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Algorithmic Performance in Power Law Graphs

Milena Mihail

Christos Gkantsidis Christos Papadimitriou

Amin Saberi

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Graphs with Heavy Tailed Degree Sequences

1 3 4 5 102 100

Interdomain Routing, WWW, P2P

E[degree] ~ constantDegrees not Concentrated around Mean

Not Erdos-Renyi

Power Law :

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Power Laws

Degree-Frequency

Rank-Degree

Eigenvalues(Adjacency Matrix)

[WWW: Kumar et al 99, Barabasi-Albert 99]

[Interdomain Routing: Faloutsos et al 99]

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How does Algorithmic Performance Scale in Power Law Graphs ?

Routing

Searching, Information Retrieval

Mechanism Design

ISPs: 900-14K Routers:500-200KWWW: 500K-3BP2P: tens Ks-2M

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How does Routing Congestion Scale?

Sprint

AT&T

Demand: , uniform. What is load of max congested link, in optimal routing ?

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Models for Power Law Graphs

One vertex at a time

New vertex attaches to existing vertices

EVOLUTIONARY : Growth & Preferential Attachment

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Models for Power Law Graphs

EVOLUTIONARYMacroscopic : 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.

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STRUCTURAL RANDOM GRAPH MODEL

Given

Choose random perfect matching over

Bollobas 80s, Molloy&Reed 90s, Chung 00s, Sigcomm/Infocom 00s

minivertices

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Given Choose random perfect matching over

STRUCTURAL RANDOM GRAPH MODEL

Bollobas 80s, Molloy&Reed 90s, Chung 00s, Sigcomm/Infocom 00s

minivertices

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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 coreserves customers from the network periphery.

Note: Why is demand ?

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Proofs, Step 1 : Reduce to ConductanceBy max multicommodity flow, Leighton-Rao 95

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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]

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Spectral ImplicationsTheorem: 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]

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Spectral ImplicationsTheorem: Eigenvalue separation

for stochastic normalization of adjacency matrix

Using matrix perturbation [Courant-Fisher Theorem] in a structural random graph model.

Rank-Degree

EigenvaluesAdjacency Matrix

On the eigenvalue Power Law [M.M. & Papadimitriou 02]

Negative implication for Information Retrieval: Principal Eigenvectors do not reveal “latent semantics”.

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How does Algorithmic Performance Scale in Power Law Graphs ?

Routing

Searching, Information Retrieval

Mechanism Design

ISPs: 900-14K Routers:500-200KWWW: 500K-3BP2P: tens Ks-2M

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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)

s te

VCG overpayment

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VCG overpayment can be arbitrarily large [Archer-Tardos 02]

s t10

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 averageVCG overpayment is ~ 30% [Feigenbaum,Papadimitriou,Sami,Shenker 02]

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Theorem [MM, Papadimitriou, Saberi 03] : The average VCG overpayment in a power law random graph arising from a structural modelis , 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.

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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.