Algorithmic Performance in Complex Networks

1. Algorithmic Performance in Complex Networks Milena Mihail Georgia Tech.

2. Outline • Metrics relevant to network function: • Expansion, Routing, • Conductance, Searching • Spectrum, in communication networks • Global Connectivity • Efficient maintenance of expansion

3. Scaling • Complex Networks • WWW 500K-3B • Internet Routing ASes: 900-15K • Routers: 500-200K • P2P tens Ks-4M • Ad-hoc (wireless, mobile, sensor) • Gene-Protein Interaction

4. How does Algorithmic Performance Scale with Number of Nodes in a Complex Communication Network? Route Mechanism design Efficient maintenance of metrics supporting the above Search

5. How does Cover Time Scale?What algorithmic primitives can improve scaling? Random walk on nodes. What is the expected time to visit all the nodes ? What is the expected time to visit a constant fraction of the nodes ? Important in WWW Crawling. Important in Searching P2P. In general, between and

6. How does Routing Congestion Scale on the Internet ? Demand: , uniform. What is load of max congested link, in optimal routing ? star expander Sparse power-law graphs ? in general Important in economics. Networks with externalities.

7. How does Routing Congestion Scale on the AS Internet ? Demand: , uniform. What is load of max congested link, in optimal routing ? star expander Sparse scale-free graphs ? in general Important in economics. Networks with externalities.

8. Edge congestion under shortest path routing on a non blocking network (regular expander). Edge congestion under shortest path routing on the Internet graph.

9. How does Capacity/Throughput/Delay Scale on an Ad-Hoc Wireless Network? Coming soon in a Workshop Near You ... Capacity of Wireless Networks, Gupta & Kumar, 2000 Mobility Increases Capacity, Grossgaluser & Tse, 20001 Capacity, Delay and Mobility in Wireless Networks, Bansal & Liu 2003 Throughput-delay Trade-off in Wireless Networks, El Gamal, Mammen, Prabhakar & Shah 2004

10. Outline • Metrics relevant to network function: • Expansion, Routing, • Conductance, Searching • Spectrum, in communication networks • Global Connectivity • Efficient maintenance of expansion

11. S S Conductance Conductance and Congestionby Leighton-Rao 95 Sparse graphs, Demand ~ degrees

12. Macroscopic Models for Scale-Free Graphs EVOLUTIONARY : Growth & Preferential Attachment One vertex at a time New vertex attaches to existing vertices Simon 55,Barabasi-Albert 99, Kumar et al 00, Bollobas-Riordan 01, Bollobas-Riordan-Spencer-Tusnady 01.

13. STRUCTURAL , aka CONFIGURATIONAL MODEL “Random” graph with “power law” degree sequence. Given Choose random perfect matching over minivertices Bollobas 80s, Molloy&Reed 90s, Aiello-Chung-Lu 00s, Sigcomm/Infocom 00s

14. STRUCTURAL MODEL Given Choose random perfect matching over minivertices

15. STRUCTURAL MODEL Given Choose random perfect matching over edge multiplicity O(log n) , a.s. connected, a.s. minivertices

16. Theorem [MM, Papadimitriou, Saberi 03]: For a random graph grown with preferential attachment with , , a.s. Previously: Cooper & Frieze 02 Bounds on Conductance Technique: Probabilistic Counting Arguments & Combinatorics. Difficulty: Non homogeneity in state-space, Dependencies. Theorem [Gkantsidis, MM, Saberi 03]: For a random graph in the structural model arising from degree sequence , , a.s. Independent: Chung,Lu,Vu 03 for a different structural random graph model

17. Structural Model, Proof Idea: Difficulty: Non homogeneity in state-space Worst case is when all vertices have degree 3.

18. Growth with Preferential Connectivity Model, Proof Idea: Difficulty: Arrival Time Dependencies Shifting Argument

19. Theorem [MM, Papadimitriou, Saberi 03]: For a random graph grown with preferential attachment with there is a poly time computable flow that routes demand between all vertices i and j with max link congestion , a.s. Note: Why is demand ? Each vertex with degree in the network core serves customers from the network periphery. Theorem [Gkantsidis,MM, Saberi 03]: For a random graph in the structural model arising from degree sequence there is a poly time computable flow that routes demand between all vertices i and j with max link congestion a.s.

20. Edge congestion under shortest path routing on a non blocking network (regular expander). Edge congestion under shortest path routing on the Internet graph.

21. Conductance and Spectrum Theorem: Eigenvalue separation for stochastic normalization of adjacency matrix follows by [Alon 86] [Jerrum-Sinclair 88] Recall: Stochastic normalizations of adjacency matrices of undirected graphs, P has n real eigenvalue-eigenvector pairs: related to “bad cuts”

22. AS Gkantsidis, MM, Saberi ‘03

24. [Gkantsidis, MM, Saberi ’03]

25. [Gkantsidis, MM, Saberi ’03]

26. Spectrum, Mixing and Cover Times Rapid Mixing of Random Walk for “mixing” in Cover Time [Broder Karlin 88] for any constant Simpler, by mixing and coupon collection

27. Cover Time with Look-Ahead One In the structural model with Theorem [MM,Saberi,Tetali 04]: can discover vertices in steps. Proof

28. Cover Time with Look-Ahead One Theorem [MM,Saberi,Tetali 04]: In the structural model with can discover vertices in steps. Proof Adamic et al ’02 Chawathe et al 03 Gkanstidis, MM, Saberi 05, Sarshar et al 05

29. HYBRID SEARCH SCHEMES: Take Advantage of Local Information to Improve Global Performance Random Walk Edge Criticality Hybrid Search Schemes Flooding Gkantsidis, MM, Saberi 04 Boyd, Diaconis, Xiao 04

30. Outline • Metrics relevant to network function: • Expansion, Routing, • Conductance, Searching • Spectrum, in communication networks • Global Connectivity • Efficient maintenance of expansion

31. P2P Network Topology Problem: A distributed resource efficient algorithm to dynamically maintain an expander. ? ? ?

32. P2P Network Topology Construction by Random Walk ? ? ? Theorem [Law & Siu ‘03]: Construct a constant expander on n vertices with overhead O( log n) per node addition.

33. P2P Network Topology Construction by Random Walk

34. P2P Network Topology Construction by Random Walk

35. P2P Network Topology Construction by Random Walk

36. ? ? ? P2P Network Topology Construction by Random Walk Theorem[Gkanstidis,MM,Saberi 04]: Construct a graph on n vertices with constant overhead per node addition where, for some constants a and b, every set of at least bn vertices has expansion a and where sets of size O( log n) also have constant expansion. Proof Technique: Taking continious samples from a Markov chain achieves Chernoff-like bounds [Ajtai,Komlos,Szemeredi 88, Zuckerman & Impagliazzo 89, Gillman 95]

38. P2P Network Topology Maintenance by 2-Link Switches Theorem [Cooper, Frieze & Greenhill 04]: The corresponding random walk on d-regular graphs is rapidly mixing. Question: How does the network “pick” a random 2-Link Switch? In reality, the links involved in a switch are within constant distance.

39. Scaling • Complex Networks • WWW 500K-3B • Internet Routing ASes: 900-15K • Routers: 500-200K • P2P tens Ks-4M • Ad-hoc (wireless, mobile, sensor) • Gene-Protein Interaction

40. Gene-Protein Interaction Networks Copying Random Graph Model: a new node v attaches with d links as follows: (1)Picks a random node u (2) For i:=1 to d with probability p, v copies the ith link of u with probability 1-p , v attaches to a uniformly random node. The exponent of the resulting Power-law graph is a function of p. [Kumar et al 01, Chung & Lu 04] For biologists, p is an indication of evolutionary fitness.

41. as a function of p, in experiment, MM & Zia ‘05 For biologists, p is an indication of evolutionary fitness.

42. Summary • Metrics relevant to network function: • Expansion, Routing, • Conductance, Searching • Spectrum, in communication networks • Global Connectivity • Efficient maintenance of expansion • Reverse engineering in bioinformatics

43. References • On the Eigenvalue Powerlaw, M. Mihail and C. Papadimitriou, RANDOM 02. • Spectral Analysis of Internet Topologies, C. Gkantsidis, M. Mihail • and E. Zegura, INFOCOM 03. • Conductance and Congestion in Powerlaw Graphs, C. Gkantsidis, M. Mihail • and A. Saberi, SIGMETRICS 03. • On Certain Connectivity Properties of the Internet Topology, M. Mihail, • C. Papadimitriou and A. Saberi, FOCS 03. • On the Random Walk Method for P2P Networks, C. Gkantsidis, M. Mihail • and A. Saberi, INFOCOM 05. • Hybrid Search Schemes in P2P Networks, C. Gkantsidis, M. Mihail • and A. Saberi, INFOCOM 05.