  Many large, real-world complex networks have rich community structure that a
network scientist seeks to understand. These communities may overlap or have
intricate internal structure. Extracting communities with particular
topological structure, even when they overlap with other communities, is a
powerful capability that would provide novel avenues of focusing in on
structure of interest. In this work we consider extracting highly-cyclic
regions of directed graphs (digraphs). We demonstrate that embeddings derived
from complex-valued eigenvectors associated with stochastic propagator
eigenvalues near roots of unity are well-suited for this purpose. We prove
several fundamental theoretic results demonstrating the connection between
these eigenpairs and the presence of highly-cyclic structure and we demonstrate
the use of these vectors on a few real-world examples.
