10-pagerank

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Unformatted text preview: Power iteration: Y! A MS Y! ½ ½ 0 A ½ 0 0 MS Set ri=1 ri=∑j Mij∙rj And iterate MS 0 ½ 0 Example: y a= m 2/7/2011 1 1 1 1 ½ ½ ¾ ½ ¼ 5/8 3/8 ¼ Jure Leskovec, Stanford C246: Mining Massive Datasets … 0 0 0 32 Teleports Follow random teleport links with probability 1.0 from dead-ends Adjust matrix accordingly 2/7/2011 Jure Leskovec, Stanford C246: Mining Massive Datasets 33 Suppose there are N pages Consider a page j, with set of outlinks O(j) We have Mij = 1/|O(j)| when j→i and Mij = 0 otherwise The random teleport is equivalent to adding a teleport link from j to every other page with probability (1-β)/N reducing the probability of following each outlink from 1/|O(j)| to β/|O(j)| Equivalent: tax each page a fraction (1-β) of its score and redistribute evenly 2/7/2011 Jure Leskovec, Stanford C246: Mining Massive Datasets 34 Construct the N x N matrix A as follows Aij = β∙Mij + (1-β)/N Verify that A is a stochastic matrix The page rank vector r is the principal eigenvector of this matrix satisfying r = A∙r Equivalently, r is the stationary distribution of the random walk with teleports 2/7/2011 Jure Leskovec, Stanford C246: Mining Massive Datasets 35 Key step is matrix-vector multiplication rnew = A∙rold Easy if we have enough main memory to hold A, rold, rnew Say N = 1 billion pages We need 4 bytes for each entry (say) 2 billion entries for vectors, approx 8GB Matrix A has N2 entries 1018 is a large number! 2/7/2011 Jure Leskovec, Stanford C246: Mining Massive Datasets 36 r = A∙r, where Aij = β Mij + (1-β)/N ri = ∑1≤j≤N Aij rj ri = ∑1≤j≤N [β Mij + (1-β)/N] rj = β ∑1≤j≤N Mij rj + (1-β)/N ∑1≤j≤N rj = β ∑1≤j≤N Mij rj + (1-β)/N, since...
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