GraphBLAS Mathematical Opportunities: Parallel Hypersparse, Matrix Based Graph Streaming, and Complex-Index Matrices

The GraphBLAS high performance library standard has yielded capabilities beyond enabling graph algorithms to be readily expressed in the language of linear algebra. These GraphBLAS capabilities enable new performant ways of thinking about algorithms that include leveraging hypersparse matrices for parallel computation, matrix-based graph streaming, and complex-index matrices. Formalizing these concepts mathematically provides additional opportunities to apply GraphBLAS to new areas. This paper formally develops parallel hypersparse matrices, matrix-based graph streaming, and complex-index matrices and illustrates these concepts with various examples to demonstrate their potential merits.