The $25 Billion Eigenvector

1. The $25 Billion Eigenvector How does Google do Pagerank?

2. The Imaginary Web Surfer: • Starts at any page, • Randomly goes to a page linked from the current page, • Randomly goes to any web page from a dangling page, • … except sometimes (e.g. 15% of the time), goes to a purely random page.

3. J A A tiny web: who should get the highest rank? B I C H D G F E

4. The associated stochastic matrix: 0.0150 0.0150 0.0150 0.0150 0.0150 0.0150 0.4400 0.0150 0.0150 0.2983 0.4400 0.0150 0.0150 0.0150 0.0150 0.0150 0.0150 0.0150 0.0150 0.0150 0.0150 0.2983 0.0150 0.0150 0.0150 0.0150 0.0150 0.0150 0.0150 0.0150 0.0150 0.2983 0.8650 0.0150 0.0150 0.0150 0.0150 0.0150 0.0150 0.0150 0.4400 0.0150 0.0150 0.8650 0.0150 0.8650 0.0150 0.0150 0.0150 0.0150 0.0150 0.2983 0.0150 0.0150 0.8650 0.0150 0.0150 0.0150 0.0150 0.0150 0.0150 0.0150 0.0150 0.0150 0.0150 0.0150 0.0150 0.8650 0.0150 0.0150 0.0150 0.0150 0.0150 0.0150 0.0150 0.0150 0.0150 0.0150 0.8650 0.2983 0.0150 0.0150 0.0150 0.0150 0.0150 0.0150 0.0150 0.0150 0.0150 0.2983 0.0150 0.0150 0.0150 0.0150 0.0150 0.0150 0.4400 0.0150 0.0150 0.0150

5. Start with equal components

6. One iteration

7. Two iterations

8. Three iterations

9. Four iterations

10. Five iterations

11. Six iterations

12. Seven iterations

13. Eight iterations

14. Nine iterations

15. Ten iterations

16. The Eigenvector

17. The Imaginary Web Surfer: • Starts at any page, • Randomly goes to a page linked from the current page, • Randomly goes to any web page from a dangling page, • … except sometimes (e.g. 15% of the time), goes to a purely random page.

18. [U,G] = surfer (‘http://google.com’, 100)

19. Pagerank Power Iteration 1 step

20. Pagerank Power Iteration 2 steps

21. Pagerank Power Iteration 3 steps

22. Pagerank Power Iteration 4 steps

23. Pagerank Power Iteration 5 steps

24. Pagerank Power Iteration the limit

25. And the winners are… 'http://www.loc.gov/standards/iso639-2' 'http://www.sil.org/iso639-3' 'http://www.loc.gov/standards/iso639-5' 'http://purl.org/dc/elements/1.1' 'http://purl.org/dc/terms' 'http://purl.org/dc' 'http://creativecommons.org/licenses/by/3.0' 'http://i.creativecommons.org/l/by/3.0/88x31.png' 'http://www.nlb.gov.sg' 'http://purl.org/dcpapers' 'http://www.nl.go.kr' 'http://purl.org/dcregistry' 'http://www.kc.tsukuba.ac.jp/index_en.html'

26. How much storage to hold this array? • Current estimate of indexed WWW: 4.7 · 1010 web pages • If placed into an array this would have 2.21 · 1021 elements • If each element is stored in 4 bytes, this would be 8.8 · 1022 bytes • Current estimate of world’s data storage capacity is 3.0 · 1018 bytes (.003% of necessary space) http://www.smartplanet.com/blog/thinking-tech/what-is-the-worlds-data-storage-capacity/6256

27. How much time to do one power step? • Current estimate of indexed WWW: 4.7 · 1010 web pages • If placed into an array this would have 2.21 · 1021 elements • Fastest current machine does 33.86 · 1015 operations per second • One step of y = Ay takes 3.68 days

28. J A How is xk+1=Axkperformed? B I C H D G F E connection = [2 5 3 4 64 5 6 5 1 10 78 1 8 9] end = [2 5 6 7 8 9 11 12 13 16]

29. How is xk+1=Axkperformed? • xk+1 = .15/n e, (where e is all 1’s) • start = 1 • for j = 1,…, n • col_tot = endj-start • for i = start,…,endj • ii = connectioni • xk+1ii =xk+1ii+.85/col_tot*xki • c) start =endj+1