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I do not see how pageRank relates to the power method. Since for the pageRank we are looking for the steady stable state (vector) for a Markov (transition) matrix and the matrix has already an eigenvalue equals to one, why multiplication is used throughout the iterations to converge into that vector. Knowing that the only things that changes during the iterations are the eigenvalues with absolute value less than 1 and their corresponding "disappearing" eigenvector.

Therefore, why this not done by factoring the Markov matrix to find the eigenvector corresponds to eigenvalue 1 instead of the iterative multiplication approach?

Alp Uzman
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Nawaf
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  • Page Rank is related to the dominant eigenvalue of a particular transitiion matrix, but mainly to the eigenvector corresponding to that eigenvalue. There is a theory about the existence and uniqueness (up to scale) of the eigenvector. The power iteration is one way to approximate the eigenvector/eigenvalue pair. Do you want to know more about the theory or about the algorithm? – hardmath Apr 10 '15 at 17:55
  • Here's a good article that describes how PageRank works and why they use the power iteration method (and why the power iteration method works): http://www.ams.org/samplings/feature-column/fcarc-pagerank (It is a very good read.) – TravisJ Apr 10 '15 at 17:58
  • In the PageRank case, the power method converges very fast... – TravisJ Apr 10 '15 at 17:58

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Finding a full eigendecomposition costs $O(n^3)$ operations and $O(n^2)$ memory space, where $n$ is the side of the matrix. The size $n$ of the Markov matrix is the number of indexed web pages, which is of the order of $10^9$, so a full factorization would be prohibitive.

Federico Poloni
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