Improving the Communication Pattern in Matrix-Vector Operations for Large Scale-Free Graphs by Disaggregation

Abstract

Matrix-vector multiplication is the key operation in any Krylov-subspace iteration method. We are interested in Krylov methods applied to problems associated with the graph Laplacian arising from large scale-free graphs. Computations with graphs of this type on parallel distributed-memory computers are challenging. This is due to the fact that scale-free graphs have a degree distribution that follows a power law, and currently available graph partitioners are not efficient for such an irregular degree distribution. The lack of a good partitioning leads to excessive interprocessor communication requirements during every matrix-vector product. 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. The matrix-vector operations for the original graph are performed via a factored triple matrix-vector product involving the embedding graph. Even though the latter graph is larger, we ...