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PageRank-1 - CS345 Data Mining Link Analysis Algorithms...

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    CS345 Data Mining Link Analysis Algorithms Page Rank Anand Rajaraman, Jeffrey D. Ullman
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Link Analysis Algorithms Page Rank Hubs and Authorities Topic-Specific Page Rank Spam Detection Algorithms Other interesting topics we won’t cover Detecting duplicates and mirrors Mining for communities Classification Spectral clustering
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Ranking web pages Web pages are not equally “important” www.joe-schmoe.com  v  www.stanford.edu Inlinks as votes www.stanford.edu  has 23,400 inlinks www.joe-schmoe.com  has 1 inlink Are all inlinks equal? Recursive question! 
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Simple recursive formulation Each link’s vote is proportional to the  importance  of its source page If page  P  with importance  x  has  n  outlinks,  each link gets  x/n  votes Page  P ’s own importance is the sum of the  votes on its inlinks
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Simple “flow” model The web in 1839 Yahoo M’soft Amazon y a m y/2 y/2 a/2 a/2 m y   =  /2 +  /2 a   =  /2 +  m m  =  /2
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Solving the flow equations 3 equations, 3 unknowns, no constants No unique solution All solutions equivalent modulo scale factor Additional constraint forces uniqueness y+a+m = 1 y = 2/5, a = 2/5, m = 1/5 Gaussian elimination method works for small  examples, but we need a better method for  large graphs
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Matrix formulation Matrix  M  has one row and one column for each web  page Suppose page j has n outlinks If j  !  i, then M ij =1/n Else M ij =0 M  is a  column   stochastic matrix Columns sum to 1 Suppose  r  is a vector with one entry per web page r i  is the importance score of page i Call it the  rank vector | r | = 1
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Example Suppose page j links to 3 pages, including i i j M r r = i 1/3
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Eigenvector formulation The flow equations can be written  =  Mr So the rank vector is an eigenvector of the  stochastic web matrix In fact, its first or principal eigenvector, with  corresponding eigenvalue 1
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Example Yahoo M’soft Amazon y 1/2 1/2 0 a 1/2 0 1 m 0 1/2 0 y a m y   =  /2 +  /2 a   =  /2 +  m m  =  /2 r = Mr y 1/2 1/2 0 y a = 1/2 0 1 a m 0 1/2 0 m
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Power Iteration method Simple iterative scheme (aka  relaxation ) Suppose there are N web pages Initialize:  r 0  = [1/N,….,1/N] T Iterate:  r k+1  =  Mr k Stop when | r k+1  r k | 1  <  ε | x | 1  = 
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