Approximate von Neumann entropy for directed graphs

Abstract

In this paper, we develop an 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 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. We illustrate the usefulness of these simplified entropy forms defined in this paper on both artificial and real-world data sets, including structures from protein databases and high energy physics theory citation networks.

Network Edge Entropy from Maxwell-Boltzmann Statistics

“…For example, the von Neumann entropy can be used as an effective characterization of network structure, commencing from a quantum analog in which the Laplacian matrix on graphs [1] plays the role of the density matrix. Further development of this idea has shown the link between the von Neumann entropy and the degree statistics of pairs of nodes forming edges in a network [2], which can be efficiently computed for both directed and undirected graphs [3]. Since the eigenvalues of the density matrix reflect the energy states of a network, this approach is closely related to the heat bath analogy in statistical mechanics.…”

“…Network entropy has been extensively used to characterize the salient features of the structure of static and dynamic of network systems arising in biology, physics, and the social sciences [1][2][3]. For example, the von Neumann entropy can be used as an effective characterization of network structure, commencing from a quantum analog in which the Laplacian matrix on graphs [1] plays the role of the density matrix.…”

“…There has been a considerable recent interest in computing the entropy associated with different types of network structure [2,3,5]. Network entropy has been extensively used to characterize the salient features of the structure of static and dynamic of network systems arising in biology, physics, and the social sciences [1][2][3].…”

“…Since the eigenvalues of the density matrix reflect the energy states of a network, this approach is closely related to the heat bath analogy in statistical mechanics. This provides a convenient route to network characterization [3,5]. By populating the energy states with particles which are in thermal equilibrium with a heat bath, this thermalization, of the occupation statistics for the energy states can be computed using the Maxwell-Boltzmann distribution [4,5].…”

Correlation Network Evolution Using Mean Reversion Autoregression

“…However, this does not generalize well for graphs since the true graph complexity cannot be accurately reflected by information such as the numbers of vertices or edges in the graph. To overcome this problem, we adopt a more meaningful measure of graph complexity, namely the von Neumann entropy, to encode the complexity of the supergraph structure (see [10] and [11] for detailed information of this entropy). Then, we have the supergraph complexity code length as follows,…”