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On Power-Law Relationships of the Internet Topology

On Power-Law Relationships of the Internet Topology. CSCI 780, Fall 2005. Outline. How does network topology look like? Random Graph? Properties of Network Topology Degree distribution Power law (this paper) Structure Hierarchical Structure. Network Topology. On Router Level

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On Power-Law Relationships of the Internet Topology

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  1. On Power-Law Relationships of the Internet Topology CSCI 780, Fall 2005

  2. Outline • How does network topology look like? • Random Graph? • Properties of Network Topology • Degree distribution • Power law (this paper) • Structure • Hierarchical Structure

  3. Network Topology • On Router Level • Topology Graph = (V, E) • Each node denotes a router • Edge is the physical link between two routers • On AS level • Topology Graph = (V, E) • Each node denotes an AS • Edge is AS pair which have a BGP session between them

  4. Two Levels of Internet Topology • Router-level: nodes are routers • AS-level: nodes are domains

  5. Why Topology Is Important? • Design Efficient Protocols • Create Accurate Model for Simulation • Derive Estimates for Topological Parameters • Study Fault Tolerance and Anti-Attack Properties

  6. Key Findings • We observe power-laws of the Internet topology • Distributions are skewed, so average can be misleading • The log-log plots are linear

  7. Power-Law, Zipf, Pareto • Power-Law (probability distribution function) P[X = x] ~ x-(k+1) = x-a • Pareto (cumulative distribution function) P[X > x] ~ x-k • Zipf ( size vs. rank ) y ~ r-b • They are different ways of looking at the same thing

  8. Internet Instances • Three Snapshots at AS-level, one at Router-level (95)

  9. Power Law Properties (degree vs. rank) • Power Law 1: (rank exponent) • The degree, dv, of a node v, is proportional to the rank of the node, rv, to the power of a constant, R: dv rvR (Rank is the index of in order of decreasing out-degree)

  10. Rank Plots • Log-Log scale graph • X axis is rank, Y axis is out-degree

  11. Power Law Properties (frequency vs. degree) • Power Law 2: (Out-degree exponent) • The frequency, fd, of an out-degree, d, is proportional to the out-degree to the power of a constant, O: fd dO

  12. Out-degree Plots • Log-log scale graph • X axis is out-degree, Y axis is frequency

  13. Out-degree Plots (cont’d)

  14. Neighborhood • Size of neighborhood within some distance • P(h): total number of pairs of nodes within h hops

  15. Hop-plot exponent • Ph hĦ

  16. Average Neighborhood Size

  17. Eigenvalue of Graph

  18. Power Law Properties(eigenvalues) • Power Law 3: • The eigenvalues, i, of a graph are proportional to the order,i,to the power of a constant, i  i • Eigenvalues of a graph are the eigenvalues for the adjacency matrix of this graph

  19. Eigenvalue plots • Log-log scale graph • X axis is the order of eigenvalue • Y axis is the eigenvalue

  20. Discussion • Describing the Internet topology • Power-low exponents are more descriptive than average • Protocol performance analysis • Estimate useful graph metrics (neighborhood) • Predication • Answer what-if questions • Realistic-graph generation

  21. Connectivity does not Mean Reachability • Now we know properties of connectivity • But connectivity DOES NOT=reachability! • Commercial agreement • Routing policy • An annotated topology….

  22. Route Propagation Policy • Constrained by contractual commercial agreements between administrative domains Regional ISP B Regional ISP A University C • e.g., An AS does not provide transit services between its providers

  23. AS Commercial Relationships • Provider-customer: • customer pays its provider for transit services • Peer-peer: • exchange traffic between customers • no exchange of money • Sibling-sibling: • have mutual transit agreement • merging ISPs, Internet connection backup • However, AS relationships are not public!

  24. AS Relationship Graph AS1 AS7 AS6 AS2 AS3 AS4 AS5 provider-to-customer edge peer-peer edge sibling-sibling edge

  25. Route Propagation Rule • An AS or a set of ASes with sibling relationship does not provide transit services between any two of its providers and peers • BGP routing table entries have certain patterns

  26. Internet Architecture • Hierarchical structure • Backbone • Edge network AS1 AS2 AS3

  27. Hierarchical Topology • Based on AS relationship • Tiers • Provider/Customer

  28. Hierarchical Topology • The number of ASes in different tiers on 2001/05, there are 11038 ASes • Tier 1: 22 (0.20%) • Tier 2: 5228 (47.37%) • Tier 3: 4193 (37.99%) • Tier 4: 1396 (12.64%) • Tier 5: 174 (1.67%) • Tier 6: 19 (0.17%) • Tier 7: 6 (0.05%)

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