Generalizing K\"orner's graph entropy to graphons

TL;DR

This paper extends K"orner's graph entropy concept from finite graphs to graphons, establishing bounds and conditions under which they coincide, and linking entropy to fractional chromatic and clique numbers.

Contribution

It introduces a graphon-based entropy framework, providing bounds and exact values for most graphons, and connects entropy to fractional graph parameters.

Findings

Bounds for graphon entropy are established.

For most graphons, bounds coincide, giving exact entropy.

Graphon entropy relates to fractional chromatic and clique numbers.

Abstract

K\"orner introduced the notion of graph entropy in 1973 as the minimal code rate of a natural coding problem where not all pairs of letters can be distinguished in the alphabet. Later it turned out that it can be expressed as the solution of a minimization problem over the so-called vertex-packing polytope. In this paper we generalize this notion to graphons. We show that the analogous minimization problem provides an upper bound for graphon entropy. We also give a lower bound in the shape of a maximization problem. The main result of the paper is that for most graphons these two bounds actually coincide and hence precisely determine the entropy in question. Furthermore, graphon entropy has a nice connection to the fractional chromatic number and the fractional clique number.