In statistical physics the mathematical description of interacting particle systems is based on what are called Gibbs ensembles. These represent a priori choices for the probability distribution on the configuration space of the system capturing physically relevant situations. One of the corner stones of statistical physics is the assumption that, in the thermodynamic limit when the system becomes very large, the microcanonical ensemble based on energy coincides with the canonical ensemble based on temperature. However, various examples of systems have been found for which these two ensembles are not equivalent. A complete theory of this intriguing phenomenon is still missing.
We show that breaking of ensemble equivalence can manifest itself also in random graphs with topological constraints. In particular, we show that if we consider a random graph in which the degrees of the nodes are fixed (= hard constraint), respectively, the degrees of the nodes are fixed on average (= soft constraint), then there is no equivalence in the limit as the graph becomes very large. We also show that the same occurs when we fix the total number of edges and triangles, in such a way that there is frustration. These facts have important consequences for how real-world networks must be modelled.