Covariance Structure Behind Breaking of Ensemble Equivalence in Random Graphs
Abstract
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 nonzero specific relative entropy of the two ensembles. In this paper we analyse a formula for the relative entropy of generic discrete random structures recently put forward 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 ultradense 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 ultradense regime. In general, we show that the degrees follow a multivariate version of the PoissonBinomial distribution in the canonical ensemble.
 Publication:

Journal of Statistical Physics
 Pub Date:
 November 2018
 DOI:
 10.1007/s109550182114x
 arXiv:
 arXiv:1711.04273
 Bibcode:
 2018JSP...173..644G
 Keywords:

 Mathematical Physics;
 Mathematics  Probability
 EPrint:
 Journal of Statistical Physics, (), 119