Complex and quantum graphs and their applications · Complex and quantum graphs and their applications Mario Arioli Combinatorial Graphs and Matrix representation I Given how can

Complex and quantum graphs and theirapplications

Mario Arioli

Visiting Professor at BMS

July 8th, 2014

Complex and quantum graphs and their applications Mario Arioli

Overview

I Motivations and Modelling (and a little bit of History)

I Complex Network

I Graphs and Random Graphs

I Metric graphs and quantum graphs

I Self-adjoint Hamiltonians and boundary conditions

I Modelling again

I Waves and eigenvalues problems

I Parabolic and time dependent problems

I Numerical issues and domain decomposition

I Open problems i.e.

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Overview

I Motivations and Modelling (and a little bit of History)

I Complex Network

I Graphs and Random Graphs

I Metric graphs and quantum graphs

I Self-adjoint Hamiltonians and boundary conditions

I Modelling again

I Waves and eigenvalues problems

I Parabolic and time dependent problems

I Numerical issues and domain decomposition

I Open problems i.e. The things I have not understood yet!!!

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Example: Naphthalene

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Example: Polystyrene

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Example: Graphene

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Example: GrapheneExample: Carbon nanostructures

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Example: spectral clustering

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Example: Human body

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Complex graphs (http://www.opte.org)

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MORE Complex graphs (http://www.opte.org)

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MORE Complex graphs (http://www.wikipidia)

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MORE Complex graphs (http://http://www.newscientist.com)

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Modelling (examples)

A fat graph (le length of edge e)

I Di�cult to have a decenttriangulation of the fat domain!

I Irregular solution (corners)

I Given an Hamiltonian on the fatgraph, to what does it convergewhen � ! 0?

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Graph (e edge and v vertex)




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Combinatorial Graphs

I A Combinatorial Graph � is defined by a set V = {vj}Nj+1 of

vertices and a set E = {ek}Mk=1 of edges connecting thevertices that can be finite or countably infinite. Each edge ecan be identified by the couple of vertices that it connects(e = (vj1 , vj2)).

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Combinatorial Graphs

I A Combinatorial Graph � is defined by a set V = {vj}Nj+1 of

vertices and a set E = {ek}Mk=1 of edges connecting thevertices that can be finite or countably infinite. Each edge ecan be identified by the couple of vertices that it connects(e = (vj1 , vj2)). NO GEOMETRY

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Combinatorial Graphs and Matrix representationI Given � how can I represent it using matrices?

1 2 3 4 5 6 7 8

0

1

2

3

4

5

I Adjacency matrix: symmetric vertex to vertex structureAij = 1 i↵ exists edge (vi , vj).

I Incidence Matrix: E is a vertex to edge matrix. Each columncorresponds to an edge and has two non zero entries 1 and�1 corresponding to first vertex and the second vertex (thesign is arbitrary and will be uninfluential for our purposes).

I There are strong relations between A and E:I EET = L� is the Laplacian of the graph andI A = diag(L�)� L�. The diag(L�)ii =

P

j Aij entries are thenumber of connections that each vertex vi has, i.e. its degreedeg(vi ).

I E can be interpreted as a discrete divergence and ET as adiscrete gradient. Moreover, Ker(ET ) is the subset of thevectors ↵e = ↵(1, 1, . . . , 1)T .

I E is also a ”Totally unimodular“ matrix (discrete optimizationjargon)

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Combinatorial Graphs and Matrix representationI Given � how can I represent it using matrices?I Adjacency matrix: symmetric vertex to vertex structure

Aij = 1 i↵ exists edge (vi , vj).

I Incidence Matrix: E is a vertex to edge matrix. Each columncorresponds to an edge and has two non zero entries 1 and�1 corresponding to first vertex and the second vertex (thesign is arbitrary and will be uninfluential for our purposes).


P




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Aij = 1 i↵ exists edge (vi , vj).I Incidence Matrix: E is a vertex to edge matrix. Each column

corresponds to an edge and has two non zero entries 1 and�1 corresponding to first vertex and the second vertex (thesign is arbitrary and will be uninfluential for our purposes).


P




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P




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P




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P




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Complex Graph? What is it!

The concept of “complex graph” (or network) is somewhat fuzzy.We are not aware of any formal definition able to sharplydistinguish within all sparse graphs what is complex from what isnot. Some concrete examples, such us the Web graph, areobviously very complex even though they can be described by thedeceptively simple concept of a (very sparse) adjacency matrix.

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In the study of unweighted graphs, the lengths of the edges areassumed to be all equal to 1. The diameter of a graph is defined asthe maximum length of all shortest paths between any pair ofnodes in �. In other terms,

diam (�) := maxu,v2V

d(u, v) ,

where d(u, v) stands for the distance between vertices u and v.

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Most real-world complex graphs are characterized by the so-calledsmall-world property, that is, the diameter of the graph is a veryslowly growing function of the total number N of vertices. Formany real graphs the diameter behaves approximately as logN oreven log logN as N ! 1. For example, it is not unusual for agraph � with N ⇡ 106 vertices to have diameter diam(�) ⇡ 10 (oreven less).

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Random properties of complex networks

Complex graphs are also usually characterized by highly skeweddegree distributions displaying fat tails. We will characterise degreedistribution in an undirected graph via the probability densityfunction p(k) that describes the number vertices in the graph withdegree k .

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For many real-world complex graphs, p(k) follows a power law:

p(k) _ k�� , � � 1.

By elementary calculus, we have that E (k), the expected value (ormean) of the degree, behaves as

E (k) ⇡ C

� � 2

⇣

1� N2��⌘

.

Therefore, in order to have sparse graphs (sparse adjacencymatrices and sparse matrices in general), we need to take � > 2 sothat E (k) remains bounded by a constant as N ! 1. (The value� = 2 is a critical value, for which the mean degree growslogarithmically with N; for 1 � < 2, the mean degree grows likea fractional power of N and the graph becomes dense in the limitof large N.)

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Many complex graphs follow a power law degree distribution with2 < � 3. and we will work under this assumption. Note that, forvalues of � between 2 and 3, the corresponding value of thevariance V (k) will approximated by

V (k) ⇡ C1

� � 3

�

1� N3��

which for N % 1 diverges.

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We observe that the fairly regular graphs obtained by meshing 2Dor 3D domains in the approximation of PDEs do not follow apower law, owing to the requirement that the triangles or thetetrahedra must preserve the minimum angle condition whichimposes a bound the maximum number of edges incident in avertex, independent of N.

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E. Estrada, The Structure of Complex Network, Oxford UniversityPress, 2011,A. Taylor and D. Higham, CONTEST: A Controllable Test MatrixToolbox for MATLAB, ACM Transactions on MathematicalSoftware, 35 (2009)

http://www.maths.strath.ac.uk/research/groups/numerical_analysis/contest

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Degree, Diameter, CentralityA from pref.m (CONTEST); Scale free random graph.

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Degree, Diameter, Centrality

A from pref.m (CONTEST); Scale free random graph.DIAMETER OF A = 5.

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Degree, Diameter, Centrality

A from pref.m (CONTEST); Scale free random graph.DIAMETER OF A = 5.Centrality: diag(exp(A))

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Degree, Diameter, CentralityA from pref.m (CONTEST); Scale free random graph.DIAMETER OF A = 5.Centrality: diag(exp(A))

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Degree, Diameter, CentralityA from pref.m (CONTEST); Scale free random graph.DIAMETER OF A = 5.Centrality: diag(exp(A))

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Why this may be not enough?

The 1318 transnational corporations that form the core of theeconomy in 2007. Superconnected companies are red 147, veryconnected companies are yellow. The size of the dot representsrevenue (Image: PLoS One) (New Scientist)

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The network of global corporate control S. Vitali, J. B. Glattfelder,and S. Battiston ETH Zurich, Kreuzplatz 5, 8032 Zurich,Switzerland.”In e↵ect, less than 1 per cent of the companies were able tocontrol 40 per cent of the entire network,” says Glattfelder.

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The network of global corporate control S. Vitali, J. B. Glattfelder,and S. Battiston ETH Zurich, Kreuzplatz 5, 8032 Zurich,Switzerland.”In e↵ect, less than 1 per cent of the companies were able tocontrol 40 per cent of the entire network,” says Glattfelder.One of the 147 (in position 34) was Lehman Brothers Holdings Inc.

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Combinatorial and metric Graphs

I A Combinatorial Graph � is defined by a set V = {vj} ofvertices and a set E = {ek} of edges connecting the verticesthat can be finite or countably infinite. Each edge e can beidentified by the couple of vertices that it connects(e = (vj1 , vj2)).

I A graph � is a ”Metric Graph ” if at each edge e is assigned alength le 2 (0,1) and a measure (normally the Lebesgueone). Each edge can be assimilated to a finite or infinitesegment of the real line (0, le) 2 R, with the naturalcoordinate se .

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I A Combinatorial Graph � is defined by a set V = {vj} ofvertices and a set E = {ek} of edges connecting the verticesthat can be finite or countably infinite. Each edge e can beidentified by the couple of vertices that it connects(e = (vj1 , vj2)). NO GEOMETRY


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I A Combinatorial Graph � is defined by a set V = {vj} ofvertices and a set E = {ek} of edges connecting the verticesthat can be finite or countably infinite. Each edge e can beidentified by the couple of vertices that it connects(e = (vj1 , vj2)).


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Remarks

I We can remove vertices of degree 2 after we fuse the 2 edgesinto one

I The graph � is a topological manifold (or a 1D simplicialcomplex) having singularities at the vertices, i.e. it is NOT adi↵erentiable manifold.

I � is provided with a global metric and the distance betweentwo points (not necessarily vertices) is the length of theshortest path between them. Thus, the points on � are thevertices and all the points on the edges. The Lebesgue’smeasure is well defined on all of � for finite graphs.

I � is NOT necessarily embedded in a Euclidean space (Rn) J.

Friedman, J-P. Tillich, Calculus on Graphs http://arxiv.org/abs/cs/0408028v1 and M.I. OstrovskiiSOBOLEV SPACES ON GRAPHS Quaestiones Mathematicae 28(2005)

Berkolaiko and Kuchment, Introduction to Quantum Graphs, AMS186, 2012

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RemarksI We can remove vertices of degree 2 after we fuse the 2 edges

into oneI The graph � is a topological manifold (or a 1D simplicial

complex) having singularities at the vertices, i.e. it is NOT adi↵erentiable manifold. Let E+ = 1

2

�

|E|+ E�

and

E� = 12

�

|E|� E�

. Let Z 2 IRN⇥n with N number of verticesand n number of co-ordinates for each vertex therepresentation of of the vertices in �. Then

Z(s) = (E+)TZ� sETZ s 2 [0, 1]

is the 1-D simplicial complex and Z(1) =�

E��TZ.





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Remarks







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Remarks







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Conditions on infinite graphs

Condition A An edge of infinite length has only one vertex. It is aray starting from a vertex.

Condition B For any positive number r and any vertex v there isonly a finite number of vertices u at a distance lessthan r from v.

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Conditions on infinite graphs

Condition A An edge of infinite length has only one vertex. It is aray starting from a vertex.

Condition B For any positive number r and any vertex v there isonly a finite number of vertices u at a distance lessthan r from v.

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Hilbert spaces

Definition-L2(�): (L2e =�

f |R

e f2ds < 1

)

L2(�) =M

e2EL2(e)

f (s) 2 L2(�) i↵ ||f ||2L2(�) =X

e2E||f ||2L2(e) < 1

Definition-H1(�) (Sobolev space) :(H1e =

⇢

f |R

e

�df

ds

�2ds < 1

�

)

H1(�) =�

M

e2EH1(e)

�

\ C 0(�)

f (s) 2 H1(�) i↵ ||f ||2H1(�) =X

e2E||f ||2H1(e) < 1

C 0(�) space of continuous functions on �.

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Quantum Graphs

Let h an operator (Hamiltonian) defined on H1(�).

A Quantum Graph is a metric graph where anHamiltonian h and boundary conditions that assure h isself-adjoint are defined.see P. Kuchment Quantum Graphs I WavesRandomMedia 14 (2004) and II J. Phys. A: Math. Gen. 38 (2005)


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Hamiltonian

Operators (s denotes the coordinate on an edge)

Second derivative f ! �d2f

ds2

Schrodinger f ! �d2f

ds2+ V (s)f

Magnetic Schrodinger f !�1

i

d

ds� A(s)

�2f + V (s)f

Others: pseudo-di↵erential, higher order derivative, etc...

A natural condition is to assume that f (e) 2 H2(e), 8e 2 E .

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Hamiltonian



ds2


ds2+ V (s)f


i

d

ds� A(s)

�2f + V (s)f



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Hamiltonian



ds2


ds2+ V (s)f


i

d

ds� A(s)

�2f + V (s)f



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Hamiltonian



ds2


ds2+ V (s)f


i

d

ds� A(s)

�2f + V (s)f



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Hamiltonian


Second derivative We will focus on f ! �d2f

ds2


ds2+ V (s)f


i

d

ds� A(s)

�2f + V (s)f



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Boundary conditions

We are interested in local conditions at the vertices. Let dv

be thedegree of vertex v. For functions fj 2 H2(e

v

) on the edgesconnected at v, we expect boundary conditions involving the valuesof the functions and their directional derivative taken in theoutgoing directions at the vertex v:

Av

F + Bv

F 0 = 0

where A 2 Rdv

⇥dv and B 2 Rd

v

⇥dv , F = (f1(v), . . . , fd

v

(v) andF 0 = (f 01(v), . . . , f

0dv

(v).

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Boundary conditions

We are interested in local conditions at the vertices. Let dv

be thedegree of vertex v. For functions fj 2 H2(e

v

) on the edgesconnected at v, we expect boundary conditions involving the valuesof the functions and their directional derivative taken in theoutgoing directions at the vertex v:

Av

F + Bv

F 0 = 0

where A 2 Rdv

⇥dv and B 2 Rd

v

⇥dv , F = (f1(v), . . . , fd

v

(v) andF 0 = (f 01(v), . . . , f

0dv

(v).

The rank of the matrices [Av

,Bv

] 2 Rd⇥2d must be equal to dv

.

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Finite graphs

TheoremLet � be a metric graph with finitely many edges. Consider the

operator h acting as � d2

ds2on each edge e 2 E , with the domain

consisting of the functions f 2 H2(e) on e and satisfying theconditions

Av

F + Bv

F 0 = 0

at each vertex v 2 V.Let

�

Av

2 Rdv

⇥dv ,B

v

2 Rdv

⇥dv |v 2 V

a collection of matricessuch that Rank(A

v

,Bv

) = dv

for all v.

h is self-adjoint i↵ 8v 2 V,Av

BTv

= Bv

ATv

(Kostrykin Schrader, 1999) (Berkolaiko and Kuchment, Introduction to Quantum Graphs, AMS186, 2012)

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A linear algebra bit (1)

We drop the subscript v for a moment

B = W⌃V T = (W1,W2)

✓

⌃1 00 0

◆✓

V T1

V T2

◆

(SVD)

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B = W⌃V T = (W1,W2)

✓

⌃1 00 0

◆✓

V T1

V T2

◆

(SVD)

A = WRV T R =

✓

R11 R12

R21 R22

◆

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B = W⌃V T = (W1,W2)

✓

⌃1 00 0

◆✓

V T1

V T2

◆

(SVD)

A = WRV T R =

✓

R11 R12

R21 R22

◆

BAT = ABT =) R21 = 0

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B = W⌃V T = (W1,W2)

✓

⌃1 00 0

◆✓

V T1

V T2

◆

(SVD)

A = WRV T R =

✓

R11 R12

R21 R22

◆

BAT = ABT =) R21 = 0

W T (A,B)

✓

V 00 V

◆

=

✓

R11 R12 ⌃1 00 R22 0 0

◆

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W T (A,B)

✓

V TFV TF 0

◆

= 0 ()✓

R11 R12 ⌃1 00 R22 0 0

◆

0

B

B

@

V T1 F

V T2 F

V T1 F 0

V T2 F 0

1

C

C

A

= 0

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W T (A,B)

✓

V TFV TF 0

◆

= 0 ()✓

R11 R12 ⌃1 00 R22 0 0

◆

0

B

B

@

V T1 F

V T2 F

V T1 F 0

V T2 F 0

1

C

C

A

= 0

Rank�

(A,B)�

= d =) R22 invertible =) V T2 F = 0 =) F 2 span(V1)

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W T (A,B)

✓

V TFV TF 0

◆

= 0 ()✓

R11 R12 ⌃1 00 R22 0 0

◆

0

B

B

@

V T1 F

V T2 F

V T1 F 0

V T2 F 0

1

C

C

A

= 0

Rank�

(A,B)�

= d =) R22 invertible =) V T2 F = 0 =) F 2 span(V1)

=) R11VT1 F + ⌃1V

T1 F 0 = 0 =) V T

1 F 0 = �⌃�11 R11V

T1 F

=) F 0 = �V1⌃�11 R11V

T1 () F 0 = LF

�

min norm solution�

⇣

(ABT = BAT⌘

=) L = LT

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A linear algebra bit (3)Let Pv = I � Qv and Qv = B+

v Bv (B+v is the Moore-Penrose

pseudo inverse) be the (symmetric) orthogonal projectors relativeto node v then

AvF + BvF0 = 0 ()

⇢

PvF = 0QvF 0 + LvF = 0

(⇤)

whereLv = B+

v AvQv

andLv = LTv .

All self-adjoint realizations of h (the negative second derivative) on�, with the vertex boundary conditions (*), satisfy the following:

8v 2 V 9Pv and Qv = Idv � Pv

(orthogonal projections) and

8v 2 V 9Lv in QvCdv .

All f 2 D(h) ⇢L

H2(e) are described by (*)at each finite vertex.

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Quadratic form

The Quadratic form h of h is

h[f , f ] =X

e2E

Z

e

�

�

�

�

df

ds

�

�

�

�

2

ds �X

v2V

X

e2E(Lv )jk fj(v)fk(v)

=X

e2E

Z

e

�

�

�

�

df

ds

�

�

�

�

2

ds �X

v2VhLvF ,F i,

where h·, ·i is the standard Hermitian inner product in Cdv . Thedomain of h consists of all f 2

L

e2E H1(e) such that PvF = 0.

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Symmetric vertex conditions: a classification

We want to classify the cases for which the conditions

⇢

PF = 0QF 0 + LF = 0

(⇤)

are invariant under the action of the symmetric group of thecoordinate permutations. ( again we drop the subscript v)

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We want to classify the cases for which the conditions

⇢

PF = 0QF 0 + LF = 0

(⇤)

are invariant under the action of the symmetric group of thecoordinate permutations. ( again we drop the subscript v) Theinvariant space of the permutation group is the one dimensionalspace generated by the vector of entries equal to one

~ =1pd(1, · · · , 1)T 2 Cd .

Then (*) are invariant under the action of this group i↵ P , Q, andL are. We have 4 possible cases.

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1. P = 0, Q = I , L = ↵~ ~ T + �II � = 0 �0-conditionsI ↵ = � = 0 Neumann-conditions

2. P = I , Q = 0 (L irrelevant) Dirichlet-conditions

3. P = I � ~ ~ T , Q = ~ ~ T , L = ↵Q �-conditions

4. P = ~ ~ T , Q = I � ~ ~ T , L = ↵P ????

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4. P = ~ ~ T , Q = I � ~ ~ T , L = ↵P ????

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4. P = ~ ~ T , Q = I � ~ ~ T , L = ↵P ????

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4. P = ~ ~ T , Q = I � ~ ~ T , L = ↵P ????

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Are these all the possible cases?

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Are these all the possible cases?

NO!!

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Examples of b.c.

�-type conditions

8

<

:

f (s) is continuous on �

8v 2 �P

e2Evdf

dse(v) = ↵v f (v)

Ev is the subset of the edges having v as a boundary point.↵v are real fixed numbersWe describe the case for a node v of degree 3 (generalization iseasy)

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Examples of b.c.

Av =

0

@

1 �1 00 1 �1

�↵v 0 0

1

A Bv =

0

@

0 0 00 0 01 1 1

1

A

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Examples of b.c.

Av =

0

@

1 �1 00 1 �1

�↵v 0 0

1

A Bv =

0

@

0 0 00 0 01 1 1

1

A

AvBTv =

0

@

0 0 00 0 00 0 �↵v

1

A

The self-adjoint condition is satisfied i↵ ↵ 2 R

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Examples of b.c.

Av =

0

@

1 �1 00 1 �1

�↵v 0 0

1

A Bv =

0

@

0 0 00 0 01 1 1

1

A

AvBTv =

0

@

0 0 00 0 00 0 �↵v

1

A


Lv =�↵v

dv

0

@

1 1 11 1 11 1 1

1

A

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Examples of b.c.

The Hamiltonian of the problem has the following h

h[f , f ] =X

e2E

Z

e

�

�

�

�

df

ds

�

�

�

�

2

ds �X

v2VhLvF ,F i

=X

e2E

Z

e

�

�

�

�

df

ds

�

�

�

�

2

ds +X

v2V↵v |f (v)|2 .

38 / 78


Examples of b.c.


h[f , f ] =X

e2E

Z

e

�

�

�

�

df

ds

�

�

�

�

2

ds �X

v2VhLvF ,F i

=X

e2E

Z

e

�

�

�

�

df

ds

�

�

�

�

2

ds +X

v2V↵v |f (v)|2 .

The case ↵v ⌘ 0 corresponds to the Neumann-Kirchho↵conditions

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Examples of b.c.


h[f , f ] =X

e2E

Z

e

�

�

�

�

df

ds

�

�

�

�

2

ds �X

v2VhLvF ,F i

=X

e2E

Z

e

�

�

�

�

df

ds

�

�

�

�

2

ds +X

v2V↵v |f (v)|2 .

The case ↵v ⌘ 0 corresponds to the Neumann-Kirchho↵conditions

8

<

:

f (s) is continuous on �

8v 2 �P

e2Evdf

dse(v) = 0

h[f , f ] =X

e2E

Z

e

�

�

�

�

df

ds

�

�

�

�

2

ds.

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Examples of b.c.

�0-type conditions

8

>

>

>

<

>

>

>

:

8v 2 �

The value of the derivativedfedse

(s) is the same 8e 2 EvP

e2Ev fe(v) = ↵df

dse(v)

Ev is the subset of the edges having v as a boundary point.↵v are real fixed numbersWe describe the case for a node v of degree 3 (generalization iseasy)

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Examples of b.c.

Bv =

0

@

1 �1 00 1 �1

�↵v 0 0

1

A Av =

0

@

0 0 00 0 01 1 1

1

A

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Examples of b.c.

Bv =

0

@

1 �1 00 1 �1

�↵v 0 0

1

A Av =

0

@

0 0 00 0 01 1 1

1

A

AvBTv =

0

@

0 0 00 0 00 0 �↵v

1

A


40 / 78


Examples of b.c.

Bv =

0

@

1 �1 00 1 �1

�↵v 0 0

1

A Av =

0

@

0 0 00 0 01 1 1

1

A

AvBTv =

0

@

0 0 00 0 00 0 �↵v

1

A

The self-adjoint condition is satisfied i↵ ↵ 2 RIf ↵v = 0 for some v then Lv = 0.

Lv = 0 8v =) h[f , f ] =X

e2E

Z

e

�

�

�

�

df

ds

�

�

�

�

2

ds.

40 / 78


Examples of b.c.

Bv =

0

@

1 �1 00 1 �1

�↵v 0 0

1

A Av =

0

@

0 0 00 0 01 1 1

1

A

AvBTv =

0

@

0 0 00 0 00 0 �↵v

1

A

The self-adjoint condition is satisfied i↵ ↵ 2 RIf ↵v 6= 0 then Bv is invertible and Pv = 0 and Qv = I .

(Lv )i ,j = � 1

↵dv8 i , j

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Examples of b.c.


h[f , f ] =X

e2E

Z

e

�

�

�

�

df

ds

�

�

�

�

2

ds +X

{v2V|↵v 6=0}

1

↵v

�

�

�

�

�

X

e2Ev

f (v)

�

�

�

�

�

2

.

The domain consists of all f (s) 2L

e H1(e) that have at each

vertex where ↵v = 0 the sum of the vertex values along all theincident edges is equal to 0.

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Examples of b.c.

Dirichlet and Neumann conditions.Dirichlet vertex conditions require that at each vertex theboundary conditions impose f (v) = 0The operator is decoupled in the sum of the negative secondderivative and

h[f , f ] =X

e2E

Z

e

�

�

�

�

df

ds

�

�

�

�

2

ds

f 2 H1(�).The spectrum �(h)

�(h) =

⇢

n2⇡2

l22|e 2 (E ), n 2 Z� 0

�

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Examples of b.c.

Dirichlet and Neumann conditions.Under Neumann vertex conditions no restriction on the value ofthe function at vertices are required. The derivative at the verticesare instead required to be zero. The operator is decoupled in thesum of the negative second derivative and

h[f , f ] =X

e2E

Z

e

�

�

�

�

df

ds

�

�

�

�

2

ds

f 2 H1(�), as for the Dirichlet case, but on a larger domain

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Modelling (Neumann Schrodinger example)

⌦d fat graph (le length of edge e)

Let ⌦d denote the fat graph and� = d ⇥ p(s) where p(s) > 0 is afunction of the arc length that canbe discontinuous at the vertices.Each vertex neighbouring iscontained in a ball of radius ⇠ d andstar-shaped with respect to a smallerball of diameter ⇠ d .

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On ⌦d we define the Schrodinger operator

Hd(A, q) =

✓

1

ir� A(s)

◆2

+ q(s)

with Neumann conditions on @⌦d (q scalar electric and A vectormagnetic potentials

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On ⌦d we define the Schrodinger operator

Hd(A, q) =

✓

1

ir� A(s)

◆2

+ q(s)

with Neumann conditions on @⌦d (q scalar electric and A vectormagnetic potentials

H(A, q)f (se) = �1

p

✓

d

dse� iAt

e(s)

◆

p

✓

d

dse� iAt

e(s)

◆

f + qe(s)f

where Ate is the tangential component of A and qe is the

restriction of q to the graph.

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Boundary conditions at the vertices

I f is continuous through each vertex

IX

{k|v2ek}

pk

✓

dfkdsk

� iAtk fk

◆

(v) = 0

pk function that gives the width of the tube around ek . The valuesof pk(v) at the same vertex can be di↵erent for di↵erent ek

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Boundary conditions at the vertices

I f is continuous through each vertex

IX

{k|v2ek}

pk

✓

dfkdsk

� iAtk fk

◆

(v) = 0

pk function that gives the width of the tube around ek . The valuesof pk(v) at the same vertex can be di↵erent for di↵erent ekTheorem For n = 1, 2, . . .

lim�!0

�n (Hd(A, q)) = �n (H(A, q)) ,

where �n is the n-th eigenvalue counted in increasing order(accounting multiplicity)

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Parabolic problems

Finally, among our goals is the analysis of parabolic problems onmetric graphs. In this case, we assume that the functions we usealso depend on a second variable t representing time, i.e. (seeRaviart-Thomas 1983),

f (t, x) : [0,T ]⇥ � ! IR.

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Parabolic problems

Let V denote either L2(�) or H1(�). Let C 0�

[0,T ];V�

be thespace of functions f (t, x) that are continuous in t with values inV , i.e, for each fixed value t⇤ of t we have that f (t⇤, ·) 2 V . Thisspace is equipped with the norm

kf kC0�

[0,T ];V� = sup

0tTkf (t, ·)kV .

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Parabolic problems

Let L2�

[0,T ];V�

be the space of functions f (t, x) that aresquare-integrable in t for the dt measure with values in V , i.e foreach fixed value t⇤ of t we have that f (t, ·) 2 V . This space isequipped with the norm

kf kL2�

[0,T ];V� =

✓

Z T

0kf (t, ·)k2V dt

◆

12

and scalar product

�

f , g�

L2�

[0,T ];V� =

Z T

0

⇣

f (t, ·), g(t, ·)⌘

Vdt.

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Parabolic problems

We note that all these definitions can be easily modified to dealwith self-adjoint operators acting on spaces of complex-valuedfunctions, as required by quantum mechanics.

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Parabolic equation

Given u0 2 H1(�) and a f 2 L2�

[0,T ], L2(�)�

find

u 2 L2�

[0,T ],H1(�)�

[ C 0�

[0,T ];H1(�)�

and8

>

>

>

>

<

>

>

>

>

:

8v 2 H1(�)d

dt

�

u(t), v�

L2�

[0,T ],L2(�)� + h

⇥

u(t), v⇤

=�

f (t), v�

L2�

[0,T ],L2(�)� on ]0,T [

u(0) = u0

h⇥

u, v⇤

=X

e2E

Z

e

@u(t, s)

@s

@v

@sds +

X

e2E

Z

em(s)u(t)v(s) ds.

Neumann-Kirchho↵ conditions

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Parabolic equation

8

>

>

<

>

>

:

@u(t, s)

@t� @2u(t, s)

@s2+mu(t, s) = f on �

u(0, s) = u0.

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Eigenvalue Problem

The analysis of the spectrum of the self-adjoint operators is moresubtle. Infinite quantum graphs can have Hamiltonian withcontinuous part of the spectrum. However, for finite quantumgraphs we can have a better situation:Kuchment, Quantum graphs: I. Some basic structures, WavesRandom Media 14 (2004)

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Eigenvalue Problem

The analysis of the spectrum of the self-adjoint operators is moresubtle. Infinite quantum graphs can have Hamiltonian withcontinuous part of the spectrum. However, for finite quantumgraphs we can have a better situation:TheoremLet � a finite quantum graph with finite edges equipped with anHamiltonian given by negative second derivative along the edgesand vertex conditions

(⇤)⇢

PvF = 0QvF 0 + LvF = 0

Then the spectrum �(h) is discrete.

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An interesting connection

hf = �f f 2 L2(�)

Let e be an edge identified by the two vertices v and w of lengthle . If � 6= n2⇡2l�2

e with n 2 Z� {0} then

fe =1

sinp�le

�

fe(v) sinp�(le � s) + fe(w) sin

p�s�

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hf = �f f 2 L2(�)



fe =1

sinp�le

�


p�s�

Substituting

f 0e (v) =lep�

sinp�

�

fe(w)� fe(v) cos lep��

in (*) and eliminating the derivatives we compute a system ofalgebraic relations

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hf = �f f 2 L2(�)



fe =1

sinp�le

�


p�s�

Substituting

f 0e (v) =lep�

sinp�

�

fe(w)� fe(v) cos lep��

in (*) and eliminating the derivatives we compute a system ofalgebraic relations

T (�)F = 0

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Theorem� 6= n2⇡2l�2

e with n 2 Z� {0} belongs to the spectrum of h i↵zero belongs to the spectrum of T (�)

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This results connects quantum graph theory to combinatorialgraph theory

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What about � = n2⇡2l�2e ?

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What about � = n2⇡2l�2e ?

If we have only Dirichlet conditions they are the only eigenvalues,otherwise ??????

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Summary

I We have defined the analytical structure of a Quantum GraphI the metric propertiesI the operatorsI the boundary conditions

I We will focus in the last part on finite graphs and we will givesome hints on:

I Solution of di↵erential equationsI Eigenvalues problems

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Summary




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Summary



I Solution of di↵erential equations

I Eigenvalues problems

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Summary




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Summary




We assume that at the vertices of degree d = 1 we have Dirichletconditions.

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Summary




We assume that at the vertices of degree d = 1 we have Dirichletconditions. Only Neumann-Kirchho↵ conditions for sake ofsimplicity

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Di↵erential equations

Given a function g(s) 2 L2(�) , we want to compute the solutionof the problem

minu

{h[u, u]� hg , ui}

with u(s) 2 H1(�).

I On each edge e 2 E we solve the local problem and we denoteby ue(se) the solution.

I Using the Neumann-Kirchho↵ conditions and the values of thederivative at the vertices, we form and solve an algebraicsystem on the vertices.

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minu

{h[u, u]� hg , ui}

with u(s) 2 H1(�).



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minu

{h[u, u]� hg , ui}

with u(s) 2 H1(�).



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minu

{h[u, u]� hg , ui}

with u(s) 2 H1(�).



We have described an abstract Domain Decomposition approach.

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Finite-element basis

On each edge of the quantum graph it is possible to use theclassical 1D finite-element method. Let e be a generic edgeidentified by two vertices, which we denote by va and vb. Theparameter x will map the edge such that for x = 0 we have thevertex va and for x = è we have the vertex vb. The first step is tosubdivide the edge in ne intervals of length he . The points

n

Ee =�

xej n�1

j=1

o

[ {vea} [ {veb}

form a chain linking the vea to veb lying on the edge e. Denoting byn

ej

on+1

j=0the standard hat basis functions we have

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8

>

>

>

>

>

>

>

>

>

>

>

<

>

>

>

>

>

>

>

>

>

>

>

:

e0(x) =

⇢

1� xh if 0 x he

0 otherwise

ej (x) =

(

1� |xj�x |h if xj�1 x xj+1

0 otherwise

ene+1(x) =

⇢

1� è�xh if è � he x è

0 otherwise

. (1)

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8

>

>

>

>

>

>

>

>

>

>

>

<

>

>

>

>

>

>

>

>

>

>

>

:

e0(x) =

⇢

1� xh if 0 x he

0 otherwise

ej (x) =

(

1� |xj�x |h if xj�1 x xj+1

0 otherwise

ene+1(x) =

⇢

1� è�xh if è � he x è

0 otherwise

. (1)

The functions ej are a basis for the finite-dimensional space

V eh =

n

w 2 H1(e); w |[xej ,xej+1]2 P1, j = 0, . . . , n + 1

o

,

where P1 is the space of linear functions.

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Finite-element framework

In practice, we subdivide each edge forming a chain made of nodeof degree 2 and we build the usual hat functions extending them tothe vertices.

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In practice, we subdivide each edge forming a chain made of nodeof degree 2 and we build the usual hat functions extending them tothe vertices.

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Globally, we can construct the space

Vh(�) =M

e2EV eh

that is a finite dimensional space of functions that belong toH1(�).

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Globally, we can construct the space

Vh(�) =M

e2EV eh

that is a finite dimensional space of functions that belong toH1(�). The continuity on � of the functions in Vh follows fromconstruction: at each vertex v we have d

v

(degree of the vertex v)linear functions that have values 1 on v each one belonging to anindependent V e

h with e 2 Ev

.

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Finite-element frameworkGlobally, we can construct the space

Vh(�) =M

e2EV eh

that is a finite dimensional space of functions that belong to H1(�).Any function fh(x) 2 Vh(�) is then a linear combination of the e

j :

fh(x) =X

e2E

n+1X

j=0

↵ej

ej (xe).

The quadratic form h of the Hamiltonian operator can be testedon all the and we have the following finite dimensional (discrete)bilinear form

hh(fh, ek) =

X

e2E

n+1X

j=0

↵ej

⇢

Z

e

d ej

dx

d ek

dxdx +

Z

eV (x) e

j ekdx

�

.

In h and hh the Kirchho↵’s conditions at each vertices are thenatural conditions and they are satisfied.

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Extended Graph

The nodes on the edges will describe a chain path between twovertices. We can then think of introducing a new (combinatorial)graph in which the nodal discretization points become additionalvertices and the edges are obtained by subdividing the edges of theoriginal (metric) graph. We call this the extended graphassociated with � and denote it by G.

Assuming for simplicity that all edges e 2 E have equal length andthat the same number n � 1 of internal nodes are used for eachedge, the new graph G will have (n � 1)⇥M + N vertices andn ⇥M edges, where N is the number of vertices and M thenumber of edges in �.

The extended graph can be huge, but it has a lot of structure.

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Extended Graph: The matrix

Reordering the nodes such that the internal nodes in a edge areconsecutive and the vertices are at the end we have that theresulting Gramian matrix H = (hh[ e

k , ek ]) is of the form

H =

H11 H12

HT12 H22

�

where H11 is block diagonal symmetric and positive definite whereeach diagonal block is of size n � 1 and tridiagonal, and H22 is adiagonal matrix with positive diagonal entries. The sparsity patternof HT

12 is that of the incidence matrix of the extended graph G.Important: We are assuming that the potential V (s) is positive.

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Extended Graph: A simple example

Figure : Example of a simple metric planar graph and of its incidencematrix.

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Figure : Example of the extension of the graph when 4 nodes chain isadded internally to each edge (left) and of its incidence matrix (right).

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Figure : The matrix H pattern where the red bullets correspond to theoriginal vertices and the blue one to the internal nodes on each edge.

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Extended Graph

In the special case V = 0 (that is, h = � d2

ds2, we obtain the

discrete (negative) Laplacian (sti↵ness matrix) L on G. When thesame number of (equidistant) discretization points is used on eachedge of �, L coincides (up to the factor h�1) with thecombinatorial graph Laplacian LG . Both the sti↵ness matrix L andthe mass matrix M = (h e

j , eki) have a block structure matching

that of H. For example, if V (s) = k (constant) then H = L+ kM.Minimization of the discrete quadratic form

Jh(uh) := h[uh, uh]� hgh, uhi, uh 2 Vh(�)

is equivalent to solving the extended linear system Huh = gh, oforder (n � 1)M + N.

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Solution of the extended linear system

The extended linear system can be solved e�ciently by block LUfactorization, by first eliminating the interior edge nodes (thisrequires solving, in parallel if one wishes, a set of M independenttridiagonal systems of order n � 1), and then solving the N ⇥ NSchur complement system

Suvh = gvh �HT12H

111g

eh ⌘ ch

for the unknowns associated with the vertices of � and

S = H22 �HT12H

111H12.

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The block LU factorization of H is given by

H =

H11 H12HT

12 H22

�

=

H11 0HT

12 S

�

I H�111 H12

0 I

�

and crucial S is SPARSE.

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Figure : The pattern of the Schur complement S.

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Theorem: The nonzero pattern of the Schur complement

S = H22 �HT12H

111H12

coincides with that of L�, the (combinatorial) graph Laplacian ofthe (combinatorial) graph �. In the special case V = 0, weactually have S = L�.

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Note that S is SPD, unless V = 0 (in which case S is only positivesemidefinite). For � not too large, we can solve the Schurcomplement system by sparse Cholesky factorization with anappropriate reordering. However, for large and complex graphs (forexample, scale-free graphs), Cholesky tends to generate enormousamounts of fill-in, regardeless of the ordering used. Hence, we needto solve the Schur complement system by iterative methods, likethe preconditioned conjugate gradient (PCG) algorithm.

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Preconditioning of matrices arising from complex graphs is anactive area of research. Some of the techniques that work well forother types of problems (like Incomple Cholesky Factorization) areuseless here. Here we consider two simple preconditioners:

I diagonal scaling with D = diag(S)

I a first degree polynomial preconditioner:

P�1 = D�1 +D�1�

D� S�

D�1.

Note: For the very sparse matrices considered here, each iterationof PCG with polynomial preconditioning costs about the same as1.5 iterations with diagonal preconditoning.

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Numerical experiments

We present first results for a simple steady-state (equilibrium)problem

�d2u

ds2+ Vu u = g on �

with Neumann-Kirchho↵ conditions at the vertices, for threedi↵erent choices of �:

I yeast, the PPI (Protein-Protein Interaction) network of beeryeast (N = 2224,M = 6609)

I drugs, a social network of drug addicts (N = 616,M = 2012)

I pref2000, a synthetic scale-free graph constructed using thepreferential attachment scheme (N = 2000,M = 3974)

In each case we assume that all edges have unit length and we use20 interior discretization points per edge (h = 1

21 ).For the potential we use V (s) = K (const.) andV (s) = K (s � 1

2)2 for K = 0.1, 1, 10.

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PPI network of Saccharomyces cerevisiae (beer yeast)

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Social network of injecting drug users in Colorado Springs

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Scale-free BarabasiAlbert graph (pref)

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The sizes of the extended system Huh = gh and of the reducedsystem Suvh = ch are, respectively:

I n = 134, 404,N = 2224 for yeast;

I n = 40, 856,N = 616 for drugs;

I n = 81, 480,N = 2000 for pref2000.

The Schur complement can be formed e�ciently since it is verysparse and we know the location of the nonzero entries in advance.Since the original graphs are very small, the Schur complementsystem is best solved by sparse Cholesky factorization, but we alsoexperiment with PCG. Without preconditioning, convergence canbe slow.For each problem we also need to solve M uncoupled tridiagonalsystems of order 20.

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Numerical Experiments

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67 / 78



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V (s) = K (s � 0.5)2

Problem K = 0.1 K = 1 K = 10It rel. error It rel. error It rel. error

yeast 13 3.3e-8 10 1.6e-8 8 2.0e-10drugs 10 1.4e-8 8 1.9e-8 6 2.8e-9

pref2000 9 2.9e-9 8 3.3e-10 6 2.5e-9

Table : Results of running pcg (TOL =peps) on Schur complement

system, diagonal preconditioner. The “exact” solution is the one returnedby backslash.

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V (s) = K (s � 0.5)2

Problem K = 0.1 K = 1 K = 10It rel. error It rel. error It rel. error

yeast 7 5.9e-8 6 4.1e-9 5 1.3e-11drugs 6 3.9e-8 5 1.7e-9 4 1.2e-10

pref2000 5 4.5e-8 5 2.4e-11 4 3.1e-11

Table : Results of running pcg (TOL =peps) on Schur complement

system, polynomial preconditioner.

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V (s) = KProblem K = 0.1 K = 1 K = 10

It rel. error It rel. error It rel. erroryeast 38 7.7e-09 14 1.4e-08 5 6.2e-09drugs 42 6.9e-09 14 9.0e-09 5 4.9e-09

pref2000 23 3.5e-09 12 1.1e-08 5 2.5e-09

Table : Results of running pcg (TOL = eps1/2) on Schur complementsystem, diagonal preconditioner.

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V (s) = KProblem K = 0.1 K = 1 K = 10

It rel. error It rel. error It rel. erroryeast 20 4.0e-09 8 2.1e-08 4 5.6e-10drugs 24 7.1e-08 9 2.3e-09 4 4.3e-10

pref2000 13 3.2e-10 8 1.2e-09 4 2.2e-10

Table : Results of running gmres (TOL = eps1/2) on Schur complementsystem, polynomial preconditioner.

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N No prec. Diagonal Polynomial2000 83 23 135000 108 23 1310000 125 23 13

Table : PCG iteration counts for pref graph, increasing N(TOL = eps1/2).

Note: Here h is constant, the size N of the graph G is increasing.The size of the extended graph G is n = 81, 480, n = 204, 360, andn = 409, 300.

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h�1 No prec. Diagonal Polynomial21 83 23 1341 83 23 1381 82 22 12101 83 21 12

Table : PCG iteration counts for pref graph, increasingV (s) = K = 0.1, N = 2000.

Note: Here h is decreasing, the size N of the graph G is fixed. Thesize of the extended graph G increases from 81, 480 to 399, 400.With diagonal or polynomial preconditioning the solution algorithmis scalable with respect to both N and h for these graphs.

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The parabolic case with FE

Space discretization using finite elements leads to the semi-discretesystem

Muh = Huh + fh, uh(0) = uh,0,

where uh = uh(t) is a vector function on the extended graph G,and the mass matrix M and Hamiltonian H are as before.A variety of methods are available for solving this linear system ofODEs: backward Euler, Crank-Nicolson, exponential integratorsbased on Krylov subspace methods, etc.Note that for large graphs and/or small h, this can be a hugesystem.We have obtained some preliminary results using Stefan Guttelscode funm kryl for evaluating the action of the matrixexponential on a vector.

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... And we can compute the solutions!!!

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... And we can compute the solutions!!!

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Conclusion

I Quantum graphs are independent from the embedding spaceand his make them a good candidate to model complexphenomena depending on many variables.

I The process of ”model reduction” can be trickyI not always self-adjoint operatorsI continuous spectrum issuesI ray edges ....

I Non linear operators: p-Laplacian,..

I Integro-di↵erential operators (fractional derivatives andvisco-elasticity, ....)

I Interesting numerical linear algebra and potential highparallelism

I Strong interaction between di↵erent expertises ...

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Conclusion







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Conclusion







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Conclusion







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Conclusion







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Conclusion







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Final

I hope the journey was pleasant, interesting, andwithout too many bumps.

Piacciavi, ... aggradir questo che vuolee darvi sol puo l’umil servo vostro.(from Ariosto (1474-1533), Orlando Furioso, Canto I)

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Final

A Final Remark from an Old Book and a Very Wise Man

Si, avons nous beau monter sur des echasses, car sur des echassesencore faut-il marcher de nos jambes. Et au plus eleve trone dumonde, si ne sommes assis que sur notre cul.(Michel Eyquem de Montaigne (1533-1592), Les Essais, Ch 13)

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Documents

Complex and quantum graphs and their applications · Complex and quantum graphs and their applications Mario Arioli Combinatorial Graphs and Matrix representation I Given how can