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

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

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

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

11. Matrix Notation = .1 .3 .2

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

13. Power Method
Initialize:
Repeat until convergence:

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

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

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 u3 u4 u5

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

22. Why does it work? From the last slide:
To get the first iterate, multiply x(0) by A.
First eigenvalue is 1.
Therefore:
Assume that lambda 1 is less than 1 and all other eigenvalues are strictly less than 1.
All less than 1

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

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
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,