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Summarizing large-scaled directed graphs into small-scale representations isa useful but less studied problem setting. Conventional clustering approaches,which based on "Min-Cut"-style criteria, compress both the vertices and edgesof the graph into the communities, that lead to a loss of directed edgeinformation. On the other hand, compressing the vertices while preserving thedirected edge information provides a way to learn the small-scalerepresentation of a directed graph. The reconstruction error, which measuresthe edge information preserved by the summarized graph, can be used to learnsuch representation. Compared to the original graphs, the summarized graphs areeasier to analyze and are capable of extracting group-level features which isuseful for efficient interventions of population behavior. In this paper, wepresent a model, based on minimizing reconstruction error with non-negativeconstraints, which relates to a "Max-Cut" criterion that simultaneouslyidentifies the compressed nodes and the directed compressed relations betweenthese nodes. A multiplicative update algorithm with column-wise normalizationis proposed. We further provide theoretical results on the identifiability ofthe model and on the convergence of the proposed algorithms. Experiments areconducted to demonstrate the accuracy and robustness of the proposed method.

Direction Matters: On Influence-Preserving Graph Summarization and Max-cut Principle for Directed Graphs