Exponential Random Graphs: mean-field approximations and limit theorems

This thesis deals with a class of Random Graphs (RGs), the Exponential Random Graph model (ERGM). RGs can reproduce emergent connectivity properties of real world networks such as social networks, populations, molecular systems, neural networks. The ERGM generalizes the Erdős–Rényi model (ERM), where every edge is drawn independently to the others with fixed probability, by introducing interactions between edges. These arise from Gibbs probability densities, which can satisfy constraints on the average of an arbitrary number of macroscopical statistics, without adding useless biases. For the ERGM, a specific ensemble is built by fixing some simple subgraphs and selecting their associated homormorphism densities as defining statistics. The main goal of this thesis is the extension of some asymptotic results derived for the Edge-Triangle model, where only edges and triangles are taken into account, to the general setting, where the model depends on a higher number of simple subgraphs. We are interested in the asymptotics of the edge density. We investigate the same asymptotic properties in a mean-field approximation of the model, which is inspired to the mean-field approximation of the Ising model. The work begins with an introduction on RGs, Large Deviations bounds and the ERM. We then present Interacting Particle Systems and we describe the ERGM, emphasizing its relations with Statistical Physics. We study the asymptotic properties of the edge density in the original model and in the mean-field approximation. We conclude presenting some numerical simulations of both models.