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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
– cardshuffling
– 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 websurfer 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 outdegrees) 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.
 Spring '13
 ErikDemaine
 Algorithms

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