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1 On the Eigenvalue Power Law Milena Mihail Georgia Tech Christos Papadimitriou U.C. Berkeley &

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2 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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3 Internet & WWW Graphs http://www.etc http://www.XXX.net http://www.YYY.com http://www.etc http://www.ZZZ.edu http://www.XXX.comhttp://www.etc Routers exchanging traffic.Web pages and hyperlinks. 10K – 300K nodes Avrg degree ~ 3

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4 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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5 Degree-Frequency Power Law degree 1345 102100 frequenc y WWW measurement: Kumar et al 99 Internet measurement: Faloutsos et al 99 E[d] = const., but No sharp concentration

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6 Degree-Frequency Power Law 1345 102100 frequenc y 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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7 Rank-Degree Power Law rank degree 1234510 Internet measurement: Faloutsos et al 99 UUNET Sprint C&WUSA AT&T BBN

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8 Eigenvalue Power Law rank eigenvalue 1234510 Internet measurement: Faloutsos et al 99

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9 This Paper: Large Degrees & Eigenvalues rank eigenvalues 1234510 UUNET Sprint C&WUSA AT&T BBN 2 3 4 2 3 4 degrees

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10 This Paper: Large Degrees & Eigenvalues

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11 Principal Eigenvector of a Star 1 1 1 1 1 1 1 1 d

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12 Large Degrees 2 3 4

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13 Large Eigenvalues 2 3 4

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14 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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15 Random Graph Model let Connectivity analyzed by Chung & Lu ‘01

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16 Random Graph Model

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17 Random Graph Model

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18 Theorem : Ffor large enough Wwith probability at least

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19 Proof : Step 1. Decomposition Vertex Disjoint StarsLR-extra RR LL LR =-

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20 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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21 Proof: Step 3: LL, RR, LR-extra LR-extra has max degree LL has edges RR has max degree

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22 Proof: Step 3: LL, RR, LR-extra LR-extra has max degree RR has max degree LL has edges

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23 Proof: Step 4: Matrix Perturbation Theory Vertex Disjoint Stars have principal eigenvalues All other parts have max eigenvalue QED

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24 Implication for Info Retrieval Spectral filtering, without preprocessing, reveals only the large degrees. Term-Norm Distribution Problem :

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25 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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26 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 :

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