  For a random graph subject to a topological constraint, the microcanonical
ensemble requires the constraint to be met by every realisation of the graph
(`hard constraint'), while the canonical ensemble requires the constraint to be
met only on average (`soft constraint'). It is known that breaking of ensemble
equivalence may occur when the size of the graph tends to infinity, signalled
by a non-zero specific relative entropy of the two ensembles. In this paper we
analyse a formula for the relative entropy of generic discrete random
structures recently conjectured by Squartini and Garlaschelli. We consider the
case of a random graph with a given degree sequence (configuration model), and
show that in the dense regime this formula correctly predicts that the specific
relative entropy is determined by the scaling of the determinant of the matrix
of canonical covariances of the constraints. The formula also correctly
predicts that an extra correction term is required in the sparse regime and in
the ultra-dense regime. We further show that the different expressions
correspond to the degrees in the canonical ensemble being asymptotically
Gaussian in the dense regime and asymptotically Poisson in the sparse regime
(the latter confirms what we found in earlier work), and the dual degrees in
the canonical ensemble being asymptotically Poisson in the ultra-dense regime.
In general, we show that the degrees follow a multivariate version of the
Poisson-Binomial distribution in the canonical ensemble.
