Statistical Mechanics of Complex Networks: Economy, Biology and Computer Networks

Albert Diaz-Guilera Universitat de Barcelona

Outline

Complex systems Topological properties of networks Complex networks in nature and society Tools Models Dynamics

Physicist out their land Multidisciplinary research Reductionism = simplicity Scaling properties Universality

Multidisciplinary research Intricate web of researchers coming from very different fields Different formation and points of view Different languages in a common framework Complexity

Complexity Challenge: “Accurate and complete description of complex systems” Emergent properties out of very simple rules – unit dynamics – interactions

Why is network anatomy important Structure always affects function The topology of social networks affects the spread of information Internet + access to the information - electronic viruses

Current interest on networks Internet: access to huge databases Powerful computers that can process this information Real world structure: – regular lattice?

– random?

– all to all?

Network complexity Structural complexity: topology Network evolution: change over time Connection diversity: links can have directions,

weights, or signs

Dynamical complexity: nodes can be complex

nonlinear dynamical systems

Node diversity: different kinds of nodes

Topological properties

Degree distribution Clustering Shortest paths Betweenness Spectrum

Degree Number of links that a node has It corresponds to the local centrality in social network analysis It measures how important is a node with respect to its nearest neighbors

Degree distribution Gives an idea of the spread in the number of links the nodes have P(k) is the probability that a randomly selected node has k links

What should we expect?

In regular lattices all nodes are identical In random networks the majority of nodes have approximately the same degree Real-world networks: this distribution has a power-law tail

P

(

k

) 

k

  “scale-free” networks

Clustering Cycles in social network analysis language Circles of friends in which every member knows each other

Clustering coefficient Clustering coefficient of a node

C i



k i

(

k i E



i

1 ) / 2 Clustering coefficient of the network

C

 1

N i N

  1

C i

What happens in real networks?

The clustering coefficient is much larger than it is in an equivalent random network

Ego-centric vs. socio-centric Focus is on links surrounding particular agents (degree and clustering) Focus on the pattern of connections in the networks as a whole (paths and distances) Local centrality vs. global centrality

Distance between two nodes Number of links that make up the path between two points 1 2 3 “Geodesic” = shortest path Global centrality: points that are “close” to many other points in the network. Global centrality defined as the sum of minimum distances to any other point in the networks

Local vs global centrality A,C Local 5 global 43 B 5 33 G,M 2 37 J,K,L 1 48 All other 1 57

Global centrality of the whole network?

Mean shortest path = average over all pairs of nodes in the network

Betweenness Measures the “intermediary” role in the network It is a set of matrices, one for ach node

B k ij

Ratio of shortest paths bewteen i and j that go through k 0 

B ij k

 1 There can be more than one geodesic between i and j Comments on Fig. 5.1

Pair dependency Pair dependency of point i on point k Sum of betweenness of k for all points that involve i Row-element on column-element

Betweenness of a point Half the sum (count twice) of the values of the columns Ratio of geodesics that go through a point Distribution (histogram) of betweenness The node with the maximum betweenness plays a central role

Spectrum of the adjancency matrix Set of eigenvalues of the adjacency matrix Spectral density (density of eigenvalues)   1

N j N

  1     

j



A symmetric and real => eigenvalues are real and the largest is not degenerate Largest eigenvalue: shows the density of links Second largest: related to the conductance of the graph as a set of resistances Quantitatively compare different types of networks

Tools

Input of raw data Storing: format with reduced disk space in a computer Analyzing: translation from different formats Computer tools have an appropriate language (matrices, graphs, ...) Import and export data

Complex networks in nature and society

NOT regular lattices NOT random graphs “simple” mathematical analysis Huge databases and computer power

Networks of collaboration Through collaboration acts Examples: – movie actor – board of directors – scientific collaboration networks (MEDLINE, Mathematical, neuroscience, e-archives,..) => Erdös number

Coauthorship network

Communication networks Hyperlinks (directed) Hosts, servers, routers through physical cables (not directed) Flow of information within a company: employees process information Phone call networks (  =2)

Internet

Networks of citations of scientific papers Nodes: papers Links (directed): citations  =3

Social networks Friendship networks (exponential) Human sexual contacts (power-law) Linguistics: words are connected if – Next or one word apart in sentences – Synonymous according to the Merrian-Webster Dictionary

Biological networks Neural networks: neurons – synapses Metabolic reactions: molecular compounds – metabolic reactions Protein networks: protein-protein interaction Protein folding: two configurations are connected if they can be obtained from each other by an elementary move Food-webs: predator-prey (directed)

C. elegans neural network

East River, CO, USA Food webs Little Rock Lake, WI, USA

Engineering networks Power-grid networks: generators, transformers, and substations; through high-voltage transmission lines Electronic circuits: electronic components (resistor, diodes, capacitors, logical gates) – wires Software engineering

Average path length random graph

Clustering

internet Degree distribution movie actors high energy coauthorship neuroscience coauthorship

Models

Random graph (Erdös-Renyi) Small world (Watts-Strogatz) Scale-free networks (Barabasi-Albert)

Random graph Binomial model: start with N nodes, every pair of nodes being connected with probability p The total number of links, n, is a random variable – E(n)=pN(N-1)/2 Probability of generating a graph, G 0 {N,n}

P

(

G

0 ) 

p n

( 1 

p

)

N

(

N

 1 ) 

n

2

Degree distribution The degree of a node follows a binomial distribution (in a random graph with p)

P

(

k i



k

) 

k

1  

p k

( 1 

p

)

N

 1 

k

Probability that a given node has a connectivity k For large N, Poisson distribution

P

(

k

) 

e



pN

(

pN

)

k k

!



e

 

k

 

k k

!



k

Mean short path Assume that the graph is homogeneous The number of nodes at distance l are l How to reach the rest of the nodes?

l rand to reach all nodes => k l =N

l

rand  ln ln 

N k

  ln ln

N pN

Clustering coefficient Probability that two nodes are connected (given that they are connected to a third)?

C rand



p

 

k N



C rand



k

  1

N

while it is constant for real networks

Small world Crossover from regular lattices to random graphs Tunable Small world network with (simultaneously): – Small average shortest path – Large clustering coefficient (not obeyed by RG)

Scale-free networks Networks grow preferentially

P(k)= exp (-k 2 /A 2 ) P(k)=k 

Dynamics

Network dynamics: – global goal – local goal Flow in complex networks: – ideas – innovations – computer viruses – problems

Global vs local optimization Design: the goal is to optimize global quantity (distance, clustering, density, ...) Evolution: decision taken at node level

Virus spreading prevalence in scale-free networks infection rate

Communication model  Communicating agents : computers, employees  Communication channels : cables, email, phone  Information packets : packets, problems  Finite capacity of the agents to deliver information

Summary