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# lect18 - More on Rankings Query-independent LAR Have an...

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More on Rankings

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Query-independent LAR Have an a-priori ordering of the web pages Q : Set of pages that contain the keywords in the query q Present the pages in Q ordered according to order π What are the advantages of such an approach?
InDegree algorithm Rank pages according to in-degree w i = |B(i)| 1. Red Page 1. Yellow Page 1. Blue Page 1. Purple Page 1. Green Page w=1 w=1 w=2 w=3 w=2

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PageRank algorithm [BP98] Good authorities should be pointed by good authorities Random walk on the web graph pick a page at random with probability 1- α jump to a random page with probability α follow a random outgoing link Rank according to the stationary distribution 1. Red Page 1. Purple Page   1. Yellow Page 1. Blue Page 1. Green Page ( 29 n q F q PR p PR p q 1 1 ) ( ) ( ) ( α - + =
Markov chains A Markov chain describes a discrete time stochastic process over a set of states according to a transition probability matrix

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Random walks Random walks on graphs correspond to Markov Chains The set of states S is the set of nodes of the graph G The transition probability matrix is the probability that we follow an edge from one node to another
An example v 1 v 2 v 3 v 4 v 5 = 2 1 0 0 0 2 1 0 0 3 1 3 1 3 1 0 0 0 1 0 1 0 0 0 0 0 0 2 1 2 1 0 P = 1 0 0 0 1 0 0 0 1 0 1 0 0 0 0 0 0 1 1 0 A

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State probability vector
An example = 0 2 1 0 0 2 1 0 0 3 1 3 1 3 1 0 0 0 1 0 1 0 0 0 0 0 0 2 1 2 1 0 P v 1 v 2 v 3 v 4 v 5 q = 1/3 q + 1/2 q t 5 q = 1/2 q t 5

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Stationary distribution A stationary distribution for a MC with transition matrix P , is a probability distribution π , such that = P π π A MC has a unique stationary distribution if it is irreducible the underlying graph is strongly connected it is aperiodic for random walks, the underlying graph is not bipartite
Computing the stationary distribution The Power Method

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The PageRank random walk Vanilla random walk make the adjacency matrix stochastic and run a random walk = 0 2 1 0 0 2 1 0 0 3 1 3 1 3 1 0 0 0 1 0 1 0 0 0 0 0 0 2 1 2 1 0 P
The PageRank random walk What about sink nodes? what happens when the random walk moves to a node without any outgoing inks? = 0 2 1 0 0 2 1 0 0 3 1 3 1 3 1 0 0 0 1 0 0 0 0 0 0 0 0 2 1 2 1 0 P

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= 0 2 1 0 0 2 1 0 0 3 1 3 1 3 1 0 0 0 1 0 5 1 5 1 5 1 5 1 5 1 0 0 2 1 2 1 0 P' The PageRank random walk Replace these row vectors with a vector v typically, the uniform vector P’ = P + dv T = otherwise 0 sink   is   i   if 1 d
- + = 5 1 5 1 5 1 5 1 5 1 5 1 5 1 5 1 5 1 5 1 5 1 5 1 5 1 5 1 5 1 5 1 5 1 5 1 5 1 5 1 5 1 5 1 5 1 5 1 5 1 2 1 0 0 0 2 1 0 0 3 1 3 1 3 1 0 0 0 1 0 5 1 5 1 5 1 5 1 5 1 0 0 2 1 2 1 0 ' P' ) 1 ( α

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## This document was uploaded on 10/05/2010.

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lect18 - More on Rankings Query-independent LAR Have an...

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