  Clustering is an important topic in algorithms, and has a number of
applications in machine learning, computer vision, statistics, and several
other research disciplines. Traditional objectives of graph clustering are to
find clusters with low conductance. Not only are these objectives just
applicable for undirected graphs, they are also incapable to take the
relationships between clusters into account, which could be crucial for many
applications. To overcome these downsides, we study directed graphs (digraphs)
whose clusters exhibit further "structural" information amongst each other.
Based on the Hermitian matrix representation of digraphs, we present a
nearly-linear time algorithm for digraph clustering, and further show that our
proposed algorithm can be implemented in sublinear time under reasonable
assumptions. The significance of our theoretical work is demonstrated by
extensive experimental results on the UN Comtrade Dataset: the output
clustering of our algorithm exhibits not only how the clusters (sets of
countries) relate to each other with respect to their import and export
records, but also how these clusters evolve over time, in accordance with known
facts in international trade.
