Complex networks and random matrices. Geoff Rodgers School of Information Systems, Computing and Mathematics

Plan 1.Introduction to scale free graphs 2.Small world networks 3.Static model of scale free graphs 4.Eigenvalue spectrum of scale free graphs 5.Results 6.Conclusions.

Scale Free Networks Many of networks in economic, physical, technological and social systems have been found to have a power-law degree distribution. That is, the number of vertices N(m) with m edges is given by N(m) ~ m - 

Examples of real networks with power law degree distributions NetworkNodesLinks/EdgesAttributes World-Wide WebWebpagesHyperlinksDirected InternetComputers and RoutersWires and cablesUndirected Actor CollaborationActorsFilmsUndirected Science CollaborationAuthorsPapersUndirected CitationArticlesCitationDirected Phone-callTelephone NumberPhone callDirected Power gridGenerators, transformers and substationsHigh voltage transmission linesDirected

Web-graph Vertices are web pages Edges are html links Measured in a massive web-crawl of 10 8 web pages by researchers at altavista Both in- and out-degree distributions are power law with exponents around 2.1 to 2.3.

Collaboration graph Edges are joint authored publications. Vertices are authors. Power law degree distribution with exponent ≈ 3. Redner, Eur Phys J B, 2001.

These graphs are generally grown, i.e. vertices and edges added over time. The simplest model, introduced by Albert and Barabasi, is one in which we add a new vertex at each time step. Connect the new vertex to an existing vertex of degree k with rate proportional to k.

For example: A network with 10 vertices. Total degree 18. Connect new vertex number 11 to vertex 1 with probability 5/18 vertex 2 with probability 3/18 vertex 7 with probability 3/18 all other vertices, probability 1/18 each.

This network is completely solvable analytically – the number of vertices of degree k at time t, n k (t), obeys the differential equation where M(t) =  kn k (t) is the total degree of the network.

Simple to show that as t   n k (t) ~ k -3 t power-law.

Small world networks Normally defined by two properties: 1.Local order: If vertices A and B are neighbours and B and C are neighbours then good chance that A and C are neighbours. 2.Finite number of steps between any pair of vertices (this is the small world effect).

Property 1 is generally associated with regular graphs e.g. 2-d square network. Property 2 is generally associated with random graphs or mean field systems.

Scale free networks are small world. But not all small world networks are scale free.

Models of small world networks Most famous due to Newman and Watts: Let n sites be connected in a circle. Each of several neighbours is connected by a unit length edge. Then each of these edges is re-wired with probability p to a randomly chosen vertex.

p = 0 is a regular ordered structure. p = 1 is an ER random graph. Small world for 0 < p < 1. Average shortest distance behaves as ~ n for p = 0 and ~ log n for p > 0.

Obviously such an approach can be generalised to any regular graph, 2-d, 3-d etc… Models are difficult to formulate analytically. Only some of the most basic properties have been obtained analytically, in contrast to both random and scale free graphs.

Local ordering property 1 Can be quantified by the clustering coefficient C, C= 3 (no. of triangles) / (no. of connected triples) This is the probability that two of one’s friends are friends themselves.

C = 1 on a fully connected graph (everyone knows everyone else). Typical values of 0.1 to 0.5 in many real world graphs.

Community Structure Sometimes called clustering, and confused with the previous property. The detection of community structure is an important part of analysing many social and technological networks. Helps to understand the interplay between network structure and function.

Traditional methods for detecting community structure involve giving each edge a weight, by some method, and arranging in a hierarchical way. Recently a new method has been introduced by Girvan and Newman. This focuses on the edges which are least central, or most between communities.

Girvan-Newman Algorithm Calculate the betweenness for all edges in the network. That is, the number of shortest paths going down each edge. Remove the edge with the highest betweenness. Recalculate the betweennesses for all edges affected by the removal. Repeat until no edges remain.

Over time, this algorithm will yield the community structure.

Static Model of Scale Free Networks An alternative theoretical formulation for a scale free graph is through the static model. Start with N disconnected vertices i = 1,…,N. Assign each vertex a probability P i.

At each time step two vertices i and j are selected with probability P i and P j. If vertices i and j are connected, or i = j, then do nothing. Otherwise an edge is introduced between i and j. This is repeated pN/2 times, where p is the average number of edges per vertex.

When P i = 1/N we recover the Erdos- Renyi graph. When P i ~ i -α then the resulting graph is power-law with exponent λ = 1+1/ α.

The probability that vertices i and j are joined by an edge is f ij, where f ij = 1 - (1-2P i P j ) pN/2 ~ 1 - exp{-pNP i P j } When NP i P j 3, then f ij ~ 2NP i P j j

Adjacency Matrix The adjacency matrix A of this network has elements A ij = A ji with probability distribution P(A ij ) = f ij δ(A ij -1) + (1-f ij )δ(A ij ).

This matrix has been studied by a number of workers Farkas, Derenyi, Barabasi & Vicsek; Numerical study ρ(μ) ~ 1/μ 5 for large μ. Goh, Kahng and Kim, similar numerical study; ρ(μ) ~ 1/μ 4. Dorogovtsev, Goltsev, Mendes & Samukin; analytical work; tree like scale free graph in the continuum approximation; ρ(μ) ~ 1/μ 2λ-1.

We will follow Rodgers and Bray, Phys Rev B (1988), to calculate the eigenvalue spectrum of the adjacency matrix.

Introduce a generating function where the average eigenvalue density is given by and denotes an average over the disorder in the matrix A.

Normally evaluate the average over lnZ using the replica trick; evaluate the average over Z n and then use the fact that as n → 0, (Z n -1)/n → lnZ.

We use the replica trick and after some maths we can obtain a set of closed equation for the average density of eigenvalues. We first define an average [ …] ,i where the index  = 1,..,n is the replica index.

The function g obeys and the average density of states is given by

Hence in principle we can obtain the average density of states for any static network by solving for g and using the result to obtain  (  ). Even using the fact that we expect the solution to be replica symmetric, this is impossible in general. Instead follow previous study, and look for solution in the dense, p   when g is both quadratic and replica symmetric.

In particular, when g takes the form

In the limit n  0 we have the solution where a(  ) is given by

Random graphs: Placing P k = 1/N gives an Erdos Renyi graph and yields as p → ∞ which is in agreement with Rodgers and Bray, 1988.

Scale Free Graphs To calculate the eigenvalue spectrum of a scale free graph we must choose This gives a scale free graph and power-law degree distribution with exponent = 1+1/ .

When  = ½ or = 3 we can solve exactly to yield where note that

General  Can show that in the limit    then

Conclusions Shown how the eigenvalue spectrum of the adjacency matrix of an arbitrary network can be obtained analytically. Again reinforces the position of the replica method as a systematic approach to a range of questions within statistical physics.

Conclusions Obtained a pair of simple exact equations which yield the eigenvalue spectrum for an arbitrary complex network in the high density limit. Obtained known results for the Erdos Renyi random graph. Found the eigenvalue spectrum exactly for λ = 3 scale free graph.

Conclusions In the tail found In agreement with results from the continuum approximation to a set of equations derived for a tree-like scale free graph.

The same result has been obtained for both dense and tree-like graphs. These can be viewed as at opposite ends of the “ensemble” of scale free graphs. This suggests that this form of the tail may be universal. Conclusions

Further details Eigenvalue spectrum Rodgers, Austin, Kahng and Kim J Phys A (2005). Spin glass Kim, Rodgers, Kahng and Kim Phys Rev E (2005).