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Ensemble nonequivalence in random graphs with modular structure. (English) Zbl 1358.82014

Summary: Breaking of equivalence between the microcanonical ensemble and the canonical ensemble, describing a large system subject to hard and soft constraints, respectively, was recently shown to occur in large random graphs. Hard constraints must be met by every graph, soft constraints must be met only on average, subject to maximal entropy. In [T. Squartini et al., “Breaking of ensemble equivalence in networks”, Phys. Rev. Lett. 115, No. 26, Article ID 268701, 5 p. (2015; doi:10.1103/PhysRevLett.115.268701)] it was shown that ensembles of random graphs are nonequivalent when the degrees of the nodes are constrained, in the sense of a non-zero limiting specific relative entropy as the number of nodes diverges. In that paper, the nodes were placed either on a single layer (uni-partite graphs) or on two layers (bi-partite graphs). In the present paper we consider an arbitrary number of intra-connected and inter-connected layers, thus allowing for modular graphs with a multi-partite, multiplex, time-varying, block-model or community structure. We give a full classification of ensemble equivalence in the sparse regime, proving that breakdown occurs as soon as the number of local constraints (i.e. the number of constrained degrees) is extensive in the number of nodes, irrespective of the layer structure. In addition, we derive an explicit formula for the specific relative entropy and provide an interpretation of this formula in terms of Poissonisation of the degrees.

MSC:

82B30 Statistical thermodynamics
82B31 Stochastic methods applied to problems in equilibrium statistical mechanics
05C80 Random graphs (graph-theoretic aspects)
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