On large deviation properties of Erdos-Renyi random graphs

TL;DR

This paper connects large deviation properties of Erdős-Rényi graphs with the free energy of the Potts model, providing explicit calculations for rare graph configurations and validating results with simulations.

Contribution

It generalizes the link between random graph properties and Potts model free energy to large deviations, including explicit rare event calculations.

Findings

Explicit formulas for rare graph configurations

Good agreement between simulations and analytical results

Alternative derivation via graph evolution dynamics

Abstract

We show that large deviation properties of Erd\"os-R\'enyi random graphs can be derived from the free energy of the $q$-state Potts model of statistical mechanics. More precisely the Legendre transform of the Potts free energy with respect to $\ln q$ is related to the component generating function of the graph ensemble. This generalizes the well-known mapping between typical properties of random graphs and the $q\to 1$ limit of the Potts free energy. For exponentially rare graphs we explicitly calculate the number of components, the size of the giant component, the degree distributions inside and outside the giant component, and the distribution of small component sizes. We also perform numerical simulations which are in very good agreement with our analytical work. Finally we demonstrate how the same results can be derived by studying the evolution of random graphs under the insertion of new vertices and edges, without recourse to the thermodynamics of the Potts model.