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On the Eigenvalue Power Law

Milena Mihail Georgia Tech

Christos PapadimitriouU.C. Berkeley

&

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Network and application studies need properties and models of:

Internet graphs & Internet Traffic.

Shift of networking paradigm: Open, decentralized, dynamic.

Intense measurement efforts. Intense modeling efforts.

Internet Measurement and Models

Routers

WWW

P2P

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Internet & WWW Graphs

http://www.etc

http://www.XXX.net

http://www.YYY.com

http://www.etc http://www.ZZZ.edu

http://www.XXX.com

http://www.etc

Routers exchanging traffic. Web pages and hyperlinks.

10K – 300K nodesAvrg degree ~ 3

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Real Internet Graphs

CAIDA http://www.caida.org

Average Degree = Constant

A Few Degrees VERY LARGE

Degrees not sharply concentrated around their mean.

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Degree-Frequency Power Law

degree1 3 4 5 102 100

freq

uen

cy

WWW measurement: Kumar et al 99Internet measurement: Faloutsos et

al 99

E[d] = const., but

No sharp concentration

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Degree-Frequency Power Law

1 3 4 5 102 100

freq

uen

cy

E[d] = const., but

No sharp concentration

degree

E[d] = const., but

No sharp concentration

Erdos-Renyi sharp concentration

Models by Kumar et al 00, x Bollobas et al 01, x Fabrikant et al 02

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Rank-Degree Power Law

rank

deg

ree

1 2 3 4 5 10

Internet measurement: Faloutsos et al 99

UUNET

SprintC&WUSA

AT&TBBN

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Eigenvalue Power Law

rank

eig

en

valu

e

1 2 3 4 5 10

Internet measurement: Faloutsos et al 99

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This Paper: Large Degrees & Eigenvalues

rank

eig

en

valu

es

1 2 3 4 5 10

UUNET

SprintC&WUSA

AT&TBBN2

34

2 3 4

deg

ree

s

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This Paper: Large Degrees & Eigenvalues

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Principal Eigenvector of a Star

11

1

11

1

1

1

d

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Large Degrees

2

3

4

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Large Eigenvalues

2

3

4

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Main Result of the Paper

The largest eigenvalues of the adjacency martix of a graph whose large degrees are power law distributed (Zipf), are also power law distributed.

Explains Internet measurements.

Negative implications for the spectral filtering method in information retrieval.

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Random Graph Model

let

Connectivity analyzed by Chung & Lu ‘01

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Random Graph Model

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Random Graph Model

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Theorem :

Ffor large enough

Wwith probability at least

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Proof : Step 1. Decomposition

Vertex Disjoint StarsLR-extra

RR

LL

LR =

-

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Proof: Step 2: Vertex Disjoint Stars

Degrees of each Vertex Disjoint Stars Sharply Concentrated around its Mean d_iHence Principal Eigenvalue Sharply Concentrated around

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Proof: Step 3: LL, RR, LR-extra

LR-extra has max degree

LL has

edges

RR has max degree

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Proof: Step 3: LL, RR, LR-extra

LR-extra has max degree

RR has max degree

LL has

edges

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Proof: Step 4: Matrix Perturbation Theory

Vertex Disjoint Stars have principal eigenvalues

All other parts have max eigenvalue QED

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Implication for Info Retrieval

Spectral filtering, without preprocessing, reveals only the large degrees.

Term-Norm Distribution Problem :

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Implication for Info Retrieval

Term-Norm Distribution Problem : Spectral filtering, without preprocessing, reveals only the large degrees.

Local information.

No “latent semantics”.

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Implication for Information Retrieval

Application specific preprocessing (normalization of degrees) reveals clusters:

WWW: related to searching, Kleinberg 97

IR, collaborative filtering, …

Internet: related to congestion, Gkantsidis et al 02

Open : Formalize “preprocessing”.

Term-Norm Distribution Problem :