cross, 55cubic, 58damaged, 242fcc, 59, 126Green’s function, 211

Green’s function, 173in probability theory, 213

hexagonal, 54, 86Green’s function, 191

honeycomb, 55, 98, 176Green’s function, 200

kagomé, 55, 111kisquadrille, 57maple leaf, 56martini, 57modified Union Jack, 93puzzle, 56ruby, 56simple cubic, 59, 122Green’s function, 206

snub hexagonal, 56snub square, 56square, 53, 67Green’s function, 177

square octagon, 55star, 55tetrakis, 57triangular, 54Union Jack, 57Union Jack modified Green’s function, 196

Laves lattice, 56left eigenvectors, 264linearsystem, 134transport, 132

linkaddition, 46, 49removal, 46weight, 136

Möbius strip, 79, 149maple leaf lattice, 56martini lattice, 57matrixblock circulant, 268circulant, 267positive definite, 263

positive semidefinite, 12, 15power, 262

Moore–PenroseGreen’s function, 164inverse, 165

multiple eigenvalue, 261multiplicityalgebraic, 261geometric, 264

nearest neighbor, 28neighborhood, 28networkdamaged, 234discontiguous, 172embedded, 130, 134, 142, 161fabricated, 41isolated, 130, 134, 142, 164, 223reinforced, 240transport, 130

Neumann boundary condition, 6Newton’s law, 79nodeclustering, 32degree, 18, 28weighed, 137

strength, 137nonlinear transport, 133

oriented incidence matrix, 30normalized, 38weighed, 139

pairwise resistance, 220mean, 228

percolation threshold, 246bond or link, 59site or node, 61

periodicboundary conditions, 13graph, 20, 69

Poiseuille law, 132Poisson equation, 1, 216polynomial, characteristic, 260positivedefinite matrix eigenvalues, 263semidefinite matrix, 12, 15, 140, 143

puzzle lattice, 56

randomgraph, 31walk, 213, 227

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resistancedistance, 228effective, 228pairwise, 220

ruby lattice, 56

Sherman–Morrison formula, 269, 273simple cubic lattice, 59, 122Green’s function, 206

snubhexagonal lattice, 56square lattice, 56

spanning tree, 36spectralexpansion, 36partitioning, 36radius, 261

spectrum of a matrix, 261square lattice, 53, 67, 145damaged, 247Green’s function, 177

square octagon lattice, 55star lattice, 55structure function, 213Sylvester’s determinant theorem, 271symmetric matrix eigenvalues, 262

tetrakis lattice, 57Toeplitz matrix, 4trace of a matrix, 261transportlinear, 132nonlinear, 133

tree, 31spanning, 36

triangularlattice, 54matrix, 261

truss, 26

unconnected graph, 30undirected graph, 30Union Jack lattice, 57, 150modified, 93Green’s function, 196

weight of an edge, 136Weyl’s theorem, 140Wigner–Seitz cell, 51Woodbury formula, 269