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sparse matrix-vector multiplication

Clustering benchmark graph. Link analysis of the web. Web page = vertexLink = directed edgeLink matrix: Aij = 1 if page i links to page j. 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 eigen1145

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sparse matrix-vector multiplication

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    1. Sparse Matrix-Vector Multiplication

    3. Link analysis of the web Web page = vertex Link = directed edge Link matrix: Aij = 1 if page i links to page j

    4. 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 8,000,000,000 by 8,000,000,000.

    5. 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)

    6. Strongly connected components Symmetric permutation to block triangular form Find P in linear time by depth-first search [Tarjan]

    7. RMAT Approximate Power-Law Graph

    8. Strongly Connected Components

    9. Sparse Adjacency Matrix and Graph Adjacency matrix: sparse array w/ nonzeros for graph edges Storage-efficient implementation from sparse data structures

    10. Breadth-First Search: Sparse mat * vec Multiply by adjacency matrix ? step to neighbor vertices Work-efficient implementation from sparse data structures

    11. Breadth-First Search: Sparse mat * vec Multiply by adjacency matrix ? step to neighbor vertices Work-efficient implementation from sparse data structures

    12. Breadth-First Search: Sparse mat * vec Multiply by adjacency matrix ? step to neighbor vertices Work-efficient implementation from sparse data structures

    13. Sparse Matrix times Sparse Matrix Shows up often as a primitive. Graphs are mostly not mesh-like, i.e. geometric locality and good separators. On a 2D processor grid, the parallel sparse algorithm looks much like the parallel dense algorithm. Redistribute to round-robin cyclic or random distribution for load balance.

    14. Load Balance Without Redistribution

    15. Load Balance With Redistribution

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