The 25 Billion Eigenvector How does Google do

The $25 Billion Eigenvector How does Google do Pagerank?

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.

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

The associated stochastic matrix: 0. 0150 0. 4400 0. 0150 0. 2983 0. 0150 0. 0150 0. 8650 0. 0150 0. 0150 0. 4400 0. 0150 0. 0150 0. 8650 0. 0150 0. 2983 0. 0150

How is yk+1=Axk performed? J A B I C H D G F E connection = [2 5 3 4 6 4 5 6 5 1 10 7 8 1 8 9] end = [2 5 6 7 8 9 11 12 13 16]

How is yk+1=Axk performed? 1. yk+1 =. 15/n e, (where e is all 1’s) 2. start = 1 3. for j = 1, …, n a) col_tot = endj-start b) for i = start, …, endj • ii = connectioni • yk+1 ii = yk+1 ii+. 85/col_tot*yki c) start =endj+1

Start with equal components

One iteration

Two iterations

Three iterations

Four iterations

Five iterations

Six iterations

Seven iterations

Eight iterations

Nine iterations

Ten iterations

The Eigenvector

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.

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

Pagerank Power Iteration 1 step

Pagerank Power Iteration 2 steps

Pagerank Power Iteration 3 steps

Pagerank Power Iteration 4 steps

Pagerank Power Iteration 5 steps

Pagerank Power Iteration the limit

And the winners are… 'http: //www. loc. gov/standards/iso 639 -2' 'http: //www. sil. org/iso 639 -3' 'http: //www. loc. gov/standards/iso 639 -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. 0/88 x 31. 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'