On the Eigenvalue Power Law

1
On the Eigenvalue Power Law

• Milena Mihail
• Georgia Tech
• Christos Papadimitriou
• U.C. Berkeley

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Internet Measurement and Models
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Internet WWW Graphs
Routers exchanging traffic.
Web pages and hyperlinks.
10K 300K nodes
Avrg degree 3
4
Real Internet Graphs
Degrees not sharply concentrated around their
mean.
Average Degree Constant A Few Degrees VERY LARGE
CAIDA http//www.caida.org
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Degree-Frequency Power Law
frequency
Ed const., but No sharp concentration
1
3
4
5
10
2
100
degree
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Degree-Frequency Power Law
Models by Kumar et al 00, x Bollobas
et al 01, x Fabrikant et al 02
Erdos-Renyi sharp concentration
Ed const., but No sharp concentration
Ed const., but No sharp concentration
frequency
1
3
4
5
10
2
100
degree
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Rank-Degree Power Law
Internet measurement Faloutsos et al 99
UUNET
Sprint
CWUSA
ATT
BBN
degree
1
2
3
4
5
10
rank
8
Eigenvalue Power Law
Internet measurement Faloutsos et al 99
eigenvalue
1
2
3
4
5
10
rank
9
This Paper Large Degrees Eigenvalues
UUNET
Sprint
degrees
CWUSA
ATT
BBN
eigenvalues
1
2
3
4
5
10
rank
10
This Paper Large Degrees Eigenvalues
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Principal Eigenvector of a Star
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Large Degrees
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Large Eigenvalues
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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
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Proof Step 1. Decomposition
Vertex Disjoint Stars
LR-extra
LR
-
LL
RR
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Proof Step 2 Vertex Disjoint Stars
Degrees of each Vertex Disjoint Stars Sharply
Concentrated around its Mean d_i
Hence Principal Eigenvalue Sharply Concentrated
around
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Proof Step 3 LL, RR, LR-extra
LR-extra has max degree
RR has max degree
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Proof Step 3 LL, RR, LR-extra
LR-extra has max degree
RR has max degree
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Proof Step 4 Matrix Perturbation Theory
QED
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Implication for Info Retrieval
Term-Norm Distribution Problem
Spectral filtering, without preprocessing,
reveals only the large degrees.
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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
Term-Norm Distribution Problem
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.