A Fuzzy Order for Graphs Based on the Continuous Entropy of Gaussian Markov Random Fields

Abstract

Gaussian Markov Random Fields over graphs have been widely used in the context of both theoretical and applied Statistics. In this paper, we study the influence of the graph on the continuous entropy of the distribution. In particular, since the continuous entropy is highly dependant on the correlations between adjacent nodes in the graph, we consider the particular case of Gaussian Markov Random Fields with uniform correlation, i.e., Gaussian Markov Random Fields in which such correlations are equal. We define a partial order relation on the set of graphs that orders the graphs according to their contribution to the continuous entropy. We also present a graded version of this order relation that allows to compare incomparable graphs with respect to the original order relation. Finally, an example for the illustration of the graded order relation is provided.