# Covariance structure behind breaking of ensemble equivalence in random graphs

Garlaschelli, Diego and den Hollander, Frank and Roccaverde, Andrea Covariance structure behind breaking of ensemble equivalence in random graphs. Working Paper arXiv (Submitted)

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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&#39;), while the canonical ensemble requires the constraint to be met only on average (soft constraint&#39;). It is known that breaking of ensemble equivalence may occur when the size of the random 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 random discrete structures recently put forward by Squartini and Garlaschelli. We consider the case of random graphs with given degree sequence (configuration model) and show that this formula correctly predicts that the specific relative entropy in the dense regime is determined by 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 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 and the ultra-dense regime, as we found in earlier work. In general, we show that the degrees follow a multivariate version of the Poisson- Binomial distribution in the canonical ensemble.