cs140-graphs

cs140-graphs - Computation on Graphs Computation on Graphs...

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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 NP-complete. 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 + (1-p)*ones(size(A)) , scaled so each column sums to 1. Importance of page i is the i-th 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 SC03) on a web crawl of mit.edu (170,000 pages) Social Network Analysis in Matlab: 1993 Social Network Analysis in Matlab: 1993 Co-author graph from 1993 Householder symposium Social Network Analysis in Matlab: 1993...
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cs140-graphs - Computation on Graphs Computation on Graphs...

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