Skew-Symmetric Adjacency Matrices for Clustering Directed Graphs

Cut-based directed graph (digraph) clustering often focuses on finding dense within-cluster or sparse between-cluster connections, similar to cut-based undirected graph clustering. In contrast, for flow-based clusterings the edges between clusters tend to be oriented in one direction and have been found in migration data, food webs, and trade data. In this paper we introduce a spectral algorithm for finding flow-based clusterings. The proposed algorithm is based on recent work which uses complex-valued Hermitian matrices to represent digraphs. By establishing an algebraic relationship between a complex-valued Hermitian representation and an associated real-valued, skew-symmetric matrix the proposed algorithm produces clusterings while remaining completely in the real field. Our algorithm is more memory efficient, requires less computation, and provably preserves solution quality. We also show the algorithm can be easily implemented using standard computational building blocks, possesses better numerical properties, and loans itself to a natural interpretation via an objective function relaxation argument.