Graph characterizations from von Neumann entropy

Lin Han, Francisco Escolano, Edwin R. Hancock, Richard C. Wilson

Abstract: In this paper we explore how the von Neumann entropy can be used as a measure of graph complexity. We also develop a simplified form for the von Neumann entropy of a graph that can be computed in terms of node degree statistics. We compare the resulting complexity with Estrada’s heterogeneity index which measures the heterogeneity of the node degree across a graph and reveal a new link between Estrada’s index and the commute time on a graph. Finally, we explore how the von Neumann entropy can be used in conjunction with thermodynamic depth. This measure has been shown to overcome problems associated with iso-spectrality encountered when using complexity measures based on spectral graph theory. Our experimental evaluation of the simplified von Neumann entropy explores (a) the accuracy of the underlying approximation, (b) a comparison with alternative graph characterizations, and (c) the application of the entropy-based thermodynamic depth to characterize protein–protein interaction networks. Highlights ► We pose the von Neumann entropy as a new graph complexity measure. ► We show how this entropy can be efficiently computed using node degree statistics. ► We reveal the link between the Estrada’s heterogeneity index and the commute time. ► We embed von Neumann entropy into the thermodynamic depth to characterize graphs.