Download - The $25 Billion Eigenvector
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The $25 Billion Eigenvector
How does Google do Pagerank?
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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.
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A tiny web: who should get the highest rank?
J A B
I C
DH
G F E
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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
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How is yk+1=Axk performed?
J A B
I C
DH
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]
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How is yk+1=Axk performed?
1. yk+1 = .15/n e, (where e is all 1’s)2. start = 13. for j = 1,…, n
a) col_tot = endj-startb) for i = start,…, endj
• ii = connectioni
• yk+1ii = yk+1
ii+.85/col_tot*yki
c) start =endj+1
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Start with equal components
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One iteration
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Two iterations
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Three iterations
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Four iterations
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Five iterations
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Six iterations
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Seven iterations
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Eight iterations
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Nine iterations
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Ten iterations
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The Eigenvector
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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.
![Page 20: The $25 Billion Eigenvector](https://reader036.vdocuments.us/reader036/viewer/2022081505/568165a8550346895dd88fe1/html5/thumbnails/20.jpg)
[U,G] = surfer (‘http://google.com’, 100)
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0.03 Pagerank Power Iteration 1 step
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0.03 Pagerank Power Iteration 2 steps
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0.035 Pagerank Power Iteration 4 steps
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0.035 Pagerank Power Iteration 5 steps
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0.09 Pagerank Power Iteration the limit
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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'