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Complex networks and random matrices.

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Complex networks and random matrices.

Geoff Rodgers

School of Information Systems, Computing and Mathematics

- Introduction to scale free graphs
- Small world networks
- Static model of scale free graphs
- Eigenvalue spectrum of scale free graphs
- Results
- Conclusions.

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

- 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.

- 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 5/18vertex 2 with probability 3/18vertex 7 with probability 3/18all 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.

- 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.

- 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

The adjacency matrix A of this network

has elements Aij = Aji with probability

distribution

P(Aij) = fijδ(Aij-1) + (1-fij)δ(Aij).

- 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.

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.

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.

where a() is given by

as p → ∞ which is in agreement with

Rodgers and Bray, 1988.

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/.

where

note that

- Can show that in the limit then

- 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.

- 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.

- 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.

- 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).