The document proposes methods to accelerate PageRank computations by extrapolating from successive PageRank iterates. It presents the power method for computing PageRank and its convergence properties. The key idea is to estimate components of the current iterate using the next few iterates, eliminating coefficients of less dominant eigenvectors to isolate the principal eigenvector corresponding to PageRank. Empirical results show quadratic extrapolation can speed up convergence, making the approach useful for related problems requiring rapidly computed principal eigenvectors.

1. Extrapolation Methods for Accelerating PageRank Computations Sepandar D. Kamvar Taher H. Haveliwala Christopher D. Manning Gene H. Golub Stanford University

4. Link Counts Linked by 2 Important Pages Linked by 2 Unimportant pages Sep’s Home Page Taher’s Home Page Yahoo! CNN DB Pub Server CS361

6. Definition of PageRank Yahoo! CNN DB Pub Server Taher Sep 1/2 1/2 1 1 0.1 0.1 0.1 0.05 0.25

7. PageRank Diagram Initialize all nodes to rank 0.333 0.333 0.333

8. PageRank Diagram Propagate ranks across links (multiplying by link weights) 0.167 0.167 0.333 0.333

9. PageRank Diagram 0.333 0.5 0.167

10. PageRank Diagram 0.167 0.167 0.5 0.167

11. PageRank Diagram 0.5 0.333 0.167

12. PageRank Diagram After a while… 0.4 0.4 0.2

14. Matrix Notation 0 .2 0 .3 0 0 .1 .4 0 .1 = .1 .3 .2 .3 .1 .1 .2 . 1 .3 .2 .3 .1 .1

15. Matrix Notation Find x that satisfies: . 1 .3 .2 .3 .1 .1 0 .2 0 .3 0 0 .1 .4 0 .1 = .1 .3 .2 .3 .1 .1 .2

20. Power Method u 1 1 u 2  2 u 3  3 u 4  4 u 5  5 Express x (0) in terms of eigenvectors of A

21. Power Method u 1 1 u 2  2  2 u 3  3  3 u 4  4  4 u 5  5  5

22. Power Method u 1 1 u 2  2  2 2 u 3  3  3 2 u 4  4  4 2 u 5  5  5 2

23. Power Method u 1 1 u 2  2  2 k u 3  3  3 k u 4  4  4 k u 5  5  5 k

24. Power Method u 1 1 u 2  u 3  u 4  u 5 

27. Power Method u 1 1 u 2  2 u 3  3 u 4  4 u 5  5 u 1 1 u 2  2  2 u 3  3  3 u 4  4  4 u 5  5  5 u 1 1 u 2  2  2 2 u 3  3  3 2 u 4  4  4 2 u 5  5  5 2

29. Our Approach u 1 u 2 u 3 u 4 u 5 Estimate components of current iterate in the directions of second two eigenvectors, and eliminate them.

31. Using Successive Iterates u 1 x (0) u 1 u 2 u 3 u 4 u 5

32. Using Successive Iterates u 1 x (1) x (0) u 1 u 2 u 3 u 4 u 5

33. Using Successive Iterates u 1 x (1) x (0) x (2) u 1 u 2 u 3 u 4 u 5

34. Using Successive Iterates x (0) u 1 x (1) x (2) u 1 u 2 u 3 u 4 u 5

35. Using Successive Iterates x (0) x’ = u 1 x (1) u 1 u 2 u 3 u 4 u 5

42. Results Quadratic Extrapolation speeds up convergence. Extrapolation was only used 5 times!

43. Results Extrapolation dramatically speeds up convergence, for high values of c (c=.99)