Introduction to Scale Free SF network

1
Introduction to Scale Free (SF) network

• The Topology of the Internet

by Chan Chi Yuk
2
Agenda

• Motivation
• Background
• Scale Free Models
• Power Laws
• Summary

3
Motivation

• Want to solve network traffic problem
• ? Need to know the topology
• The Internet has done a great job
• ? But how?

4
Possible Applications

• Provide realistic models for
• Simulations
• Protocols design
• Network system design
• Traffic engineering
• Estimate fault-tolerance
• Predict network evolution

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Background
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ER model

• Exponential Random Graph
• Predicted by Erdös and Rényi
• Pconnect 2 node pER
• percolation threshold pc 1/N
• pER c/N, c lt 1 ? isolated trees
• pER 1/N, i.e. c 1 ? cycles of all order
  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.
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WS model

• Small World Network
• Predicted by Watts and Strogatz
• Begins with 1D lattice of N nodes with links
  between the nearest and next nearest neighbors (n
  2)
• PRewire pWS
• pWS 0 ? highly clustered, ltlgt N, P(k)
  d(k-z), z 2n
• 0 lt pWS lt 0.01 ?small world property, P(k) peak
  around z, but boarder
• pWS 1 ? random graph, poorly clustered, ltlgt
  log N, pER z/N

D. J. Watts, S. H. Strogatz, Nature, 393 (1998),
pp.440.
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Scale Free Models
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Scale Free Models

• Scale Free (SF) Network
• Self-similarities
• Power law
• Heavy-tailed distribution
• P(Xgtx) x-a, 0ltalt2
• Zipf distribution / Zeta distribution
• P(k) Ck-(a1)
• Pareto distribution
• f(x) abax-(a1)

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 independently,
  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

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.
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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 Ptwo 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. 251262. 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

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Questions Answers

• Thank you.