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Unformatted text preview: 1 Evaluating the Web PageRank Hubs and Authorities 2 PageRank r Intuition : solve the recursive equation: a page is important if important pages link to it. r In highfalutin terms: i m p o r t a n c e = the principal eigenvector of the stochastic matrix of the Web. R A few fixups needed. 3 Stochastic Matrix of the Web r Enumerate pages. r Page i corresponds to row and column i . r M [ i , j ] = 1/ n if page j links to n pages, including page i ; 0 if j does not link to i . R M [ i , j ] is the probability well next be at page i if we are now at page j . 4 Example i j Suppose page j links to 3 pages, including i 1/3 5 Random Walks on the Web r Suppose v is a vector whose i th component is the probability that we are at page i at a certain time. r If we follow a link from i at random, the probability distribution for the page we are then at is given by the vector M v . 6 Random Walks  (2) r Starting from any vector v , the limit M ( M ( M ( M v ) )) is the distribution of page visits during a random walk. r Intuition : pages are important in proportion to how often a random walker would visit them. r The math : limiting distribution = principal eigenvector of M = PageRank . 7 Example: The Web in 1839 Yahoo Msoft Amazon y 1/2 1/2 a 1/2 0 1 m 0 1/2 y a m 8 Simulating a Random Walk r Start with the vector v = [1,1,,1] representing the idea that each Web page is given one unit of i m p o r t a n c e . r Repeatedly apply the matrix M to v , allowing the importance to flow like a random walk. r Limit exists, but about 50 iterations is sufficient to estimate final distribution. 9 Example r Equations v = M v : y = y /2 + a /2 a = y /2 + m m = a /2 y a = m 1 1 1 1 3/2 1/2 5/4 1 3/4 9/8 11/8 1/2 6/5 6/5 3/5 . . . 10 Solving The Equations r Because there are no constant terms,...
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This note was uploaded on 01/31/2011 for the course CS 345 taught by Professor Dunbar,a during the Fall '07 term at UC Davis.
 Fall '07
 Dunbar,A

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