**Abstract: **

In statistical physics, it is often assumed that the canonical and microcanonical ensembles are asymptotically equivalent. This assumption does not always hold, for example for certain random network models. This phenomenon is called breaking of ensemble equivalence (BEE). In the literature, it was found that BEE is the rule rather than the exception for two classes of network models: sparse random graphs where the constraints are on the degree of every vertex, and dense random graphs where the constraints are on the densities of two or more subgraphs that are frustrated.

In this talk, we show that BEE is possible even if there is only one constraint on the density of a single subgraph. It turns out that BEE occurs only in a certain range of choices for the density and the number of edges of the subgraph, which we refer to as the BEE-phase. We discuss several properties of the phase diagram. We also analyse the behaviour of the two ensembles in the BEE-phase in an effort to better understand what causes BEE.

This talk is based on joint work with Frank den Hollander.

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