Complex Networks

Third Lecture

I. A few examples of Complex Networks

II. Basic concepts of graph theory and network theory

III. Models

IV. Communities

Program

Three main goals:

1) Identify the “universality classes” of graphs.

2) Identify the “microsopic rules” which generate a particular class of

3) Predict the behaviour of the system when one changes the boundary conditions.

Universality

• Two main classes with different behaviours of the connectivity: exponential graphs and power law graphs

• random graphs are exponential

• Almost all the biological networks are instead of the power law type

Topological heterogeneityStatistical analysis of centrality measures:

P(k)=Nk/N=probability that a randomly chosen node has degree kalso: P(b), P(w)….

Two broad classes•homogeneous networks: light tails•heterogeneous networks: skewed, heavy tails

Topological heterogeneityStatistical analysis of centrality measures

Broad degree distributions

Power-law tailsP(k) ~ k-typically 2< <3

Topological heterogeneityStatistical analysis of centrality measures:

Poisson vs.Power-law

log-scale

linear scale

Exp. vs. Scale-FreePoisson distribution

Exponential Network

Power-law distribution

Scale-free Network

ConsequencesPower-law tailsP(k) ~ k-

Average=< k> = k P(k)dkFluctuations< k2 > = k2 P(k) dk ~ kc

3-

kc=cut-off due to finite-sizeN 1 => diverging degree fluctuations for < 3

Level of heterogeneity:

1) Random graph2) Small world3) Preferential attachment4) Copying model

Models

Usual random graphs: Erdös-Renyi model (1960)

N points, links with probability p:static random graphs

Average number of edges: <E > = pN(N-1)/2

Average degree: < k > = p(N-1)

p=c/N to havefinite average degree

Erdös-Renyi model (1960)

<k> < 1: many small subgraphs

< k > > 1: giant component + small subgraphs

Erdös-Renyi model (1960)Probability to have a node of degree k•connected to k vertices, •not connected to the other N-k-1

P(k)= CkN-1 pk (1-p)N-k-1

Large N, fixed pN=< k > : Poisson distribution

Exponential decay at large k

Erdös-Renyi model (1960)

Small clustering: < C > =p =< k > /N

Short distances l=log(N)/log(< k >)(number of neighbors at distance d: < k >d )

Poisson degree distribution

Generalized random graphs

Desired degree distribution: P(k)

• Extract a sequence ki of degrees taken from P(k)

• Assign them to the nodes i=1,…,N

• Connect randomly the nodes together, according to their given degree

Small-world networks

Watts & Strogatz,

Nature 393, 440 (1998)

N = 1000

•Large clustering

coeff. •Short typical path

N nodes forms a regular lattice. With probability p, each edge is rewired randomly

=>Shortcuts

Statistical physics approachMicroscopic processes of the

many component units

Macroscopic statistical and dynamical properties of the system

Cooperative phenomenaComplex topology

Natural outcome of the dynamical evolution

Find microscopic mechanisms

Microscopic mechanism: An example

(1) The number of nodes (N) is NOT fixed. Networks continuously expand

by the addition of new nodesExamples: WWW: addition of new documents Citation: publication of new papers

(2) The attachment is NOT uniform.A node is linked with higher probability to a

node that already has a large number of links.Examples : WWW : new documents link to well known sites (CNN, YAHOO, NewYork Times, etc) Citation : well cited papers are more likely to be cited again

(1) GROWTH : At every timestep we add a new node with m edges (connected to the nodes already present in the system).

(2) PREFERENTIAL ATTACHMENT : The probability Π that a new node will be connected to node i depends on the connectivity ki of that node

A.-L.Barabási, R. Albert, Science 286, 509 (1999)

jj

ii k

kk

)(

Microscopic mechanism: An example

BA network

Connectivity distribution

Problem with directed graphs

Natural extension:

(kiin )

kiin

jk jin

What happens if kiin = 0?

(kiin ) 0!

Nodes with zero indegree will never receivelinks! Bad!

Linear preferential attachment

Microscopic mechanism:

S. N. Dorogovtev, J. F. F. Mendes, A. N. Samukhin, Phys. Rev. Lett. 85, 4633 (2000)

(ki) ki k0

j (k j k0)

(1) GROWTH : At every timestep we add a new node with m edges (connected to the nodes already present in the system).

(2) PREFERENTIAL ATTACHMENT : The probability Π that a new node will be connected to node i depends on the connectivity ki of that node and a constant k0 (attractivity), with -m < k0 < ∞

P(k) ~ k (3k0 /m )

Degree distribution:

Extension to directed graphs:

Problem of nodes with zero indegree solved!

(kiin )

kiin k0

j (k jin k0)

P(k in ) ~ (k in ) (2k0 /m )

Microscopic mechanism:

P.L. Krapivsky, S. Redner, F. Leyraz, Phys. Rev. Lett. 85, 4629 (2000)

(ki) ki

jki

Non-linear preferential attachment

(1)α<1: P(k) has exponential decay!

(2)α>1: one or more nodes is attached to a macroscopic fraction of nodes (condensation); the degree distribution of the other nodes is exponential

(3)α=1: P(k) ~ k-3

Copying model

Microscopic mechanism:

a. Selection of a vertexb. Introduction of a new vertexc. The new vertex copies m linksof the selected oned. Each new link is kept with proba , rewiredat random with proba 1-

Growing network:

J. M. Kleinberg, S. R. Kumar, P. Raghavan, S. Rajagopalan, A. Tomkins, Proc. Int. Conf. Combinatorics & Computing, LNCS 1627, 1 (1999)

Copying model

Microscopic mechanism:

Probability for a vertex to receive a new link at time t:

•Due to random rewiring: (1-)/t

•Because it is neighbour of the selected vertex: kin/(mt)

effective preferential attachment, withouta priori knowledge of degrees!

Copying model

Microscopic mechanism:

Degree distribution:

=> model for WWW and evolution of genetic networks

=> Heavy-tails