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Iterative Methods and Partitioning Techniques for Large Sparse Problems in Network Analysis

The analysis of networks is an important aspect in
many fields. Here we consider three different topics: the numerical
solution of Markov chains, ranking of genes, and parallel
computations with large scale-free graphs. First, additive Schwarz
methods are a class of domain decomposition methods that are
suitable for the solution of large linear systems in serial as well
as in parallel mode. We adapt the Restricted Additive Schwarz (RAS)
method to the computation of the stationary probability
distribution vector of large, sparse, irreducible stochastic
matrices. The convergence properties are analyzed and extensive
numerical experiments aimed at assessing the effect of varying the
number of subdomains and the choice of overlap are discussed. Next,
the ranking of genes plays an important role in the identification
of key genes for a specific disease. A modification of the PageRank
algorithm that is used to rank web pages is the GeneRank algorithm.
We assessed the performance of additive Schwarz methods as well as
that of Chebyshev iteration for the solution of the GeneRank
problem. Finally, many large networks are scale-free. That is, the
degree distribution follows a power-law. Currently available graph
partitioners are not efficient for such an irregular degree
distribution. The lack of a good partitioning leads to excessive
inter-processor communication requirements during each
matrix-vector product on parallel distributed-memory computers. We
present an approach to alleviate this problem based on embedding
the original irregular graph into a more regular one by
disaggregating (splitting up) vertices in the original graph. Even
though the latter graph is larger, we are able to decrease the
communication requirements considerably and improve the performance
of the matrix-vector product.