Ensemble nonequivalence in random graphs with modular structure

TL;DR

This paper investigates when the microcanonical and canonical ensembles of random graphs with modular structures are equivalent, revealing that non-equivalence occurs if the number of constrained degrees grows extensively with the number of nodes.

Contribution

It generalizes previous results to graphs with multiple layers, providing a full classification of ensemble equivalence based on the extensiveness of local constraints.

Findings

Ensemble non-equivalence occurs if the number of constrained degrees is extensive.

Derived a formula for the specific relative entropy in modular graphs.

Provided an interpretation of the entropy formula via Poissonisation of degrees.

Abstract

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 Squartini et al. (2015) it was shown that ensembles of random graphs are non-equivalent 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, block-model or community structure. We give a full classification of ensemble equivalence, proving that breakdown occurs if and only if 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 a formula for the specific relative entropy and provide an interpretation of this formula in terms of Poissonisation of the degrees.