Introduction to complex networks Part I: Structure

Introduction to complex networksIntroduction to complex networksPart I: StructurePart I: Structure

Ginestra BianconiPhysics Department,Northeastern University, Boston,USA

NetSci 2010 Boston, May 10 2010

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Complex networks

describe

the underlying structure of interacting complex

Biological, Social and Technological systems.


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Identify your network !QuickTime™ and a decompressor








Why working on networks?Why working on networks?Because

NETWORKS

encode for the ORGANIZATION,

FUNCTION,ROBUSTENES

AND DYNAMICAL BEHAVIOR of the entire complex system

Types of networksSimple Each link is either existent or non existent, the links do not have directionality

(protein interaction map, Internet,…)

Directed The links have directionality,i.e., arrows(World-Wide-Web, social networks…)

Signed The links have a sign(transcription factor networks, epistatic networks…)



Types of networksWeighted The links are associated to a real number

indicating their weight(airport networks, phone-call networks…)With features of the nodes The nodes might

have weight or color(social networks, diseasome, ect..)




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Bipartite networks: Bipartite networks: ex. Metabolic networkex. Metabolic network

Reaction pathway Bipartite Graph

A+B C+DA+D EB+C E

A B C D E

1 2 3

A E

B

C

D

Metabolites projection

Reactions (enzymes) projection

1

2

3

1

2 3

Adjacency matrix

€

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

⎛

⎝

⎜ ⎜ ⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟ ⎟ ⎟

1

2 3

45

Network:A set of labeled nodes and links between them

Adjacency matrix:The matrix of entriesa(i,j)=1 if there is a ink

between node i and j

a(i,j)=0 otherwise

Total number of nodes Nand links L

The total number of nodes Nand the total number of links L

are the most basic characteristics of a network

Degree distribution

0

1

2

3

1 2 3

P(k)

1

2 3

45

€

ki = aijj

∑Degree if node i:Number of links of node i Network: Degree distribution:

P(k): how many nodes have degree k

Random graphs

Binomial Poisson distribution distribution

G(N,p) ensembleG(N,p) ensemble

Graphs with N nodesEach pair of nodes linked

with probability p

G(N,L) ensembleG(N,L) ensemble

Graphs with exactly N nodes and

L links

€

P(k) =N −1k

⎛ ⎝ ⎜

⎞ ⎠ ⎟pk (1 − p)N −1−k

P(k) =1k!c ke−c

Universalities:Scale-free degree distribution

)exp()(~)( 00

τ

γ

kkkkkkP +

−+ −

€

P(k)∝ k −γ γ ∈ (2,3)

k

Actor networks WWW Internet

∞→2k

finitek

kFaloutsos et al. 1999Barabasi-Albert 1999

Topology of the yeast protein Topology of the yeast protein networknetwork

€

P(k) ~ (k + k0)−γ e−k+k0

kτ

γ ≈ 2.5

H. Jeong et al. Nature (2001)

Scale-free networksScale-free networks • Technological networks:

– Internet, World-Wide Web

• Biological networks :Biological networks : – Metabolic networks,

– protein-interaction networks,– transcription networks

• Transportation networks:Transportation networks:

– Airport networks

• Social networks:Social networks: – Collaboration networks

– citation networks

• Economical networks:Economical networks: – Networks of shareholders

in the financial market – World Trade Web

Why this universality?

• Growing networks:Growing networks:– Preferential attachment

Barabasi & Albert 1999,Dorogovtsev Mendes 2000,Bianconi & Barabasi 2001,

etc.

• Static networks:Static networks:– Hidden variables mechanism

Chung & Lu 2002, Caldarelli et al. 2002, Park & Newman 2003

Scale-free networksScale-free networks

€

γ > 3

γ−∝ kkP )(

with

21 <γ<with

with

32 <γ<€

k finite

k 2 finite

€

k finite

k 2 →∞

€

k →∞

k 2 →∞

Hyppocampus functional neural network

€

P(k) ∝ k−γ

€

γ∈ (1,2]k →∞

k 2 →∞



Bonifazi et al . Science 2009Dense scale-free networks

Social network of phone calls

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€

P(k) =a

(k + k0)γe−k / kc

γ = 8.4

J. P Onnela PNAS 2007



Finite scale network

Clustering coefficient

€

Ci =|#of closed triangles |

ki(ki −1) /2

€

C2 =13

C3 =23

1

2 3

45

Definition of local clusteringcoefficient

Network Clustering coefficient of nodes 2,3

Shortest distance

The shortest distance between two nodes is the minimal number of links than a path must hop to go from the source to the destination

1

3

4

The shortest distancebetween node 4 and node 1 is 3between node 3 and node 1 is 2

2

5

Diameter and average distance

The diameter D of a network is the maximal length of the shortest distance between any pairs of nodes in the network

The average distance is the average length of the shortest distance between any pair of nodes in the network

€

D ≥ l

€

l

Universality: Small worldUniversality: Small world

Ci =# of links between 1,2,…ki neighbors

ki(ki-1)/2

Networks are clustered (large average Ci, i.e.C)

but have a small characteristic path length

(small L).

Network C Crand L N

WWW 0.1078 0.00023 3.1 153127

Internet 0.18-0.3 0.001 3.7-3.76 3015-6209

Actor 0.79 0.00027 3.65 225226

Coauthorship 0.43 0.00018 5.9 52909

Metabolic 0.32 0.026 2.9 282

Foodweb 0.22 0.06 2.43 134

C. elegance 0.28 0.05 2.65 282

Ki

i

Watts and Strogatz (1999)

Small world in social systems

1967 Milligran experimentPeople from Nebraska and Kansas where ask to

contact a person in Boston though their network of acquaintances

The strength of weak ties(Granovetter 1973)

Weak ties help navigate the social networks

Small-world model (Watts & Strogatz 1998)

The model for coexistence of high clustering and small distances in complex networks

Diameter and average lengthDiameter and average lengthDiameterDiameter

Average distanceAverage distance

Poisson Poisson networksnetworks

Scale-free Scale-free networksnetworks

)log(ND≈

)log()log(><

≈ kND

)log()log(><

≈ kNl

€

l ≈log(N)

log(log(N))if γ = 3

log(log(N)) if γ < 3

⎧ ⎨ ⎪

⎩ ⎪

Small world propertySmall world property Ultra small world propertyUltra small world property

Chung & Lu 2004 Cohen &Havlin 2003

Giant componentGiant componentA connected component of a network is a subgraph such that for each pair of nodes in the subgraph there is at least one path linking themThe giant component is the connected component of the network which contains a number of nodes of the same order of magnitude of the total number of links

Molloy-Reed principleMolloy-Reed principleA network has a giant A network has a giant

component ifcomponent if 22

≥><><

kk

Second moment <kSecond moment <k22>> Molloy-Reed condition

Poisson Poisson networksnetworks

Scale-free Scale-free networksnetworks

><+>>=<< kkk 22 1>≥<k

33)log(

2/)3(2

<γ=γ

⎩⎨⎧

>≈< γ− ifif

NNk 3≤γ

Molloy-Reed 1995

RobustnessRobustnessComplex systems maintain their basic functions

even under errors and failures

Node failure

fc

0 1Fraction of removed nodes, f

1

S

Robustness of scale-free networksRobustness of scale-free networks

1

S

0 1ffc

Attacks

FailuresTopological

error tolerance

R. Albert et al. 2000Cohen et al. 2000Cohen et al. 2001

RobustnessRobustness

Failure=Attack Robust against FailureWeak against Attack

R. Albert et al. 2000

Betweenness CentralityThe betweenness centrality of a node v

is given by

where st is the total number of shortest paths between s and t

st(v) are the number of such paths passing through v



€

B(v) =s,t≠v∑ σ st (v)

σ st

The nodes A and B are also called bridges and have high betweeness

Link probability in Link probability in uncorrelated networksuncorrelated networks

In uncorrelated networks the probability that a node i is linked to a node j is given by

€

pij =kik j

< k > N

i

j

€

pij = 23k N

€

PR (k,k ') =kk'k N

P(k)P(k ')

Degree correlationsDegree correlations

€

P(k,k ')PR (k,k ')

The map of

reveals correlations in the protein interaction map S. Maslov and K. Sneppen Science 2002

Link probability between nodes of degree classes k and k’

Specific characteristic of networks: Degree correlations

log(

k nn(k

))

α≈kkknn )(

Assortative networks α>0

Uncorrelated networks α=0

Disassortative networks α<0

kkiNjj

inn

i

kk

kk=∈

∑=)(

)( 1

log (k)

Average degree of a neighbor of a node of degree k

Degree-degree correlations in Degree-degree correlations in the Internet at the AS level the Internet at the AS level

€

knn (k) ≈ kα α < 0the network is the network is disassortativedisassortative

Vazquez et al. PRL 2001

€

knn (k) =1ki

k jj∈N ( i)∑

ki =k

Average degree of a neighbor of a node of degree k

log(k)

log(

C(k

)) Uncorrelated

networks =0

Modular networks >0

Correlations and Correlations and clustering coefficientclustering coefficient

€

C(k) ≈ k−δ

€

C(k) =ai, ja jrar,i

i, j,r∑k(k −1)

i k( i) =k

Average clustering coefficient C(k) of nodes of degree k

Clustering coefficient of Clustering coefficient of metabolic networksmetabolic networks

€

C(k) ≈ k−δ

There are important correlations in the network!The network is not ‘random’Highly connected nodes link more “distant” nodes of the networkRavasz,et al. Science (2002).

Loops or cycles of size LA loop or a cycle of size L is a path of the network that start at one point and ends after hopping L links on the same point without crossing the intermediate nodes more than one time

2 3

45

1

This network has 2 loops of size 3and 1 loop of size 4

Small loops in the Internet at the AS level

The number of loops of size L= 3,4,5

grows with the network size N as

In Poisson networks instead they are a fixed number

Independent on N

€

ℵL ∝ N ξ (L )

G. Bianconi et al.PRE 2005

Cliques of size c

Clique of size cis a fully connected subgraph of the network of

c nodes and c(c-1)/2 links

2 3

45

1This network contains 4 cliques of size 3 (triangles)and 1 clique of size 4

Small subgraphs appear Small subgraphs appear abruptly when we increase abruptly when we increase

the average number of the average number of links in the random graphslinks in the random graphs

€

c =LN

≈ Nα

∞−α

€

c ≈ Nα

-1 -1/2 -1/3 -1/4 0 1/3 1/2

Subgraph thresholdsSubgraph thresholds

Finite average connectivity

Poisson distribution

Diverging average connectivity

∞

2/)3(

21 L

L NL

γ−≈ℵ

γ−≈k)k(P

γ 3 2 1

Small subgraphs in SF networksSmall subgraphs in SF networks

Cmax=3

Small loopsOf length L

Cliques of Size C

γγ /)3(

21 L

L NL

−≈ℵ

c

c ccN

⎟⎟⎠⎞

⎜⎜⎝⎛

−≈ℵ

−

)(

2/)3(

γ

γc

c ccN

⎟⎟⎠⎞

⎜⎜⎝⎛

−≈ℵ

)(

)/1(

γ

γ

∞⏐⏐ →⏐ ∞→Nmaxc ∞⏐⏐ →⏐ ∞→N

maxc

finiteLℵ

G. Bianconi and M. Marsili (2004), (2005)

€

k →∞

€

k finite

K-core of a networkK-core of a networkA A K-coreK-core of a network of a network

is the subgraph of a network obtained by is the subgraph of a network obtained by removing the nodes with connectivity kremoving the nodes with connectivity kii<K<K

iteratively until the network iteratively until the network has only nodes with has only nodes with

K-core of the InternetDIMES Internet dataVisualization:Alvarez-Hamelin et al. 2005

€

ki ≥K

K-coreson trees with given degree

distributions

Scale-free networks

γγ −−≈ 32max cutKpmK

1

max )(−

⎟⎟⎠⎞

⎜⎜⎝⎛

≈γ

cuτKmpKM

S. Dorogovtsev, et al. PRL (2006) Carmi et al. PNAS (2007)

Maximal K of the K-cores

Size of the minimal K-core

How to build a null modelHow to build a null modelform a given network:form a given network:swap of connectionsswap of connections

ChooseChoose two two random links random links linking four distinct linking four distinct nodesnodes

If possible (not If possible (not already existing already existing links) links) swap the swap the ends of the linksends of the links

Maslov & Sneppen 2002

MotifsMotifsThe motifs are subgraphssubgraphs which appear with higher frequency in real networks than in randomized networks

In biological networks the motifs are told to be motifs are told to be selected selected by evolutionby evolutionand are relevant to and are relevant to understand the function of understand the function of the network.the network.

Motifs in the transcriptome network of e.coli.S.S. Shen-Orr, et al., 2002

Milo et al. 2002

Motifs of size 3 in directed networks



The number of distinguishable possible motifs increases exponentially with the motif size limiting the extension of this method to large subgraphs

Specific characteristics of a network:

communities

A community of a network define a set of nodes with similar connectivity pattern.

Dolphins social network High-school dating networks

S. Fortunato Phys. Rep. 2010

Girvan and Newman algorithm

1. The betweenness of all existing edges in the network is calculated first.

2. The edge with the highest betweenness is removed.

3. The betweenness of all edges affected by the removal is recalculated.

4. Steps 2 and 3 are repeated until no edges remain.

Girvan and Newman PNAS 2002

The algorithm

ModularityAssign a community si=1,2,…K to each

node, The modularity Q is given by

Tight communities can be found by maximizing the modularity function

€

Q =1k N

aij −kik j

k N

⎡ ⎣ ⎢

⎤ ⎦ ⎥

ij∑ δ (si,s j )

Newman PNAS 2006

Cliques Cliques for community for community detectiondetection

Overlapping set of cliques can be

helpful to identify community

structures in different networks

For this method to work systematically networks must have many cliques (ex. Scale-free networks)

Palla et al. Nature (2005)

Weight distribution and weight-Weight distribution and weight-degree correlationsdegree correlations

The weights of a network might correlate with the degree of the nodes which are linked by them.

Strength Sk

Is a measure of the inhomogeneities of the distribution of the weights among the nodes of the network.

Disparity Y2k

Is a measure of the inhomogeneities of the weights ending at a node of degree k.

Strength of the node and Strength of the node and weight-degree correlationsweight-degree correlations

The strength of the nodes is the average weight of links connecting that node

€

Sk = wijj

∑ki =k

∝k

k1+η

⎧ ⎨ ⎪

⎩ ⎪

All weights distribute equally to the strength of the nodes

Weight-degree correlations:Highly weighted links are connected to highly connected nodes

Coauthorship networksCoauthorship networks

The coauthorship network is a scale-free network

Power-law strength distribution and

Linear dependence of strength versus degree

A. Barrat et al. PNAS 2004

Airport networkAirport network

The airport network is a scale-free network

With power-law strength distribution and

The strength versus k show weight-degree correlations.

A. Barrat et al. PNAS 2004

The DisparityThe Disparity

€

Y 2k =

wi, j

si

⎛ ⎝ ⎜

⎞ ⎠ ⎟

j∑

2

ki =k

≈1/k....1

⎧ ⎨ ⎪

⎩ ⎪

All weights contribute equally

There is one dominating link

In many networks we observeIn many networks we observe

indicating a hierarchy of weightsindicating a hierarchy of weights

€

Y 2(k) =1/kα with α ∈(0,1)

Metabolic NetworkMetabolic NetworkThe degree distribution of the metabolite projection of the metabolic network is scale-free.

The distribution of the fluxes in metabolic networks (e.coli in succinate substrate) is power-law.

The Y2(k) predicted by flux-balance analysis techniques is non trivial

€

Y 2(k) ≈ k −0.27Almaas et al. Nature 2004

Extraction of sector Extraction of sector information in financial information in financial

marketsmarkets

Minimal-Spanning-Trees Planar maximally filtered graphs

NYSE daily returns USA equity market 1995-98 Bonanno et al. (2003) Tumminello et al. (2007)

Why working on networks?Why working on networks?Because

NETWORKS

encode for the ORGANIZATION,

FUNCTION,ROBUSTENES

AND DYNAMICAL BEHAVIOR of the entire complex system

Documents

Introduction to complex networks Part I: Structure