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.