The entropy of random-free graphons and properties

TL;DR

This paper investigates the entropy growth of random-free graphons, showing it can be arbitrarily close to quadratic, but is bounded by O(n log n) for those in the closure of a random-free property, and provides a counterexample to a conjecture.

Contribution

It proves that the entropy of random-free graphons can grow as fast as any subquadratic function and constructs a non-stepfunction example with linear entropy, refuting a prior conjecture.

Findings

Entropy of random-free graphons can be arbitrarily close to quadratic growth.

Entropy is O(n log n) for graphons in the closure of a random-free property.

Constructed a non-stepfunction random-free graphon with linear entropy.

Abstract

Every graphon defines a random graph on any given number $n$ of vertices. It was known that the graphon is random-free if and only if the entropy of this random graph is subquadratic. We prove that for random-free graphons, this entropy can grow as fast as any subquadratic function. However, if the graphon belongs to the closure of a random-free graph property, then the entropy is $O(n \log n)$. We also give a simple construction of a non-stepfunction random-free graphon for which this entropy is linear, refuting a conjecture of Janson.