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

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 In general, between and 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 ? How does Cover Time Scale? What algorithmic primitives can improve scaling? Important in WWW Crawling. Important in Searching P2P.

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

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

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

9 How does Capacity/Throughput/Delay Scale on an Ad-Hoc Wireless Network? Capacity of Wireless Networks, Gupta & Kumar, 2000 Mobility Increases Capacity, Grossgaluser & Tse, 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 Conductance Sparse graphs, Demand ~ degrees S S Conductance and Congestion by Leighton-Rao 95

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

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

14 STRUCTURAL MODEL Given Choose random perfect matching over minivertices

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

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

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

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

19 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. 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. Each vertex with degree in the network core serves customers from the network periphery. Note: Why is demand ?

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

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

22 AS Gkantsidis, MM, Saberi ‘03

23

24 [Gkantsidis, MM, Saberi ’03]

25 [Gkantsidis, MM, Saberi ’03]

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

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

28 Proof In the structural model with Cover Time with Look-Ahead One Theorem [MM,Saberi,Tetali 04]: can discover vertices in steps. 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 Flooding Random Walk Edge Criticality Hybrid Search Schemes 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]

37

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

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

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.