Lecture14 - Approximate Page Rank 1 Approximate Page Rank...

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Approximate Page Rank 1 Approximate Page Rank Let pr α,s = pr ( α, s ) denote the page rank of a graph. Then pr α,s is a fixed point for the equation p = αs + (1 - α ) pZ . We can compute pr α,s approximately using the recurrence relation p t +1 = αs + (1 - α ) p t Z . Recall that we can rewrite our page rank equation as p = sαI I - (1 - α ) Z = X k =0 (1 - α ) k Z k While s is defined to be a probability distribution, i.e || s || 1 = 1 , we shall relax that constraint to s ( v ) 0 for all v V ( G ) , when we are computing approximate page rank. Fact 1. pr α,s ( S ) + pr α,s 0 ( S ) = pr α,s + s 0 ( S ) for any S V ( G ) Fact 2. pr ( α, s ) Z = pr ( α, sZ ) Fact 3. pr ( α, s ) = αs + (1 - α ) pr ( α, sZ ) Definition 1. p’ is an -approximate of pr α,s if p 0 = pr α,s - r for a residue function r : V R + ∪ { 0 } such that r ( v ) · d v for all v V Definition 2. The support of vector p 0 is the number of non-zero values in p 0 , denoted supp ( p 0 ) . Theorem 1. For any seed s , with || s || 1 1 , s 0 , and for any > 0 , we can compute an -approximate page rank p 0 where vol ( supp ( p 0 )) O ( 1 (1 - α ) ) .
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