  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 methods. 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 uses less memory and
asymptotically less computation while provably preserving 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.
