Connections between Classical and Parametric Network Entropies

Matthias Dehmer,Frank Emmert-Streib,Abbe Mowshowitz

doi:10.1371/journal.pone.0015733

Abstract

This paper explores relationships between classical and parametric measures of graph (or network) complexity. Classical measures are based on vertex decompositions induced by equivalence relations. Parametric measures, on the other hand, are constructed by using information functions to assign probabilities to the vertices. The inequalities established in this paper relating classical and parametric measures lay a foundation for systematic classification of entropy-based measures of graph complexity.

Highlights

Information theory has proven to be a useful tool in the analysis and measurement of network complexity [1]
In addition to the use of measures on graphs to analyze biological or chemical systems, information theory has been employed in network physics, see [1,9,10]
Arnand et al [1] provide a comprehensive review of Shannon entropy measures applied to network ensembles

Summary

Introduction

Information theory has proven to be a useful tool in the analysis and measurement of network complexity [1]. Common to all such research is the use of Shannon’s [19] classical measure to derive entropies of the underlying graph topology interpreted as the structural information content of a graph. Special graph invariants (e.g., number of vertices, edges, degrees, distances etc.) and equivalence relations have given rise to special measures of information contents [11,12,15]. This gives rise to general information inequalities between measures; on the other hand, bounds for special classes of graphs can be obtained.

Results

Conclusion