Entropy versus heterogeneity for graphs

Abstract

In this paper we explore and compare two contrasting graph characterizations. The first of these is Estrada's heterogeneity index, which measures the heterogeneity of the node degree across a graph. Our second measure is the the von Neumann entropy associated with the Laplacian eigenspectrum of graphs. Here we show how to approximate the von Neumann entropy by replacing the Shannon entropy by its quadratic counterpart. This quadratic entropy can be expressed in terms of a series of permutation invariant traces, which can be computed from the node degrees in quadratic time. We compare experimentally the effectiveness of the approximate expression for the entropy with the heterogeneity index. © 2011 Springer-Verlag Berlin Heidelberg.