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Scaling Properties of the Internet Graph

Scaling Properties of the Internet Graph. Aditya Akella With Shuchi Chawla, Arvind Kannan and Srinivasan Seshan PODC 2003. Internet Evolution. AS-level graph. AS interconnects: varied capacities. Internet Evolution. Say, network doubles in size. Internet Evolution.

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Scaling Properties of the Internet Graph

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  1. Scaling Properties of the Internet Graph Aditya Akella With Shuchi Chawla, Arvind Kannan and Srinivasan Seshan PODC 2003

  2. Internet Evolution AS-level graph AS interconnects: varied capacities

  3. Internet Evolution Say, network doubles in size

  4. Internet Evolution Double all capacities? Moore’s-law like scaling sufficient? If so, good scaling!

  5. Internet Evolution Plain doubling not enough? Moore’s-law like scaling insufficient?

  6. Internet Evolution Plain doubling not enough? Congested hot-spots If so, poor scaling!!

  7. Key Questions • How does the worst congestion grow? • O(n)?O(n2)? • How much of this is due to… • Power-law structure? • Other distributions • Routing algorithm? • BGP-Policy routing • Traffic demand matrix? • What can be done? • Redesign the network? • Change routing?

  8. Outline • Analysis Overview • Results from simulation • Discussion of results, network design • Conclusion

  9. Outline • Analysis Overview • Outline key observations • Results from simulation • Discussion of results, network design • Conclusion

  10. Analysis • To understand scaling properties of power-law graphs • Sanity check the (more realistic) simulation results • Simple evolutionary model • Preferential Connectivity • Known to yield power-law graphs • Unit traffic between all node-pairs • Routed along the shortest path • How does maximum congestion depend on n, the number of vertices? • Congestion on an edge == number of shortest path routes using the edge • Analysis mainly for intuition; simulation results have the final say.

  11. Key Observations (I) e* -- edge between the top two degree nodes s1 and s2. Observation 1:A significant fraction of single-source shortest path trees (W(n) trees) in the graph contain e*. e* occurs in both trees S1 S1 e* e* S2 S2

  12. Key Observations (II) Observation 2:In at least a constant fraction of the W(n) shortest path trees, s1 and s2 retain at least a constant fraction of their degrees. S1 ,S2 retain most of their degrees 5/5 4/5 S1 S1 e* e* S2 S2 4/4 3/4

  13. S1 e* S2 Key Observations (III) Observation 3:The degrees of s1 and s2 are W(n1/a). And In each tree that e* belongs to, congestion on e* min{degtree(s1), degtree(s2)}. Congestion(e*)  3 So…

  14. Key Result Theorem: The expected maximum edge congestion is W(n1+1/a) (shortest path routing, any-2-any).  W(n1.8) or worse for the Internet. Bad Scaling!

  15. Outline • Analysis Overview • Results from simulation • Discussion of results, network design • Conclusion

  16. Outline • Analysis Overview • Results from simulation • Methodology • A few plots • Discussion of results, network design • Conclusion

  17. Methodology: Outline • Topology • Power-law • Real AS-level topologies • Inet-3.0 generated synthetic • Exponential • Inet-3.0 generated; density same as similar-sized Inet power-law graphs • Tree-like • Grown from the preferential connectivity model

  18. Methodology: Outline • Routing algorithm • Shortest-path • BGP routing • Policy-based, valley-free • Synthetic graphs: heuristically classify edges before imposing policy routing

  19. Methodology: Outline • Traffic matrix • Uniform demands: Any-2-any • Between all pairs • Non-uniform: Clout model • Between “leaves” or “stubs” • Popularity: average degree of the neighbors • Stub identification

  20. Methodology: Outline Topology X Routing X Traffic matrix We seek  Max edge congestion as a function of n

  21. Shortest-Path Routing (Any-2-any) • Exponential >> Power law graphs > Power-law trees

  22. Policy Routing (Any-2-Any) • Poor scaling just like shortest path, but…

  23. Policy Routing vs. Shortest Path Any-2-Any Synthetic Graphs Real Graphs • Policy routing is never worse!

  24. The Clout Model • Scaling is even worse • Same true for policy… • But policy routing is better again!

  25. Outline • Analysis overview • Results from simulation • Discussion of results, network design • Conclusion

  26. Discussion • Scaling according to Moore’s law insufficient • Congested hot-spots in the “core” • May have to alter routing or the macroscopic structure • Routing: Diffuse demand in a centralized manner • Structure: Add additional edges to the graph

  27. Adding Parallel Links • Intuition: Congestion higher on edges with higher avg degree

  28. Adding Parallel Links • #parallel links is dependant on degrees of nodes at the ends of the edge • Candidate functions • Minimum, Maximum, Sum and Product of degrees • Shortest path routing, any-2-any • New edge congestion = edge congestion/#parallel links

  29. Parallel Links • Even min yields Q(n) scaling! Desirable extent of AS-AS peering

  30. Conclusion • Congestion scales poorly in Internet-like graphs • Policy-routing does not worsen the congestion • Alleviation possible via simple, straight-forward mechanisms

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