Approximate entropy of network parameters

TL;DR

This paper introduces a structural and dynamical approach to measure approximate entropy in networks, revealing how it reflects network complexity and can distinguish different dynamical processes, with applications in biological data analysis.

Contribution

It defines a new entropy measure based on approximate entropy for network structures and dynamics, linking it to network complexity and applications in biological data.

Findings

Entropy converges to binary Shannon entropy with network size.

Approximate entropy distinguishes networks generated by different dynamical processes.

Applications demonstrate relevance to biological data, such as cancer genomics.

Abstract

We study the notion of approximate entropy within the framework of network theory. Approximate entropy is an uncertainty measure originally proposed in the context of dynamical systems and time series. We firstly define a purely structural entropy obtained by computing the approximate entropy of the so called slide sequence. This is a surrogate of the degree sequence and it is suggested by the frequency partition of a graph. We examine this quantity for standard scale-free and Erd\H{o}s-R\'enyi networks. By using classical results of Pincus, we show that our entropy measure converges with network size to a certain binary Shannon entropy. On a second step, with specific attention to networks generated by dynamical processes, we investigate approximate entropy of horizontal visibility graphs. Visibility graphs permit to naturally associate to a network the notion of temporal correlations, therefore providing the measure a dynamical garment. We show that approximate entropy distinguishes visibility graphs generated by processes with different complexity. The result probes to a greater extent these networks for the study of dynamical systems. Applications to certain biological data arising in cancer genomics are finally considered in the light of both approaches.