Is breaking of ensemble equivalence monotone in the number of constraints?

TL;DR

This paper investigates how the breaking of ensemble equivalence in random graphs is affected by the number of degree constraints, showing that the relative entropy decreases monotonically as constraints are relaxed.

Contribution

It proves that the relative entropy is monotone with respect to the number of degree constraints and connects the asymptotic behavior to Dirac and Gaussian distributions.

Findings

Relative entropy decreases monotonically with fewer constraints.

In the dense regime, degrees follow asymptotic Dirac and Gaussian distributions.

The formula applies to generic discrete random structures.

Abstract

Breaking of ensemble equivalence between the microcanonical ensemble and the canonical ensemble may occur for random graphs whose size tends to infinity, and is signaled by a non-zero specific relative entropy of the two ensembles. In [3] and [4] it was shown that breaking occurs when the constraint is put on the degree sequence (configuration model). It is not known what is the effect on the relative entropy when the number of constraints is reduced, i.e., when only part of the nodes are constrained in their degree (and the remaining nodes are left unconstrained). Intuitively, the relative entropy is expected to decrease. However, this is not a trivial issue because when constraints are removed both the microcanonical ensemble and the canonical ensemble change. In this paper a formula for the relative entropy valid for generic discrete random structures, recently formulated by Squartini and Garlaschelli, is used to prove that the relative entropy is monotone in the number of constraints when the constraint is on the degrees of the nodes. It is further shown that the expression for the relative entropy corresponds, in the dense regime, to the degrees in the microcanonical ensemble being asymptotically multivariate Dirac and in the canonical ensemble being asymptotically Gaussian.