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

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

3. Internet Evolution Say, network doubles in size

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

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

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

7. Key Questions How does the worst congestion grow?  O(n)? O(n 2 )? 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 s 1 and s 2. Observation 1: A significant fraction of single-source shortest path trees (  n) trees) in the graph contain e*. S1S1 S2S2 e*e* S1S1 S2S2 e*e* e * occurs in both trees

12. Key Observations (II) Observation 2: In at least a constant fraction of the  (n) shortest path trees, s 1 and s 2 retain at least a constant fraction of their degrees. S1S1 S2S2 e*e* 4/4 4/5 S1S1 S2S2 e*e* 5/5 3/4 S 1,S 2 retain most of their degrees

13. Key Observations (III) Observation 3: The degrees of s 1 and s 2 are  (n 1/  ). And In each tree that e* belongs to, congestion on e*  min{deg tree (s1), deg tree (s2)}. S1S1 S2S2 e*e* So… Congestion(e*)  3

14. Key Result Theorem: The expected maximum edge congestion is  (n 1+1/  ) (shortest path routing, any-2-any).    (n 1.8 ) 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  (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