L09_-_MCMC

All outgoing links from a webpage w are equally

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

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.

Ask a homework question - tutors are online