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Algorithmic Performance in Power Law Graphs

Explore how algorithmic performance scales in power law graphs, particularly in routing, searching, and information retrieval mechanisms. Learn about structural random graph models, evolutionary growth patterns, and spectral implications related to degree sequences.

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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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