Complex networks and random matrices. Geoff Rodgers School of Information Systems, Computing and Mathematics
Plan • Introduction to scale free graphs • Small world networks • Static model of scale free graphs • Eigenvalue spectrum of scale free graphs • Results • 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 -
Network Nodes Links/Edges Attributes World-Wide Web Webpages Hyperlinks Directed Internet Computers and Routers Wires and cables Undirected Actor Collaboration Actors Films Undirected Science Collaboration Authors Papers Undirected Citation Articles Citation Directed Phone-call Telephone Number Phone call Directed Power grid Generators, transformers and substations High voltage transmission lines Directed Examples of real networks with power law degree distributions
Web-graph • Vertices are web pages • Edges are html links • Measured in a massive web-crawl of 108 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, nk(t), obeys the differential equation where M(t) = knk(t) is the total degree of the network.
Simple to show that as t nk(t) ~ k-3 t power-law.
Small world networksNormally defined by two properties: Local order: If vertices A and B are neighbours and B and C are neighbours then good chance that A and C are neighbours. 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.
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 Pi.
At each time step two vertices i and j are selected with probability Pi and Pj. • 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 Pi = 1/N we recover the Erdos-Renyi graph. When Pi ~ 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 fij, where fij = 1 - (1-2PiPj)pN/2 ~ 1 - exp{-pNPiPj} When NPiPj <<1 for all i ≠ j, and when 0 < α < ½, or λ > 3, then fij ~ 2NPiPj j
Adjacency Matrix The adjacency matrix A of this network has elements Aij = Aji with probability distribution P(Aij) = fijδ(Aij-1) + (1-fij)δ(Aij).
This matrix has been studied by a number of workers • Farkas, Derenyi, Barabasi & Vicsek; Numerical study ρ(μ) ~ 1/μ5 for large μ. • Goh, Kahng and Kim, similar numerical study; ρ(μ) ~ 1/μ4. • Dorogovtsev, Goltsev, Mendes & Samukin; analytical work; tree like scale free graph in the continuum approximation; ρ(μ) ~ 1/μ2λ-1.
We will follow Rodgers and Bray, Phys Rev B 37 3557 (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 Zn and then use the fact that as n → 0, (Zn-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 Pk = 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.
Conclusions • 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.
Further details • Eigenvalue spectrum Rodgers, Austin, Kahng and Kim J Phys A 38 9431 (2005). • Spin glass Kim, Rodgers, Kahng and Kim Phys Rev E 71 056115 (2005).
