  Summarizing large-scaled directed graphs into small-scale representations is
a useful but less studied problem setting. Conventional clustering approaches,
which based on "Min-Cut"-style criteria, compress both the vertices and edges
of the graph into the communities, that lead to a loss of directed edge
information. On the other hand, compressing the vertices while preserving the
directed edge information provides a way to learn the small-scale
representation of a directed graph. The reconstruction error, which measures
the edge information preserved by the summarized graph, can be used to learn
such representation. Compared to the original graphs, the summarized graphs are
easier to analyze and are capable of extracting group-level features which is
useful for efficient interventions of population behavior. In this paper, we
present a model, based on minimizing reconstruction error with non-negative
constraints, which relates to a "Max-Cut" criterion that simultaneously
identifies the compressed nodes and the directed compressed relations between
these nodes. A multiplicative update algorithm with column-wise normalization
is proposed. We further provide theoretical results on the identifiability of
the model and on the convergence of the proposed algorithms. Experiments are
conducted to demonstrate the accuracy and robustness of the proposed method.
