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Lecture12

# Lecture12 - May 5 2010 Yuh-Jie(Eunice Chen Local PageRank...

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May 5, 2010 Yuh-Jie (Eunice) Chen Local PageRank Properties 1 Introduction In this lecture, we will talk about some local PageRank properties. That is, given a graph G ( V, E ) , starting vector u, and 0 < α 1 , we will investigate the behaviors of pr α,u on subset S V. 2 Review Recall in the previous lecture we have extended the definition of a probability function f : V R + { 0 } to 1. f : S R + { 0 } , where S V 2. f : R + R + { 0 } and the following fact Fact 1. Let Z denote a lazy work. Given function f : R + R + , order the vertices in h according to function f. Then fZ ( x ) f ( x (1 - h f ) ) + f ( x (1 + h f ) ) 2 . 3 Local PageRank Properties 3.1 Stationary distribution Recall that the stationary distribution is defined as π ( x ) = { d x vol ( G ) if x V 0 otherwise . 1

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Theorem 1. For any starting vector u, set of vertices S V, and integer k 0 , pr α,u ( S ) - π ( S ) 1 - (1 - α ) k + vol ( S )(1 - γ 2 8 ) k (1 - α ) k , where γ = min h ( S j ) and vertices in S is ordered according to pr α,u .
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Lecture12 - May 5 2010 Yuh-Jie(Eunice Chen Local PageRank...

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