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Unformatted text preview: 1 Evaluating the Web PageRank Hubs and Authorities 2 PageRank Intuition : solve the recursive equation: a page is important if important pages link to it. In highfalutin terms: importance = the principal eigenvector of the stochastic matrix of the Web. A few fixups needed. 3 Stochastic Matrix of the Web Enumerate pages. Page i corresponds to row and column i . M [ i , j ] = 1/ n if page j links to n pages, including page i ; 0 if j does not link to i . 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 Suppose v is a vector whose i th component is the probability that we are at page i at a certain time. 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) Starting from any vector v , the limit M ( M ( M ( M v ) )) is the distribution of page visits during a random walk. Intuition : pages are important in proportion to how often a random walker would visit them. 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 Start with the vector v = [1,1,,1] representing the idea that each Web page is given one unit of importance . Repeatedly apply the matrix M to v , allowing the importance to flow like a random walk. Limit exists, but about 50 iterations is sufficient to estimate final distribution. 9 Example 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 Because there are no constant terms, these 3 equations in 3 unknowns do not...
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 Spring '09

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