Finding cliques in directed weighted graphs using complex Hermitian adjacency matrices

Abstract

The objective of this paper is to present the adaptation of the well known class of spectral graph partitioning algorithms to a new class of adjacency matrices. By the use of complex Hermitian adjacency matrices for asymmetric weighted digraphs and the subsequent application of an enhanced spectral partitioning algorithm, a better understanding of patterns within such digraphs becomes possible. This approach was used to find cliques within online communities. To validate our approach we benchmarked against existing implementations of spectral partitioning algorithms. The major result of our research is the application of spectral partitioning in an asymmetric communication environment. The practical implication of our work is the broadening of the use of a spectral partitioning algorithm to a new class of adjacency matrices that are able to model asymmetric weighted communication streams such as email exchange or market transactions.