cs240a-graphs

# cs240a-graphs - Computation on Graphs Computation Graph...

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Computation on Graphs Computation on Graphs

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2 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
Clustering benchmark graph Clustering benchmark graph

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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 Matrix entries are edge weights Diagonal contains self-edges Number of nonzeros per row is the vertex degree
Example: Web graph and matrix Example: Web graph and matrix 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

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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 SC’03) on a web crawl of mit.edu (170,000 pages)

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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 Which author has the most collaborators? >>[count,author] = max(sum(A))

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## This note was uploaded on 12/27/2011 for the course CMPSC 240A taught by Professor Gilbert during the Fall '09 term at UCSB.

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cs240a-graphs - Computation on Graphs Computation Graph...

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