Properties of Growing Networks
Geoff Rodgers
School of Information Systems, Computing
and Mathematics
Plan
1. 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 5/18
vertex 2 with probability 3/18
vertex 7 with probability 3/18
all other vertices, probability 1/18 each.
10
9
3
2
4
8
1
7
5
6
This network is completely solvable
analytically – the number of vertices of
degree k at time t, nk(t), obeys the
differential equation


dn (t )
1
k

(k  1)n
 kn   k1
k 1
k
dt
M (t )
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.
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 ErdosRenyi 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
 

g     Pi exp   i      1
i
 ,i
  
and the average density of states is given
by
1
1
   
Re
n
N
  


N
i 1
2
,i
• 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
1
2
g    a  
2

In the limit n  0 we have the solution
1
     Re

N
1
N
1

k 1 i  pNPk a
where a() is given by
N
Pk
a 
k 1 iμ  pNPk a
Random graphs: Placing Pk = 1/N gives an
Erdos Renyi graph and yields
1
   
2 p
4p  2
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
Pk  1   N
 1 
k
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
 8 sin 3  sin    cos 
    
2
2
2
p sin    sin 2   
where
   2 2
 cot   log 
   1  0
 sin   p
note that





d


1


General 
• Can easily show that in the limit    then
   ~
1

2  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
   ~
1

2  1
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.