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Algorithmic Performance in Power Law Graphs

Algorithmic Performance in Power Law Graphs. Milena Mihail Christos Gkantsidis Christos Papadimitriou Amin Saberi. Graphs with Heavy Tailed Degree Sequences. E[degree] ~ constant. Power Law :. Degrees not Concentrated around Mean. Not Erdos-Renyi. 1. 2. 3. 4.

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Algorithmic Performance in Power Law Graphs

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

  2. Graphs with Heavy Tailed Degree Sequences E[degree] ~ constant Power Law : Degrees not Concentrated around Mean Not Erdos-Renyi 1 2 3 4 5 10 100 Interdomain Routing, WWW, P2P

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

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

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

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

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

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

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

  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 Previously known [Cooper-Frieze 02] Technical: Establish conductance by counting arguments. Difficulties arise from inhomogeneity of underlying state space. Need invariants and/or worst case characterizations.

  13. Spectral Implications Theorem: Eigenvalue separation for stochastic normalization of adjacency matrix follows by [Jerrum-Sinclair 88] 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]

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

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

  17. VCG overpayment can be arbitrarily large [Archer-Tardos 02] 1 VCG pays 1 + (10-5) = 6 to each edge of cost 1 1 1 1 1 s t 10 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. Theorem [MM, Papadimitriou, Saberi 03] : The average VCG overpayment in a sparse near-regular random graph (structural model, uniform degrees) is , w.h.p. 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:

  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.

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