Graph characterizations from von Neumann entropy

Authors:
Lin Han;Francisco Escolano;Edwin R. Hancock;Richard C. Wilson
Affiliations:
Department of Computer Science, The University of York, YO10 5GH, UK;Department of Computer Science and Artificial Intelligence, Alicante University, Spain;Department of Computer Science, The University of York, YO10 5GH, UK;Department of Computer Science, The University of York, YO10 5GH, UK
Venue:
Pattern Recognition Letters
Year:
2012

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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.