dynamical processes on complex networks · books and reviews on processes on complex networks •...

Dynamical processes oncomplex networks

Marc BarthélemyCEA, IPhT, France

Lectures IPhT 2010

Outline I. Introduction: Complex networks

1. Example of real world networks and processes

II. Characterization and models of complex networks

1. Tool: Graph theory and characterization of large networks Diameter, degree distribution, correlations (clustering, assortativity), modularity

2. Some empirical results

3. Models Fixed N: Erdos-Renyi random graph, Watts-Strogatz, Molloy-Reed algorithm,Hidden variables Growing networks: Barabasi-Albert and variants (copy model, weighted, fitness)

III. Dynamical processes on complex networks

1. Overview: from physics to search engines and social systems

2. Percolation

3. Epidemic spread: contact network and metapopulation models

• Statistical mechanics of complex networksReka Albert, Albert-Laszlo BarabasiReviews of Modern Physics 74, 47 (2002)cond-mat/0106096

• The structure and function of complex networksM. E. J. Newman, SIAM Review 45, 167-256 (2003)cond-mat/0303516

• Evolution of networksS.N. Dorogovtsev, J.F.F. Mendes, Adv. Phys. 51, 1079 (2002)cond-mat/0106144

References: reviews on complex networks

References: books on complex networks

• Evolution and structure of the Internet: A statistical PhysicsApproachR. Pastor-Satorras, A. VespignaniCambridge University Press, 2004.

• Evolution of networks: from biological nets to the Internet andWWWS.N. Dorogovtsev, J.F.F. MendesOxford Univ. Press, 2003.

• Scale-free networksG. CaldarelliOxford Univ. Press, 2007.

Books and reviews on processes oncomplex networks

• Complex networks: structure and dynamicsS. Boccaletti et al.Physics Reports 424, 175-308 (2006).

• Dynamical processes on complex networksA. Barrat, M. Barthélemy, A. VespignaniCambridge Univ. Press, 2008.

What is a network ?

Network=set of nodes joined by links G=(V,E)

- very abstract representation- very general

- convenient to describe many different systems:biology, infrastructures, social systems, …

Networks and Physics

• Complex

Most networks of interest are:

• Very large

Statistical tools needed ! (see next chapter)

Studies on complex networks (1998- )

• 1. Empirical studies Typology- find the general features

• 2. Modeling Basic mechanisms/reproducing stylized facts

• 3. Dynamical processesImpact of the topology on the properties of dynamicalprocesses: epidemic spread, robustness, …

Empirical studies: Unprecedentedamount of data…..

Transportation infrastructures (eg. BTS)

Census data (socio-economical data)

Social networks (eg. online communities)

Biological networks (-omics)

Empirical studies: sampling issues

Social networks: various samplings/networks Transportation network: reliable data Biological networks: incomplete samplings Internet: various (incomplete) mapping processes WWW: regular crawls …

possibility of introducing biases in themeasured network characteristics

Networks characteristics

Networks: of very different origins

Do they have anything in common?Possibility to find common properties?

- The abstract character of the graph representationand graph theory allow to give some answers…- Important ingredients for the modeling

Modeling complex networks

Microscopical processes

Properties at the macroscopic level

Statistical physics

• many interacting elements• dynamical evolution• self-organisation

Non-trivial structureEmergent properties, cooperative

phenomena

Networks inInformation technology

Networks in Information technology:processesImportance of Internet and the web

Congestion Virus propagation Cooperative/social phenomena (online

communities, etc.) Random walks, search (pagerank algo,…)

- Nodes=routers- Links= physical connections

different granularities

Internet

Many mapping projects (topology and performance): CAIDA, NLANR, RIPE, …(http://www.caida.org)

Internet mapping

• continuously evolving and growing• intrinsic heterogeneity• self-organizing

Largely unknown topology/properties

Nodes: Computers, routersLinks: physical lines

Large-scale visualization

Internet backbone

Internet-map

World Wide Web

Over 1 billion documents

ROBOT: collects all URL’sfound in a document andfollows them recursively

Nodes: WWW documentsLinks: URL links

Virtual network to find and shareinformations

Social networks

Social networks: processes

Many social networks are the support of somedynamical processes

Disease spread Rumor propagation Opinion/consensus formation Cooperative phenomena …

Scientific collaboration network

Nodes: scientistsLinks: co-authored papers

Weights: depending on• number of co-authored papers• number of authors of each paper• number of citations…

Citation network Nodes: papers

Links: citations

Science citation indexS. Redner

Hopfield J.J., PNAS1982

Actor’s network Nodes: actorsLinks: cast jointly

N = 212,250 actors〈k〉 = 28.78

John Carradine

The Sentinel

(1977)

The story of

Mankind (1957)

http://www.cs.virginia.edu/oracle/star_links.html

Ava Gardner Groucho Marx

distance(Ava, Groucho)=2

Distance ?

Character network Nodes: characters Links: co-appearance in a scene

Les Miserables-V. HugoNewman & Girvan, PRE (2004)-> Community detection problem

The web of Human sexualThe web of Human sexualcontactscontacts

Liljeros et al., Nature (2001)

Transportation networks

Transporting energy, goods or individuals

- formation and evolution

- congestion, optimization, robustness

- disease spread

NB: spatial networks (planar): interesting modeling problems

Nodes: airportsLinks: direct flight

Transporting individuals: global scale(air travel)

Transporting individuals: intra city

TRANSIMS project

Nodes: locations(homes, shops,offices, …) Links: flow ofindividuals

Chowell et al Phys. Rev. E (2003) Nature (2004)

Transporting individuals: intra city

Transporting goods

State of Indiana (Bureau of Transportation statistics)

Nodes: powerplants, transformers,etc,…) Links: cables

Transporting electricity

New York state power grid

Transporting electricity US power grid

Transporting gas European pipelines

Transporting water

Nodes: intersections, auxinssourcesLinks: veins

Example of aplanar network

Networks in biology

(sub-)cellular level: Extracting usefulinformation from the huge amount ofavailable data (genome, etc). Link structure-function ?

Species level: Stability of ecosystems,biodiversity

Neural Network Nodes: neuronsLinks: axons

Metabolic Network

Nodes: proteinsLinks: interactions

Protein Interactions

Nodes: metabolitesLinks:chemical reactions

Genetic (regulatory) network

Nodes: genesLinks: interaction

Genes are colored according to theircellular roles as described in the YeastProtein Database

Understanding these networks: one of themost important challenge in complexnetwork studies

Food webs Nodes: species Links: feeds on

N. Martinez

Lectures IPhT 2010

2. Percolation

II. 1 Graph theory basics and statistical tools

Graph theory: basics

Graph G=(V,E)

V=set of nodes/vertices i=1,…,N E=set of links/edges (i,j)

Undirected edge:

Directed edge:

Bidirectional communication/interaction

Maximum number of edges Undirected: N(N-1)/2 Directed: N(N-1)

Complete graph :

(all-to-all interaction/communication)

Planar graph: can be embedded in the plane, i.e., it canbe drawn on the plane in such a way that its edges mayintersect only at their endpoints.

planar Non planar

Adjacency matrix

N nodes i=1,…,N

1 if (i,j) E0 if (i,j) ∉ E

011131011211011111003210 0

Symmetricfor undirected networks

Adjacency matrix

N nodes i=1,…,N

1 if (i,j) E0 if (i,j) ∉ E

011030010200001101003210

Non symmetricfor directed networks

Sparse graphs

Density of a graph D=|E|/(N(N-1)/2)

Number of edges

Maximal number of edgesD=

Sparse graph: D <<1 Sparse adjacency matrix

Representation: lists of neighbours of each node

l(i, V(i))

V(i)=neighbourhood of i

G=(V,E)Path of length n = ordered collection of n+1 vertices i0,i1,…,in V n edges (i0,i1), (i1,i2)…,(in-1,in) E

i2i0 i1

Cycle/loop = closed path (i0=in)

Paths and connectedness

G=(V,E) is connected if and only if there exists apath connecting any two nodes in G

is connected

• is not connected• is formed by two components

G=(V,E)In general: different components with different sizes

Giant component= component whosesize scales with the number of vertices N

Existence of agiant component

Macroscopic fraction of thegraph is connected

Paths and connectedness

Paths and connectedness: directed graphs

Tendrils Tube Tendril

Giant SCC: Strongly Connected Component Giant OUT

ComponentGiant IN Component

Disconnectedcomponents

Paths can be directed (eg. WWW)

Paths and connectedness: directed graphs

“Bow tie” diagram for the WWW (Broder et al,2000)

Shortest paths

Shortest path between i and j: minimum numberof traversed edges

distance l(i,j)=minimum number ofedges traversed on a pathbetween i and j

Diameter of the graph:

Average shortest path:

Shortest paths

Complete graph:

Regular lattice:

“Small-world” network

Degree of a node

k>>1: Hubs

How many friends do you have ?

Statistical characterization

• List of degrees k1,k2,…,kN Not very useful!

• Histogram: Nk= number of nodes with degree k

• Distribution: P(k)=Nk/N=probability that a randomly chosen node has degree k

• Cumulative distribution: probability that a randomly chosen node has degree at least k (reduced noise)

Statistical characterization

• Average degree:

• Second moment (variance):

Sparse graph:

Heterogeneous graph:

Degree distribution P(k) : probability that a node has k links

Degree distribution

Poisson/Gaussian distribution Power-law distribution

Centrality measuresHow to quantify the importance of a node?

Degree=number of neighbours=∑j aij

Closeness centrality

gi= 1 / ∑j l(i,j)

(directed graphs: kin, kout)

Betweenness Centrality

σst = # of shortest paths from s to tσst(ij)= # of shortest paths from s to t via (ij)

kij: large centrality

jk: small centrality

P(k): not enough to characterize a network

Large degree nodes tend toconnect to large degree nodesEx: social networks

Large degree nodes tend toconnect to small degree nodesEx: technological networks

Multipoint degree correlations

Measure of correlations:P(k’,k’’,…k(n)|k): conditional probability that a node of degreek is connected to nodes of degree k’, k’’,…

Simplest case:P(k’|k): conditional probability that a node of degree k isconnected to a node of degree k’

Case of random uncorrelated networks

• P(k’|k) independent of k

• proba that an edge points to a node of degree k’:

proportional to kʼ

number of edges from nodes of degree k’number of edges from nodes of any degree

2-points correlations: Assortativity

Are your friends similar to you ?

• P(k’|k): difficult to handle and to represent

Exemple:

k=3k=7

k=4k=4

ki=4knn,i=(3+4+4+7)/4=4.5

Assortativity

Correlation spectrum:Putting together nodes whichhave the same degree:

class of degree k

Assortativity

Also given by:

Assortativity

Assortative behaviour: growing knn(k)Example: social networksLarge sites are connected with large sites

Disassortative behaviour: decreasing knn(k)Example: internetLarge sites connected with small sites, hierarchical structure

# of links between neighbors

k(k-1)/2

Do your friends know each other ?

3-points: Clustering coefficient

• P(k’,k’’|k): cumbersome, difficult to estimate from data

Correlations: Clustering spectrum

• Average clustering coefficient

= average over nodes withvery different characteristics

• Clustering spectrum:

putting together nodes whichhave the same degree class of degree k

(link with hierarchical structures)

Real networks are fragmented into group or modules

Modularity vs.Fonctionality ?

More on correlations: communities and modularity

Extract relevant anduseful information inlarge complex network

More links “inside”than “outside”

More on correlations: communities and modularity

Mesoscopic objects:communities(or modules)

Applications

Biological networks: function ?

Social networks: social groups

Geography/regional science: urbancommunities

Neurophysiology (?)

How can we compare differentpartitions?

Modularity

Partition P1

Partition P2

- Modularity: quality function Q- P1 is better than P2 if Q(P1)>Q(P2)

Modularity [Newman & Girvan PRE 2004]

Number of modules in P Number of links in module s Number of links in the network Total degree of nodes in module s

= # links in module s

Average number oflinks in module s for arandom network

= Total degree of nodes in module s

probability that a half-link randomlyselected belongs in module s:

probability that a link begins in sand ends in s’:

average number of links internal tos:

Motifs

• Find n-node subgraphs in real graph.

• Find all n-node subgraphs in a set of randomized graphs with

the same distribution of incoming and outgoing arrows.

• Assign Z-score for each subgraph.

• Subgraphs with high Z-scores are denoted as Network

Motifs.

randreal NNZ

Milo et al, Science (2002)

Summary: characterization of large networks

Statistical indicators:-degree distribution

- diameter

- clustering

- assortativity

- modularity

Internet, Web, Emails: importance of traffic

Ecosystems: prey-predator interaction

Airport network: number of passengers

Scientific collaboration: number of papers

Metabolic networks: fluxes hererogeneous

Are:- Weighted networks

- With broad distributions of weights

Beyond topology: weights

Weighted networks

General description: weights

i jwij

aij: 0 or 1wij: continuous variable

usually wii=0symmetric: wij=wji

Weighted networks

Weights: on the links

Strength of a node:

Naturally generalizes the degree to weighted networks

Quantifies for example the total traffic at a node

Weighted clustering coefficient

si=16ci

w=0.625 > ci

ki=4ci=0.5

si=8ci

w=0.25 < ci

Random(ized) weights: C = Cw

C < Cw : more weights on cliquesC > Cw : less weights on cliques

k(wjk)

Average clustering coefficientC=∑i C(i)/NCw=∑i Cw(i)/N

Clustering spectra

Weighted clustering coefficient

Weighted assortativity

ki=5; knn,i=1.8

ki=5; si=21; knn,i=1.8 ; knn,iw=1.2: knn,i > knn,i

ki=5; si=9; knn,i=1.8 ; knn,iw=3.2: knn,i < knn,i

Participation ratio “disparity”

1/ki if all weights equal

close to 1 if few weights dominate

Participation ratio “disparity”

In general:

Heterogeneous

Homogeneous (all equal)

Lectures IPhT 2010

2. Percolation

II. 3 Models of complexnetworks

Some models of complex networks

Static networks: N nodes from the beginning- The archetype: Erdos-Renyi random graph (60’s)- A small-world: Watts-Strogatz model (1998)- Fitness models (hidden variables)

Dynamic: the network is growing- Scale-free network: Barabasi-Albert (1999)- Copy model- Weighted network model

Optimal networks

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.

Erdös-Renyi (1959): recap

Some properties:

- Average number of edges

- Average degree

Finite average degree

Erdös-Renyi model: recap

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:

: 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

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

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

• Degree

• Degree distribution

• 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 are morelikely to be cited again

(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)

!=" )(

P(k) ~k-3

Barabási & Albert, Science 286, 509 (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

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

Mean-field 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) = α d(i,j) + V(j)d= euclidean distanceV(j)= measure of centrality

Optimization of conflicting objectives

Lectures IPhT 2010

1. Tool: Graph theory and characterization of large networks

3. Models Erdos-Renyi random graph Watts-Strogatz small-world Barabasi-Albert and variants

1. Percolation

2. Epidemic spread

3. Other processes: from physics to social systems

III. 1 Percolation, randomfailures, targetedattacks

Consequences of the topologicalheterogeneity

Robustness and vulnerability

RobustnessComplex systems maintain their basic functions

even under errors and failures(cell: mutations; Internet: router breakdowns)

node failure

0 1Fraction of removed nodes, f

SS: fraction of giantcomponent

Case of Case of Scale-free NetworksScale-free Networks

Random failure fc =1 (2 < γ ≤ 3)

Attack =progressive failure of the mostconnected nodes fc <1

R. Albert, H. Jeong, A.L. Barabasi, Nature 406 378 (2000)

Random failures: percolation

p=probability of anode to be presentin a percolationproblem

Question: existence or not of a giant/percolating cluster, i.e. of a connected cluster of nodes of size O(N)

f=fraction ofnodes removedbecause of failure

Percolation in complex networks

q=probability that a randomly chosen link does notlead to a giant percolating cluster

Proba that none of theoutgoing k-1 links leadsto a giant cluster

Proba that the link leadsto a node of degree k

Average overdegrees

q=probability that a randomly chosen link does not lead to agiant percolating cluster

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