Properties of Growing Networks

Geoff RodgersSchool of Information Systems, Computing

and Mathematics

Plan1. Introduction to growing networks

2. Static model of scale free graphs

3. Eigenvalue spectrum of scale free graphs

4. Results

5. Conclusions.

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 

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

 

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 5/18vertex 2 with probability 3/18vertex 7 with probability 3/18all other vertices, probability 1/18 each.

1

23

4

5

7

9

8

10

6

This network is completely solvableanalytically – 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.

k1 1)1()(

1)( kknknk

tMdt

tkdn

Simple to show that as t

nk(t) ~ k-3 t

power-law.

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

Adjacency Matrix

The adjacency matrix A of this network has elements Aij = Aji with probability

distribution

P(Aij) = fij δ(Aij-1) + (1-fij)δ(Aij).

The adjacency matrix of complex networks 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

1 exp ,

i ii iPg

N

iiNn 1

,21Re1

• 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

2

21 ag

In the limit n 0 we have the solution

where a() is given by

N

k k apNPiN 1

11Re1

N

1

k k

k

apNPiμPa

Random graphs: Placing Pk = 1/N gives an Erdos Renyi graph and yields

as p → ∞ which is in agreement with Rodgers and Bray, 1988.

242

1

pp

Scale Free Graphs

To calculate the eigenvalue spectrum of a scale free graph we must choose

kNPk11

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

222

3

2sinsincossinsin8

p

012sin

logcot 2

p

note that

1

d

General

• Can easily show that in the limit then

12

1 ~

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.

12

1 ~

• 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