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Properties of Growing Networks
Geoff RodgersSchool of Information Systems, Computing
and Mathematics
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Plan1. Introduction to growing networks
2. Static model of scale free graphs
3. Eigenvalue spectrum of scale free graphs
4. Results
5. Conclusions.
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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
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
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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 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
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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
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Simple to show that as t
nk(t) ~ k-3 t
power-law.
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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-α then the resulting graph is power-law with exponent λ = 1+1/ α.
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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 <<1 for all i ≠ j, and when 0 < α < ½, or λ > 3, then fij ~ 2NPiPj
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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).
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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.
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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 <…> 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
1 exp ,
i ii iPg
N
iiNn 1
,21Re1
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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
2
21 ag
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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
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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
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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/.
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When = ½ or = 3 we can solve exactly to yield
where
222
3
2sinsincossinsin8
p
012sin
logcot 2
p
note that
1
d
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General
• Can easily show that in the limit then
12
1 ~
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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 λ = 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.
12
1 ~
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• 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