properties of growing networks
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Properties of Growing Networks. Geoff Rodgers School of Information Systems, Computing and Mathematics. Plan. Introduction to growing networks Static model of scale free graphs Eigenvalue spectrum of scale free graphs Results Conclusions. Networks. Many of networks in economic, physical, - PowerPoint PPT PresentationTRANSCRIPT
Properties of Growing Networks
Geoff Rodgers
School 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
2
3
4
5
7
9
8
10
6
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.
k1 1)1(
)(
1)(
kkn
knk
tMdt
tk
dn
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 i
i iPg
N
iiNn 1
,
21Re
1
• 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
2
1 ag
In the limit n 0 we have the solution
where a() is given by
N
k k apNPiN 1
11Re
1
N
1
k k
k
apNPiμ
Pa
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
p
p
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
2sinsin
cossinsin8
p
012
sinlogcot 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