Lecture Notes 18

Lecture Notes 18 - Lecture18 Lecture 18: Application of...

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Lecture18 1 Outline: 1) Markov-Matrices and the Iterative map p k+1 =Mp k 2) Example: 2x2 Markov chain 3) Eigenvalues and eigenvectors of Markov Matrices 4) Google as world's largest Eigen problem: The PageRank Algorithm A) Overview of Search Engine requirements B) Construction of the Google Matrix G (a really big Markov Matrix) C) Iteration and the power method 5) Matlab Demo Lecture 18: Application of EigenProblems: Markov Matrices and the page-rank algorithm Definition: a Markov Matrix M contains all positive elements (M ij >0) and has column sums=1. 2x2 Example: M=[ .6 .8 ; .4 .2 ]; Markov Matrices: M describes the discrete transitional probabilities between states in a Markov Chain given by the iterative map p k+1 =Mp k where p is a vector containing the discrete probability of being in state p i 2 state example: with p 0 =[1 0]', M=[ .6 .8 ; .4 .2]=[3 4 ; 2 1]/5 Question: What is the most probable state after n iterations? Is there a steady-state "fixed point" such that p =Mp ? Markov Matrices: Example: M=[ .6 .8 ; .4 .2 ] ( can show for M=[ a b ; (1-a) (1-b) ]) Markov Matrices: EigenValues and EigenVectors
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Lecture18 2 Properties of Markov Matrices (Perron-Frobenius Theorem)
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This note was uploaded on 06/02/2010 for the course APMA APMA E3101 taught by Professor Spiegelman during the Fall '07 term at Columbia.

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Lecture Notes 18 - Lecture18 Lecture 18: Application of...

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