Entropic graph embedding via multivariate degree distributions

Although there are many existing alternative methods for using structural characterizations of undirected graphs for embedding, clustering and classification problems, there is relatively little literature aimed at dealing with such problems for directed graphs. In this paper we present a novel method for characterizing graph structure that can be used to embed directed graphs into a feature space. The method commences from a characterization based on the distribution of the von Neumann entropy of a directed graph with the in and out-degree configurations associated with directed edges. We start from a recently developed expression for the von Neumann entropy of a directed graph, which depends on vertex in-degree and out-degree statistics, and thus obtain a multivariate edge-based distribution of entropy. We show how this distribution can be encoded as a multi-dimensional histogram, which captures the structure of a directed graph and reflects its complexity. By performing principal components analysis on a sample of histograms, we embed populations of directed graphs into a low dimensional space. Finally, we undertake experiments on both artificial and real-world data to demonstrate that our directed graph embedding method is effective in distinguishing different types of directed graphs.