Introduction to Scale Free (SF) network
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Transcript of Introduction to Scale Free (SF) network
Introduction to Scale Free (SF) network
The Topology of the Internet
by Chan Chi Yuk
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Agenda
Motivation Background Scale Free Models Power Laws Summary
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Motivation
Want to solve network traffic problem Need to know the topology
The Internet has done a great job But how?
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Possible Applications
Provide realistic models for Simulations Protocols design Network system design Traffic engineering
Estimate fault-tolerance Predict network evolution
Background
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ER model
Exponential Random Graph Predicted by Erdös and Rényi P[connect 2 node] = pER percolation threshold: pc = 1/N
pER ~ c/N, c < 1 isolated trees pER ~ 1/N, i.e. c = 1 cycles of all ord
er appear Poisson distribution:
P. Erdös and A. Rényi, “On the Evolution of Random Graphs” Publications of the Mathematical Institute of the Hungarian Academy of Science 5. (1960), pp.17-61.
,
( )!
k
P k ek
11(1 )k N k
ER ER
Np p
k
,
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WS model
Small World Network Predicted by Watts and Strogatz Begins with 1D lattice of N nodes with l
inks between the nearest and next nearest neighbors (n = 2)
P[Rewire] = pWS pWS = 0 highly clustered, <l> ~ N, P(k) =
δ(k-z), z = 2n 0 < pWS < 0.01 small world property, P(k)
peak around z, but boarder pWS = 1 random graph, poorly clustered,
<l> ~ log N, pER = z/N
D. J. Watts, S. H. Strogatz, Nature, 393 (1998), pp.440.
Scale Free Models
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Scale Free Models
Scale Free (SF) Network Self-similarities Power law
Heavy-tailed distribution P(X>x) ~ x-a, 0<a<2
Zipf distribution / Zeta distribution P(k) = Ck-(a+1)
Pareto distribution f(x) = abax-(a+1)
A.-L. Barabási, R. Albert, and H. Jeong, “Scale-free characteristics of random networks: The topology of the world wide web,” Physical A., 281, 2000, pp.69-77.
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Scale Free Models
Models For random graph, edges are chosen independen
tly, and thus the distribution of degree decays exponentially
Therefore, for power law degree distribution, the choice of edge must be correlated.
Barabási and Albert (BA) model Kumar model Stochastic model Optimization model
W. Aiello, F. Chung, and L. Lu, “Random evolution in massive graphs,” Proceedings of the Fourty-Second Annual IEEE Symposium on Foundations of Computer Science, (FOCS 2001), pp.510-519.
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BA model
Growth Start with m0 nodes, and then add a
node with m edges at every time step.
m≦m0 Preferential Attachment
It is a simple model but… Fixed exponent = 3
( ) ii
jj
kkk
A.-L. Barabási, R. Albert, and H. Jeong, “Mean-field theory for scale-free random networks,” Physical A., 272, 1999, pp.173-187.
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Kumar model
Growth Add a node wt at every time step.
Attachment Node u (v) is chosen according to out(in)-degree P(join u to v) = ab P(join wt to v) = (1-a)b P(join u to wt) = a(1-b) P(join wt to wt) = (1-a)(1-b)
The exponents can be controlled but… Density is restricted to 1
R. Kumar, P. Raghavan, S. Rajagopalan, D. Sivakumar, A. Tomkins, and E. Upfal, “Stochastic models for the web graph”
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Other models
Stochastic model Urn transfer model Also has growth and attachment, but
different probabilities Optimization model
Simultaneous minimization of link density and path
Use the statistics in software engineering as an example
M. Levene, T. Fenner, G. Loizou, and R. Wheeldon, “A Stochastic Model for the Evolution of the Web”S. Valverde, R. Ferror Cancho, and R. V. Sole, “Scale-free Networks from Optimal Design,” cond-mat/0204344, April 2002.
Power Laws
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Power Laws
Degree (connectivity) Number of links connected to the node
Eigenvalues Eigenvalues of the adjacency matrix
Distance Number of nodes within H hops
Betweenness (Load) Number of shortest path passing through the node
Clustering coefficient Average P[two neighbors are connected]
M. Faloutsos, P. Faloutsos, and C. Faloutsos, “On Power-Law Relationships of the Internet Topology,” Proceedings of ACM Sigcomm, August/Sept. 1999, pp. 251–262. A. Vázquez, R. Pastor-Satorra, and A. Vespignani, “Internet topology at the router and autonomous system level,” cond-mat/0206084, v1, June 2002.
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Out-Degree vs. Rank
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Frequency of Out-Degree
Robust but fragile
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Frequency of Out-Degree
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Eigenvalues
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Nodes within H hops
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Betweenness
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Clustering coefficient
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Summary
Internet is a complex network that cannot be modeled in the past
Scale Free models are proposed Many properties follows power law Application of Scale Free model can
be further studied
Questions & Answers
Thank you.