On large deviation properties of Erdos-Renyi random graphs

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.