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The document proposes methods to accelerate PageRank computations using extrapolation techniques. It discusses how PageRank works and is typically computed using an iterative power method. The authors' approach is to use successive PageRank vectors to estimate the components in the directions of the first few eigenvectors, subtracting them to remove their influence and speed convergence. Empirical results show quadratic extrapolation can significantly speed up PageRank convergence, though not enough for truly personalized computations. The extrapolation techniques may help accelerate other similar problems.
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The document proposes methods to accelerate PageRank computations using extrapolation techniques. It discusses how PageRank works and is typically computed using an iterative power method. The authors' approach is to use successive PageRank vectors to estimate the components in the directions of the first few eigenvectors, subtracting them to remove their influence and speed convergence. Empirical results show quadratic extrapolation can significantly speed up PageRank convergence, though not enough for truly personalized computations. The extrapolation techniques may help accelerate other similar iterative methods.
Extrapolation
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 speeds up convergence, especially for high damping factors, though not enough for true personalized PageRank. The techniques provide a general approach for accelerating power method computations.
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- 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)
Editor's Notes
- Why does the Power Method Work?
- Assume that lambda 1 is less than 1 and all other eigenvalues are strictly less than 1.
- Here, talk about in the past, how lambda 2 is often close to 1, so the power method is not useful. However, in our case,
- Note : derivation given here is slightly different from what’s in the paper the one here is perhaps more intuitive the one in the paper is more compact