Covariance Structure Behind Breaking of Ensemble Equivalence in Random Graphs

Diego Garlaschelli,Frank den Hollander,Andrea Roccaverde

doi:10.1007/s10955-018-2114-x

Diego Garlaschelli, Frank den Hollander

Open Access

https://doi.org/10.1007/s10955-018-2114-x

Copy DOI

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 non-zero 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 ultra-dense 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 ultra-dense regime. In general, we show that the degrees follow a multivariate version of the Poisson–Binomial distribution in the canonical ensemble.

Highlights

Introduction and Main Results1.1 Background and OutlineFor most real-world networks, a detailed knowledge of the architecture of the network is not available and one must work with a probabilistic description, where the network is assumed to be a random sample drawn from a set of allowed configurations that are consistent with a set of known topological constraints [7]
In the present paper we take a fresh look at breaking of ensemble equivalence by analysing a formula for the relative entropy, based on the covariance structure of the canonical ensemble, recently put forward by Squartini and Garlaschelli [6]
We consider the case of a random graph with a given degree sequence and show that this formula correctly predicts that the specific relative entropy is determined by the scaling of the determinant of the covariance matrix of the constraints in the dense regime, while it requires an extra correction term in the sparse regime and the ultra-dense regime

Summary

Background and Outline

For most real-world networks, a detailed knowledge of the architecture of the network is not available and one must work with a probabilistic description, where the network is assumed to be a random sample drawn from a set of allowed configurations that are consistent with a set of known topological constraints [7]. If the constraint is on the total numbers of edges and triangles, with values different from what is typical for the Erdos–Renyi random graph in the dense regime, the natural scale turns out to be αn = n2 [2] (in which case sα∞ is the specific relative entropy ‘per edge’). Such a severe breaking of ensemble equivalence comes from ‘frustration’ in the constraints. Except for the symbols Gn and Sn(Pmic | Pcan), we suppress the n-dependence from the notation

Constraint on the Degree Sequence

Relevant Regimes

Linking Ensemble Nonequivalence to the Canonical Covariances

Discussion and Outline

Poisson–Binomial Degrees in the General Case

Poisson Degrees in the Sparse Regime

Poisson Degrees in the Ultra-Dense Regime

Gaussian Degrees in the Dense Regime

Crossover Between the Regimes

Other Constraints

Outline

Preparatory Lemmas