# Ensemble equivalence for dense graphs

## TL;DR

This paper investigates the conditions under which micro-canonical and canonical ensembles of dense graphs are equivalent, using large deviation theory to analyze the relative entropy and identifying frustration of constraints as a key factor.

## Contribution

It provides a rigorous analysis of ensemble equivalence in dense graphs, highlighting the role of constraint frustration and employing large deviation theory for graphons.

## Key findings

- Ensemble equivalence holds when constraints are not frustrated.
- Breaking of ensemble equivalence occurs under frustrated constraints.
- The analysis applies large deviation principles to graphons in dense regimes.

## Abstract

In this paper we consider a random graph on which topological restrictions are imposed, such as constraints on the total number of edges, wedges, and triangles. We work in the dense regime, in which the number of edges per vertex scales proportionally to the number of vertices $n$. Our goal is to compare the micro-canonical ensemble (in which the constraints are satisfied for every realisation of the graph) with the canonical ensemble (in which the constraints are satisfied on average), both subject to maximal entropy. We compute the relative entropy of the two ensembles in the limit as $n$ grows large, where two ensembles are said to be \emph{equivalent} in the dense regime if this relative entropy divided by $n^2$ tends to zero. Our main result, whose proof relies on large deviation theory for graphons, is that breaking of ensemble equivalence occurs when the constraints are \emph{frustrated}. Examples are provided for three different choices of constraints.