Complex networks and random matrices'

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Complex networks and random matrices.

• Geoff Rodgers
• School of Information Systems, Computing and
  Mathematics

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Plan

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

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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 -?

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Examples of real networks with power law degree
distributions  
 
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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.

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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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• 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.

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For exampleA 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
• 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 ? ?
• nk(t) k-3 t
• power-law.

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Small world networks 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.

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

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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.

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• p 0 is a regular ordered structure.
• p 1 is an ER random graph.
• Small world for 0 lt p lt 1.
• Average shortest distance behaves as
• n for p 0
• and log n for p gt 0.

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• 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.

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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 ones friends
  are friends themselves.

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

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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.

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• 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.

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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.

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• Over time, this algorithm will yield the
  community structure.

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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.

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• 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.

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• When Pi 1/N we recover the Erdos-Renyi graph.
• When Pi i-a then the resulting graph is
  power-law with exponent ? 11/ a.

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• 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 ltlt1 for all i ? j, and when 0 lt a lt
  ½, or ? gt 3, then fij 2NPiPj

j
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Adjacency Matrix

• The adjacency matrix A of this network
• has elements Aij Aji with probability
• distribution
• P(Aij) fij d(Aij-1) (1-fij)d(Aij).

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This matrix has been studied by a number of
workers

• Farkas, Derenyi, Barabasi Vicsek Numerical
  study ?(µ) 1/µ5 for large µ.
• Goh, Kahng and Kim, similar numerical study ?(µ)
  1/µ4.
• Dorogovtsev, Goltsev, Mendes Samukin
  analytical work tree like scale free graph in
  the continuum approximation ?(µ) 1/µ2?-1.

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• We will follow Rodgers and Bray, Phys Rev B 37
  3557 (1988), to calculate the eigenvalue spectrum
  of the adjacency matrix.

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Introduce a generating function

• where the average eigenvalue density is given
• by

and ltgt denotes an average over the disorder in
the matrix A.
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• 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.

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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.

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The function g obeys

• and the average density of states is given
• by

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• 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.

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In particular, when g takes the form
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In the limit n ? 0 we have the solution

• where a(?) is given by

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

• as p ? 8 which is in agreement with
• Rodgers and Bray, 1988.

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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 ? 11/?.
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When ? ½ or ? 3 we can solve exactly to yield

• where

note that
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General ?

• Can show that in the limit ? ? ? then

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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.

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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 ? 3
  scale free graph.

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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.
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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.

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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).