Entropic Weighted Rank Function

TL;DR

This paper investigates when weighted rank functions of matroids are entropic, revealing their position on the submodularity cone boundary and identifying conditions for entropic cases, especially for graphic and representable matroids.

Contribution

It characterizes the entropic weighted rank functions of matroids, especially for certain classes like graphic and representable matroids, extending previous results on matroid rank functions.

Findings

Weighted rank functions lie on the boundary of the submodularity cone.

Integer-valued weighted rank functions are entropic for representable matroids over characteristic 2 fields.

Constant weight rank functions are entropic for graphic matroids under a necessary and sufficient condition.

Abstract

It is known that the entropy function over a set of jointly distributed random variables is a submodular set function. However, not any submodular function is of this form. In this paper, we consider a family of submodular set functions, called weighted rank functions of matroids, and study the necessary or sufficient conditions under which they are entropic. We prove that weighted rank functions are located on the boundary of the submodularity cone. For the representable matroids over a characteristic 2 field, we show that the integer valued weighted rank functions are entropic. We derive a necessary condition for constant weight rank functions to be entropic and show that for the case of graphic matroids, this condition is indeed sufficient. Since these functions generalize the rank of a matroid, our findings generalize some of the results of Abbe et. al. about entropic properties of the rank function of matroids.