Properties of the Graph Modularity Matrix and Its Applications

Abstract We study the popular modularity matrix and respective functional used in connection with graph clustering and derive some properties useful when performing vertex aggregation of the associated graph. These properties are employed in the derivation of a multilevel parallel pairwise aggregation algorithm. Comparative performance results of the studied algorithm applied to graph clustering tested against the popular Louvain algorithm are presented. Some illustrative examples show that the resulting aggregates if used in an adaptive algebraic multigrid (AMG) are able to follow strong direction of anisotropy in finite element problems.