Breaking of ensemble equivalence for dense random graphs under a single constraint

TL;DR

This paper investigates the conditions under which ensemble equivalence breaks down in dense random graphs with a single density constraint, revealing a specific phase where spectral and structural differences emerge.

Contribution

It establishes the occurrence of breaking of ensemble equivalence (BEE) for dense graphs with one constraint, identifying the BEE-phase and spectral signatures, and analyzing the ensembles' asymptotic behaviors.

Findings

BEE occurs in a specific density and edge range for dense graphs.

Spectral gap exists between ensembles in the BEE-phase.

Microcanonical behaves like Erdős-Rényi, canonical like a mixture of two Erdős-Rényi graphs.

Abstract

Two ensembles are often used to model random graphs subject to constraints: the microcanonical ensemble (= hard constraint) and the canonical ensemble (= soft constraint). It is said that breaking of ensemble equivalence (BEE) occurs when the specific relative entropy of the two ensembles does not vanish as the size of the graph tends to infinity. The latter means that it matters for the scaling properties of the graph whether the constraint is met for every single realisation of the graph or only holds as an ensemble average. In the literature, it was found that BEE is the rule rather than the exception for two classes: sparse random graphs when the number of constraints is of the order of the number of vertices and dense random graphs when there are two or more constraints that are frustrated. In the present paper we establish BEE for a third class: dense random graphs with a single constraint, namely, on the density of a given finite simple graph. We show that BEE occurs only in a certain range of choices for the density and the number of edges of the simple graph, which we refer to as the BEE-phase. We show that, in part of the BEE-phase, there is a gap between the scaling limits of the averages of the maximal eigenvalue of the adjacency matrix of the random graph under the two ensembles, a property that is referred to as spectral signature of BEE. Proofs are based on an analysis of the variational formula on the space of graphons for the limiting specific relative entropy derived by Den Hollander et al. (2018), in combination with an identification of the minimising graphons and replica symmetry arguments. We show that in the replica symmetric region of the BEE-phase, as the size of the graph tends to infinity, the microcanonical ensemble behaves like an Erd\H{o}s-R\'enyi random graph, while the canonical ensemble behaves like a mixture of two Erd\H{o}s-R\'enyi random graphs.