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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.

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Complex networks and random matrices l.jpg

Complex networks and random matrices.

Geoff Rodgers

School of Information Systems, Computing and Mathematics


Slide2 l.jpg
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 l.jpg
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 -


Slide4 l.jpg

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 l.jpg
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 l.jpg
Collaboration graph

  • Edges are joint authored publications.

  • Vertices are authors.

  • Power law degree distribution with exponent ≈ 3.

  • Redner, Eur Phys J B, 2001.


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For example: added over time. 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.


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This network is completely solvable added over time.

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.


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Simple to show that as t added over time.  

nk(t) ~ k-3 t

power-law.


Small world networks normally defined by two properties l.jpg

Small world networks added over time. Normally 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).


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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.



Models of small world networks l.jpg
Models of small world networks 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.


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  • p = 0 is a regular ordered structure. networks are scale free.

  • 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.


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Static model of scale free networks l.jpg
Static Model of Scale Free Networks graph, 2-d, 3-d etc…

  • 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.


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Slide19 l.jpg

When P probability 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.


Slide20 l.jpg

j


Adjacency matrix l.jpg
Adjacency Matrix is f

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 l.jpg
This matrix has been studied by a number of workers is f

  • 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.


Slide23 l.jpg


Introduce a generating function l.jpg
Introduce a generating function 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.


Slide25 l.jpg

Normally evaluate the average over lnZ to calculate the eigenvalue spectrum of the adjacency matrix.

using the replica trick; evaluate the

average over Zn and then use

the fact that as n → 0, (Zn-1)/n → lnZ.


Slide26 l.jpg

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 l.jpg
The function g obeys a set of closed equation for the average density of eigenvalues. We first define an average [ …]

and the average density of states is given

by


Slide28 l.jpg

  • 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 l.jpg
In particular, when g takes the form states for any static network by solving for g and using the result to obtain


In the limit n 0 we have the solution l.jpg
In the limit n states for any static network by solving for g and using the result to obtain  0 we have the solution

where a() is given by


Random graphs placing p k 1 n gives an erdos renyi graph and yields l.jpg
Random graphs: Placing P states for any static network by solving for g and using the result to obtain k = 1/N gives an Erdos Renyi graph and yields

as p → ∞ which is in agreement with

Rodgers and Bray, 1988.


Scale free graphs l.jpg
Scale Free Graphs states for any static network by solving for g and using the result to obtain

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 or 3 we can solve exactly to yield l.jpg
When  = ½ or  = 3 we can solve exactly to yield states for any static network by solving for g and using the result to obtain

where

note that


General l.jpg
General states for any static network by solving for g and using the result to obtain 

  • Can show that in the limit    then


Conclusions l.jpg
Conclusions states for any static network by solving for g and using the result to obtain

  • 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.


Conclusions37 l.jpg
Conclusions states for any static network by solving for g and using the result to obtain

  • 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.


Conclusions38 l.jpg
Conclusions states for any static network by solving for g and using the result to obtain

  • 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.


Conclusions39 l.jpg
Conclusions states for any static network by solving for g and using the result to obtain

  • 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 l.jpg
Further details states for any static network by solving for g and using the result to obtain

  • 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).


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