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Unformatted text preview: Computation on Graphs Computation on Graphs Graphs and Sparse Matrices Graphs and Sparse Matrices 1 1 1 2 1 1 1 3 1 1 1 4 1 1 5 1 1 6 1 1 1 2 3 4 5 6 3 6 2 1 5 4 • Sparse matrix is a representation of a (sparse) graph 3 Graph partitioning Graph partitioning • Assigns subgraphs to processors • Determines parallelism and locality. • Tries to make subgraphs all same size (load balance) • Tries to minimize edge crossings (communication). • Exact minimization is NPcomplete. edge crossings = 6 edge crossings = 10 Link analysis of the web Link analysis of the web • Web page = vertex • Link = directed edge • Link matrix: A ij = 1 if page i links to page j 1 2 3 4 7 6 5 1 5 2 3 4 6 7 1 5 2 3 4 6 7 Web graph: PageRank (Google) Web graph: PageRank (Google) [Brin, Page] • Markov process: follow a random link most of the time; otherwise, go to any page at random. • Importance = stationary distribution of Markov process. • Transition matrix is p*A + (1p)*ones(size(A)) , scaled so each column sums to 1. • Importance of page i is the ith entry in the principal eigenvector of the transition matrix. • But the matrix is 1,000,000,000,000 by 1,000,000,000,000. An important page is one that many important pages point to. A Page Rank Matrix A Page Rank Matrix • Importance ranking of web pages • Stationary distribution of a Markov chain • Power method: matvec and vector arithmetic • Matlab*P page ranking demo (from SC’03) on a web crawl of mit.edu (170,000 pages) Social Network Analysis in Matlab: 1993 Social Network Analysis in Matlab: 1993 Coauthor graph from 1993 Householder symposium Social Network Analysis in Matlab: 1993...
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This note was uploaded on 12/27/2011 for the course CMPSC 140 taught by Professor Gilbert during the Fall '11 term at UCSB.
 Fall '11
 GILBERT

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