5 Understanding Page Rank

Understanding Google’s PageRank™ Amy Langville, Carl Meyer, Google’s Page Rank and Beyond: The Science of Search Engine Rankings. Princeton University Pres, 2006

An Elegant Formula ,[object Object],[object Object],[object Object],[object Object]

 π  π   S + (1-  ) E) ,[object Object],[object Object],[object Object],r (P j ) |P j | _____  P j  B Pi

 π  π   S + (1-  ) E) ,[object Object],[object Object],[object Object],[object Object],r (P j ) |P j | _____ P j  B Pi  r(P i ) : the rank of a given page P j  B pi : the ranks of the set of back- linking pages r (P j ) : the rank of a given page |P j | : the number of out-links on a page

 π  π   S + (1-  ) E) ,[object Object],1 2 3 5 4 6

 π  π   S + (1-  ) E) ,[object Object],1 2 3 6 4 5

But there’s a problem ,[object Object],r (P j ) |P j | _____ P j  B Pi  r(P i ) : the rank of a given page P j  B pi : the ranks of the set of back- linking pages r (P j ) : the rank of a given page |P j | : the number of out-links on a page ,[object Object],1 2 3 6 4 5

The solution. . . sort of ,[object Object],[object Object],[object Object],1 2 3 6 4 5

Directed graph iterative node values ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],1 2 3 6 4 5

CMS matrix This can’t go on forever Some values are equivalent (ties). In the interest of speed and efficiency, we need to know if the ranks converge—that is, will we break all ties, or will we keep doing this indefinitely and never have a decisive ranking? To determine this, the formula must be transformed using binary adjacency transformation, and Markov chain theory. 1 2 3 6 4 5

Convert the iterative calculation to a matrix calculation using binary adjacency transformation for a 1Xn matrix ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[ ]

Now, you can treat a row as a vector, or set of values P 1 P 2 P 3 P 4 P 5 P 6 P 1 0 ½ ½ 0 0 0 P 2 0 0 0 0 0 0 P 3 1/3 1/3 0 0 1/3 0 P 4 0 0 0 0 ½ ½ P 5 0 0 0 ½ 0 ½ P 6 0 0 0 1 0 0 [ ]   

This is a sparse matrix. That’s good. P 1 P 2 P 3 P 4 P 5 P 6 P 1 0 ½ ½ 0 0 0 P 2 0 0 0 0 0 0 P 3 1/3 1/3 0 0 1/3 0 P 4 0 0 0 0 ½ ½ P 5 0 0 0 ½ 0 ½ P 6 0 0 0 1 0 0 [ ]

 π  π   S + (1-  ) E) ,[object Object],[object Object],[object Object],r (P j ) |P j | _____ P j  B Pi  ,[object Object]

 π  π   S + (1-  ) E) ,[object Object],[object Object],[object Object]

5 Understanding Page Rank

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