1. 1 Algorithmic Performance in Power Law Graphs Milena Mihail Christos Gkantsidis Christos Papadimitriou Amin Saberi

2. 2 Graphs with Heavy Tailed Degree Sequences 1345102100 Interdomain Routing, WWW, P2P E[degree] ~ constant Degrees not Concentrated around Mean Not Erdos-Renyi Power Law :

3. 3 Power Laws Degree-Frequency Rank-Degree Eigenvalues (Adjacency Matrix) [WWW: Kumar et al 99, Barabasi-Albert 99] [Interdomain Routing: Faloutsos et al 99]

4. 4 How does Algorithmic Performance Scale in Power Law Graphs ? Routing Searching, Information Retrieval Mechanism Design ISPs: 900-14K Routers:500-200K WWW: 500K-3B P2P: tens Ks-2M

5. 5 How does Routing Congestion Scale? Sprint AT&T Demand:, uniform. What is load of max congested link, in optimal routing ?

6. 6 Models for Power Law Graphs One vertex at a time New vertex attaches to existing vertices EVOLUTIONARY: Growth & Preferential Attachment

7. 7 Models for Power Law Graphs EVOLUTIONARY Macroscopic : Growth & Preferential Attachment Simon 55, Barabasi-Albert 99, Kumar et al 00, Bollobas-Riordan 01. Microscopic : Growth & Multiobjective Optimization, QoS vs Cost Fabrikant-Koutsoupias-Papadimitriou 02. STRUCTURAL, aka CONFIGURATIONAL “Random” graph with “power law” degree sequence.

8. 8 STRUCTURAL RANDOM GRAPH MODEL Given Choose random perfect matching over Bollobas 80s, Molloy&Reed 90s, Chung 00s, Sigcomm/Infocom 00s minivertices

9. 9 Given Choose random perfect matching over STRUCTURAL RANDOM GRAPH MODEL Bollobas 80s, Molloy&Reed 90s, Chung 00s, Sigcomm/Infocom 00s minivertices

10. 10 Theorem [Gkantsidis,MM, Saberi 02]: 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 ?

11. 11 Proofs, Step 1 : Reduce to Conductance By max multicommodity flow, Leighton-Rao 95

12. 12 Lemma [MM, Papadimitriou, Saberi 03]: For a random graph grown with preferential attachment with,, a.s. Lemma [Gkantsidis, MM, Saberi 02]: For a random graph in the structural model arising from degree sequence,, a.s. Proofs, Step 2 : Bounds on Conductance Technical: Establish conductance by counting arguments. Difficulties arise from inhomogeneity of underlying state space. Need invariants and/or worst case characterizations. Previously known [Cooper-Frieze 02]

13. 13 Spectral Implications Theorem: Eigenvalue separation for stochastic normalization of adjacency matrix follows by Further Algorithmic Performance Implications: Random Walk Trajectory ~ Independent Samples Cover Time ~ Coupon Collection (WWW, P2P crawling) see also [Cooper-Frieze 02] Chernoff-like Bounds (P2P searching) see also [Cohen et al 02, Shenker et al 03] [Jerrum-Sinclair 88]

14. 14 Spectral Implications Theorem: Eigenvalue separation for stochastic normalization of adjacency matrix Using matrix perturbation [Courant-Fisher Theorem] in a structural random graph model. Rank-Degree Eigenvalues Adjacency Matrix On the eigenvalue Power Law [M.M. & Papadimitriou 02] Negative implication for Information Retrieval: Principal Eigenvectors do not reveal “latent semantics”.

15. 15 How does Algorithmic Performance Scale in Power Law Graphs ? Routing Searching, Information Retrieval Mechanism Design ISPs: 900-14K Routers:500-200K WWW: 500K-3B P2P: tens Ks-2M

16. 16 Incentive Compatible Mechanism Design VCG mechanism for shortest path routing [Nissan-Ronen 99] Pay(e) = cost(e) + cost(st shortest path in G-e) – cost(st shortest path in G) st e VCG overpayment

17. 17 VCG overpayment can be arbitrarily large [Archer-Tardos 02] st 10 1 1 1 1 1 VCG pays 1 + (10-5) = 6 to each edge of cost 1 This is “inherent” in any truthful mechanism [Elkind,Sahai,Steiglitz 03] In the real Interdomain Internet graph, with unit link costs, the average VCG overpayment is ~ 30% [Feigenbaum,Papadimitriou,Sami,Shenker 02]

18. 18 Theorem [MM, Papadimitriou, Saberi 03] : The average VCG overpayment in a power law random graph arising from a structural model is, w.h.p. Conjecture: Theorem [MM, Papadimitriou, Saberi 03] : The average VCG overpayment in a sparse near-regular random graph (structural model, uniform degrees) is, w.h.p.

19. 19 Some Open Problems Routing: integral shortest paths. Routing & Searching: incentives to share resources, particularly relevant to P2P applications. Maintain “good connectivity” (e.g. an expander) in a distributed, dynamic setting.