Lyon Lecture IIb

Statistical physics of complexnetworks

Marc BarthelemyCEA, FranceEHESS-CAMS, France

IXXI Lyon July 2008

Outline I. Introduction: Complex networks

1. Complex systems and networks

2. Graph theory and characterization of large networks: tools

3. Characterization of large networks: results

4. Models

II. Dynamical processes

1. Resilience and vulnerability

2. Epidemiology

III. Advanced topics

1. Global disease spread

2. Community detection

3. Evolution and formation of the urban street network in cities

I. 4 Models

Models: static vs. dynamic

• Static: N nodes from the beginning, connection rules- Erdos-Renyi- Generalized random graphs- Watts-Strogatz model- Fitness models (hidden variables)

• Dynamic: the network is growing

- Barabasi-Albert- Copy model- Weighted network model

Simplest model of random graphs:Erdös-Renyi (1959)

N nodes, connected with probability p

Paul Erdős and Alfréd Rényi(1959)"On Random Graphs I" Publ.Math. Debrecen 6, 290–297.

Simplest model of random graphs:Erdös-Renyi (1959)

Some properties:

- Average number of edges

- Average degree

Finite average degree

Erdös-Renyi model: degree distribution

Proba to have a node of degree k=• connected to k vertices, • not connected to the other N-k-1

Large N, fixed : Poisson distribution

Exponential decay at large k

• N points, links with proba p:

Erdös-Renyi model: clustering and averageshortest path

• Neglecting loops, N(l) nodes at distance l:

For

: many small subgraphs

: giant component + small subgraphs

Erdös-Renyi model: components

Erdös-Renyi model: summary

- Small clustering

- Small world

- Poisson degree distribution

Generalized random graphs

Desired degree distribution: P(k)

Extract a sequence ki of degrees taken fromP(k)

Assign them to the nodes i=1,…,N

Connect randomly the nodes together,according to their given degree

Generalized random graphs

Average clustering coefficient

Average shortest path

Small-world and randomness

Watts-Strogatz (1998)

Lattice Random graphReal-Worldnetworks*

Watts & Strogatz, Nature 393, 440 (1998)

* Power grid, actors, C. Elegans

Watts-Strogatz (1998)

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

• Large clustering coeff.• Short typical path

MB and Amaral, PRL 1999

Barrat and Weight EPJB 2000

N = 1000

Watts-Strogatz (1998)

Fitness model (hidden variables)

Erdos-Renyi: p independent from the nodes

• For each node, a fitness

Soderberg 2002Caldarelli et al 2002

• Connect (i,j) with probability

• Erdos-Renyi: f=const

Fitness model (hidden variables)

• Degree

• Degree distribution

Fitness model (hidden variables)

• If power law -> scale free network

• If and

Generates a SF network !

Barabasi-Albert (1999) Everything’s fine ?

Small-world network

Large clustering

Poisson-like degree distribution

Except that for

Internet, Web

Biological networks

…

Power-law distribution:

Diverging fluctuations !

Internet growth

Moreover - dynamics !

Barabasi-Albert (1999)

(1) The number of nodes (N) is NOT fixed.Networks continuously expand by theaddition of new nodes Examples:

WWW : addition of new documentsCitation : publication of new papers

(2) The attachment is NOT uniform.A node is linked with higher probability to a node thatalready has a large number of links: ʻʼRich get richerʼʼ

Examples :WWW : new documents link to wellknown sites (google, CNN, etc)Citation : well cited papers aremore likely to be cited again

Barabasi-Albert (1999)

(1) GROWTH : At every time step we add a new node with medges (connected to the nodes already present in thesystem).(2) PREFERENTIAL ATTACHMENT :The probability Π that a new node will be connected to node idepends on the connectivity ki of that node

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

jj

ii

k

kk

!=" )(

P(k) ~k-3

Barabási & Albert, Science 286, 509 (1999)

jj

ii

k

kk

!=" )(

Barabasi-Albert (1999)

Barabasi-Albert (1999)

Barabasi-Albert (1999)

Clustering coefficient

Average shortest path

Copy model

a. Selection of a vertex

b. Introduction of a new vertex

c. The new vertex copies m linksof the selected one

d. Each new link is kept with proba 1-α, rewiredat random with proba α

1−α

α

Growing network:

Copy model

Probability for a vertex to receive a new link at time t (N=t):

• Due to random rewiring: α/t

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

effective preferential attachment, withouta priori knowledge of degrees!

Copy model

Degree distribution:

model for WWW and evolution of genetic networks

=> Heavy-tails

Preferential attachment: generalization

Rank known but not the absolute value

Who is richer ?

Fortunato et al, PRL (2006)

Preferential attachment: generalization

Rank known but not the absolute value

Scale free network even in the absence of the value of the nodesʼ attributes

Fortunato et al, PRL (2006)

Weighted networks

• Topology and weights uncorrelated

• (2) Model with correlations ?

Weighted growing network

• Growth: at each time step a new node is added with m links tobe connected with previous nodes

• Preferential attachment: the probability that a new link isconnected to a given node is proportional to the nodeʼs strength

Barrat, Barthelemy, Vespignani, PRL 2004

Redistribution of weights

New node: n, attached to iNew weight wni=w0=1Weights between i and its other neighbours:

The new traffic n-i increases the traffic i-j

Onlyparameter

n i

j

Evolution equations

Evolution equations

Correlations topology/weights:

Numerical results: P(w), P(s)

(N=105)

Another mechanism:Heuristically Optimized Trade-offs (HOT)

Papadimitriou et al. (2002)

New vertex i connects to vertex j by minimizing the function Y(i,j) = a d(i,j) + V(j)d= euclidean distanceV(j)= measure of centrality

Optimization of conflicting objectives

Analytical results

Correlations topology/weights: wij ~ min(ki,kj)a , a=2d/(2d+1)

•power law growth of s

•k proportional to s