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 Structure Hierarchical 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, d v, of a node v, is proportional to the rank of the node, r v, to the power of a constant, R: d v  r v R ( 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, f d, of an out-degree, d, is proportional to the out-degree to the power of a constant, O: f d  d O

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 P h  h Ħ

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 A Regional ISP B 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 AS3AS2 AS5AS4 AS7 AS6 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 AS2 AS1 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%)