On the Eigenvalue Power Law

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

2. P2P WWW • 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. Routers Internet Measurement and Models

3. http://www.etc http://www.etc http://www.ZZZ.edu http://www.XXX.com http://www.etc http://www.XXX.net http://www.YYY.com 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

5. WWW measurement: Kumar et al 99 Internet measurement: Faloutsos et al 99 Degree-Frequency Power Law frequency E[d] = const., but No sharp concentration 1 3 4 5 2 10 100 degree

6. Degree-Frequency Power Law Models by Kumar et al 00, x Bollobas et al 01, x Fabrikant et al 02 Erdos-Renyi sharp concentration E[d] = const., but No sharp concentration E[d] = const., but No sharp concentration frequency 1 3 4 5 2 10 100 degree

7. Rank-Degree Power Law Internet measurement: Faloutsos et al 99 UUNET Sprint C&WUSA AT&T 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. 2 2 4 4 3 3 This Paper: Large Degrees & Eigenvalues UUNET Sprint degrees C&WUSA AT&T BBN eigenvalues 1 2 3 4 5 10 rank

10. This Paper: Large Degrees & Eigenvalues

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

12. 2 4 3 Large Degrees

13. 2 4 3 Large Eigenvalues

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.

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

16. Random Graph Model

17. Random Graph Model

18. Wwith probability at least Ffor large enough Theorem :

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

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

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

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

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

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

25. Implication for Info Retrieval Term-Norm Distribution Problem : Spectral filtering, without preprocessing, reveals only the large degrees. Local information. No “latent semantics”.

26. 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”.