Network analysis using entropy component analysis

Structural complexity measures have found widespread use in network analysis. For instance, entropy can be used to distinguish between different structures. Recently we have reported an approximate network von Neumann entropy measure, which can be conveniently expressed in terms of the degree configurations associated with the vertices that define the edges in both undirected and directed graphs. However, this analysis was posed at the global level, and did not consider in detail how the entropy is distributed across edges. The aim in this paper is to use our previous analysis to define a new characterization of network structure, which captures the distribution of entropy across the edges of a network. Since our entropy is defined in terms of vertex degree values defining an edge, we can histogram the edge entropy using a multi-dimensional array for both undirected and directed networks. Each edge in a network increments the contents of the appropriate bin in the histogram, indexed according to the degree pair in an undirected graph or the in/out-degree quadruple for a directed graph. We normalize the resulting histograms and vectorize them to give network feature vectors reflecting the distribution of entropy across the edges of the network. By performing principal component analysis (PCA) on the feature vectors for samples, we embed populations of graphs into a low-dimensional space. We explore a number of variants of this method, including using both fixed and adaptive binning over edge vertex degree combinations, using both entropy weighted and raw bin-contents, and using multi-linear principal component analysis (MPCA), aimed at extracting the tensorial structure of high-dimensional data, as an alternative to classical PCA for component analysis. We apply the resulting methods to the problem of graph classification, and compare the results obtained to those obtained using some alternative state-of-the-art methods on real-world data.