  Graph clustering is a basic technique in machine learning, and has widespread
applications in different domains. While spectral techniques have been
successfully applied for clustering undirected graphs, the performance of
spectral clustering algorithms for directed graphs (digraphs) is not in general
satisfactory: these algorithms usually require symmetrising the matrix
representing a digraph, and typical objective functions for undirected graph
clustering do not capture cluster-structures in which the information given by
the direction of the edges is crucial. To overcome these downsides, we propose
a spectral clustering algorithm based on a complex-valued matrix representation
of digraphs. We analyse its theoretical performance on a Stochastic Block Model
for digraphs in which the cluster-structure is given not only by variations in
edge densities, but also by the direction of the edges. The significance of our
work is highlighted on a data set pertaining to internal migration in the
United States: while previous spectral clustering algorithms for digraphs can
only reveal that people are more likely to move between counties that are
geographically close, our approach is able to cluster together counties with a
similar socio-economical profile even when they are geographically distant, and
illustrates how people tend to move from rural to more urbanised areas.
