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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?
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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 - + =
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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
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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
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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
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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
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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
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- + = 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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lect18 - More on Rankings Query-independent LAR Have an...

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