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Link Counts

GOOGLE Page Rank engine needs speedup. Link Counts. Taher’s Home Page. Sep’s Home Page. CS361. DB Pub Server. CNN. Yahoo!. Linked by 2 Unimportant pages. Linked by 2 Important Pages. adapted from G. Golub et al. importance of page i. importance of page j.

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Link Counts

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  1. GOOGLE Page Rank engine needs speedup Link Counts Taher’s Home Page Sep’s Home Page CS361 DB Pub Server CNN Yahoo! Linked by 2 Unimportant pages Linked by 2 Important Pages adapted from G. Golub et al

  2. importance of page i importance of page j number of outlinks from page j pages j that link to page i Definition of PageRank • The importance of a page is given by the importance of the pages that link to it.

  3. 1/2 1/2 1 1 0.05 0.25 0.1 0.1 0.1 Definition of PageRank Sep Taher DB Pub Server CNN Yahoo!

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

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

  6. PageRank Diagram 0.5 0.333 0.167

  7. PageRank Diagram 0.167 0.5 0.167 0.167

  8. PageRank Diagram 0.333 0.5 0.167

  9. PageRank Diagram 0.4 0.4 0.2 After a while…

  10. importance of page i importance of page j number of outlinks from page j pages j that link to page i Computing PageRank • Initialize: • Repeat until convergence:

  11. .1 .3 .2 .3 .1 .1 .1 .3 .2 .3 .1 .1 = 0 .2 0 .3 0 0 .1 .4 0 .1 .2 Matrix Notation

  12. .1 .3 .2 .3 .1 .1 .1 .3 .2 .3 .1 .1 0 .2 0 .3 0 0 .1 .4 0 .1 = .2 Matrix Notation Find x that satisfies:

  13. Power Method • Initialize: • Repeat until convergence:

  14. Find x that satisfies: Find x that satisfies: A side note • PageRank doesn’t actually use PT. Instead, it uses A=cPT + (1-c)ET. • So the PageRank problem is really: not:

  15. Power Method • And the algorithm is really . . . • Initialize: • Repeat until convergence:

  16. Power Method Express x(0) in terms of eigenvectors of A u1 1 u2 a2 u3 a3 u4 a4 u5 a5

  17. Power Method u1 1 u2 a22 u3 a33 u4 a44 u5 a55

  18. Power Method u1 1 u2 a222 u3 a332 u4 a442 u5 a552

  19. Power Method u1 1 u2 a22k u3 a33k u4 a44k u5 a55k

  20. Power Method u1 1 u2 0 u3 0 u4 0 u5 0

  21. Then, you can write any n-dimensional vector as a linear combination of the eigenvectors of A. u1 1 u2 a2 u3 a3 u4 a4 u5 a5 Why does it work? • Imagine our n x n matrix A has n distinct eigenvectors ui.

  22. All less than 1 Why does it work? • From the last slide: • To get the first iterate, multiply x(0) by A. • First eigenvalue is 1. • Therefore:

  23. u1 1 u2 a22 u3 a33 u4 a44 u5 a55 u1 1 u2 a222 u3 a332 u4 a442 u5 a552 Power Method u1 1 u2 a2 u3 a3 u4 a4 u5 a5

  24. Convergence • The smaller l2, the faster the convergence of the Power Method. u1 1 u2 a22k u3 a33k u4 a44k u5 a55k

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