1. Analysis of the Internet Topology Michalis Faloutsos, U.C. Riverside (PI) Christos Faloutsos, CMU (sub- contract, co-PI) DARPA NMS, no. 00-1-8936

2. Analyze and model the Internet topology preferably with a few simple numbers Power-law: Frequency of degree vs. degree Goal: Find Order in Chaos A real Internet instance, Oct 1995

3. Motivation Intuitively: “ You can’t resolve the traffic jam problem of a city without looking at the street layout.” The Internet community has little knowledge of the topology

4. What is the novelty? Power-laws to describe skewed distributions, i.e. y = a x p. Spectral-analysis to capture the “finger-print” of the graph Multi-fractals to identify 80-20 distributions i.e. 20% nodes have 80% of the edges Use the above to define and identify a hierarchy

5. Deliverables New graph metrics A model for the Internet topology A software tool to analyze graphs

6. Road Map Current Research: –Four Power-laws Future Directions –Graph metrics –Applications Conclusion

7. Our Power-laws I. Outdegree of nodes vs. rank II. Frequency of outdegree III. Eigenvalues of adj. matrix IV. Pairs of nodes within h hops Accuracy: correlation coeff. > 0.97

8. I. Power-law: rank R The plot is a line in log-log scale Exponent = slope R = -0.74 R outdegree Rank: nodes in decreasing outdegree order Dec’98

9. II.Power-law: outdegree O The plot is linear in log-log scale O = -2.15 Exponent = slope Outdegree Frequency Nov’97

10. III. Eigenvalues Let A be the adjacency matrix of graph The eigenvalue is: – A v = v, where v some vector Eigenvalues are related to topological properties

11. III.Power-law: eigen exponent E Find the eigenvalues of the adjacency matrix Eigenvalues in decreasing order (first 20) E = -0.48 Exponent = slope Eigenvalue Rank of decreasing eigenvalue Dec’98

12. IV. Power-law: hopplot H Pairs of reachable nodes as a function of hops Simpler: neighborhood size vs hops No. of Pairs Hops Router level ’95 H = 2.83

13. Road Map Current Research: –Four Power-laws Future Directions –Graph metrics –Applications Conclusion

14. Looking For More Patterns Analyze the dynamic nature of the Internet Study the eigenvalues of the adjacency matrix Identify multi-fractal relationships Identify a hierarchy

15. Practical Applications and our Patterns Provide realistic models for simulations –Faster simulations Improve the design of protocols –Estimate useful network properties Facilitate traffic engineering Predict network evolution, “what-if” scenarios Estimate fault-tolerance of the topology Applications will inspire and guide the search for an Internet model and its metrics

16. Conclusions The Internet has more structure than we thought! Our tools look very promising; they characterize concisely the topology Modeling the Internet will have significant practical impact

17. Analysis of the Internet Topology IMPACT : SCHEDULE : NEW IDEAS: 1. Set of topological metrics 2. Comprehensive model of the Internet topology 3. Tool for analyzing the topology of graphs YEAR 1 YEAR 2YEAR 3 A model for the Internet topology will: Improve the design of routing protocols Help explain the behavior of traffic Improve the validity of network simulations Estimate the vulnerability of topology to malicious users U.C.Riverside: Michalis Faloutsos Frequency distribution of node degree Internet Topology, 1995 Use power-laws Use the eigenvalue analysis of the adjacency matrix Use multi-fractal analysis Describe the topology concisely i.e. with a few simple numbers

18. I. Estimating with the rank exponent R Lemma: Given the nodes N, and an estimate for the rank exponent R, we predict the edges E: See paper for details [SIGCOMM’99]

19. IV. The Average Neighborhood Size Avg. Neighborhood size versus h hops Before: with avg. degree d: d^h Now: with hopplot: h^H Qualitative: sphere in H-dimensions, H = 4.86 With hopplot Real With avg. degree Avg. Neighborhood Hops

20. Time Evolution: rank R The rank exponent has not changed! Domain level