L09_-_MCMC

# All outgoing links from a webpage w are equally

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Unformatted text preview: has eigenvalue 1 (with multiplicity 1), and there is a unique positive vector x such that: Note that if a limiting distribution x exists, then it –  x ⋅ A = A (*) definitely needs to satisfy (*), o.w. it wouldn’t be a limiting ￿ distribution. The point of the theorem is that the Markov –  i xi = 1 (**) Chain indeed converges to the unique eigenvector of A satisfying (*) and (**), no matter what x0 is. Theorem: For any x0, xt→ x as t→∞. •  x is called the “stationary distribution of G” Two obvious Questions: –  why is x∞ interesting? –  how fast does xt→x∞ ? Menu •  Random walks on graphs •  Markov Chains •  Examples: –  pagerank –  card-shuffling –  colorings Pagerank •  No better proof that something is useful than having cool applications  •  It turns out that Markov Chains have a famous one: PageRank. •  PageRank of a webpage w ≈ Probability that a web-surfer starting from some central page (e.g. Yahoo!) and clicking random links arrives at webpage w. •  How to compute this probability? •  Form graph G = the hyperlink graph; •  Namely, G has a node for every webpage, and there is an edge from webpage w1 to webpage w2 iff there is a hyperlink from w1 to w2. •  All outgoing links from a webpage w are equally probable. •  Compute stationary distribution x∞, i.e. the left eigenvector of the transition matrix A of G, corresponding to eigenvalue 1. •  Pagerank of page w = x∞(w). Computing Pagerank •  Graph G = the hyperlink graph   Compute stationary distribution of G, i.e. the left eigenvector of the (normalized by out-degrees) adjacency matrix A of G, corresponding to eigenvalue 1. •  How to compute stationary distribution? •  Idea 1: Crawl the web, create giant A, solve eigenvalue problem. •  Runtime O(n3) using Gaussian elimination •  this is too much for n = size of the web. •  Idea 2: Simulate the walk sufficiently many times (theory meets practice) –  Start at some...
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## This note was uploaded on 09/07/2013 for the course EECS 18.410 taught by Professor Erikdemaine during the Spring '13 term at MIT.

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