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Approximate von Neumann entropy for directed graphs

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JournalPhysical Review E
DatePublished - 12 May 2014
Issue number5
Volume89
Number of pages14
Original languageEnglish

Abstract

In this paper, we develop a novel entropy measure for assessing the structural complexity of 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 an alternative technique that is applicable to directed graphs. We commence by using Chung’s generalization 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 node in-degree and outdegree 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. We illustrate the usefulness of these simplified entropy forms defined in this paper on both artificial and real-world datasets, including structures from protein databases and high energy physics theory citation networks.

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