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Introduction to Scale Free (SF) network

Introduction to Scale Free (SF) network. The Topology of the Internet. by Chan Chi Yuk. Agenda. Motivation Background Scale Free Models Power Laws Summary. 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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Introduction to Scale Free (SF) network

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  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

  5. Background

  6. 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 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.

  7. 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) • 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.

  8. Scale Free Models

  9. 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.

  10. 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.

  11. 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.

  12. 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”

  13. 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.

  14. Power Laws

  15. 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.

  16. Out-Degree vs. Rank

  17. Frequency of Out-Degree • Robust but fragile

  18. Frequency of Out-Degree

  19. Eigenvalues

  20. Nodes within H hops

  21. Betweenness

  22. Clustering coefficient

  23. 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

  24. Questions & Answers Thank you.

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