Entropy and Heterogeneity Measures for Directed Graphs

Abstract

In this paper, we aim to develop novel methods for measuring the structural complexity for directed graphs. Although there are many existing alternative measures for quantifying the structural properties of undirected graphs, there are relatively few corresponding measures for directed graphs. To fill this gap in the literature, we explore a number of alternative techniques that are applicable to directed graphs. We commence by using Chung’s generalisation of the Laplacian of a directed graph to extend the computation of von Neumann entropy from undirected to directed graphs. We provide a simplified form of the entropy which can be expressed in terms of simple vertex in-degree and out-degree statistics. Moreover, we find approximate forms of the von Neumann entropy that apply to both weakly and strongly directed graphs, and that can be used to characterize network structure. Next we explore how to extend Estrada’s heterogeneity index from undirected to directed graphs. Our measure is motivated by the simplified von Neumann entropy, and involves measuring the heterogeneity of differences in in-degrees and out-degrees. Finally, we perform an analysis which reveals a novel linear relationship between heterogeneity and resistance distance (commute time) statistics for undirected graphs. This means that the larger the difference between the average commute time and shortest return path length between pairs of vertices, the greater the heterogeneity index. Based on this observation together with the definition of commute time on a directed graph, we define an analogous heterogeneity measure for directed graphs. We illustrate the usefulness of the measures defined in this paper for datasets describing Erdos-Renyi, ’small-world’, ’scale-free’ graphs, Protein-Protein Interaction (PPI) networks and evolving networks.