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Breaking of ensemble equivalence between the microcanonical ensemble and thecanonical 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 onthe degree sequence (configuration model). It is not known what is the effecton the relative entropy when the number of constraints is reduced, i.e., whenonly part of the nodes are constrained in their degree (and the remaining nodesare left unconstrained). Intuitively, the relative entropy is expected todecrease. However, this is not a trivial issue because when constraints areremoved both the microcanonical ensemble and the canonical ensemble change. Inthis paper a formula for the relative entropy valid for generic discrete randomstructures, recently formulated by Squartini and Garlaschelli, is used to provethat the relative entropy is monotone in the number of constraints when theconstraint is on the degrees of the nodes. It is further shown that theexpression for the relative entropy corresponds, in the dense regime, to thedegrees in the microcanonical ensemble being asymptotically multivariate Diracand in the canonical ensemble being asymptotically Gaussian.

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