Asymptotics for Sparse Exponential Random Graph Models

TL;DR

This paper investigates the asymptotic behavior of sparse exponential random graph models, providing precise estimates for key probabilistic measures and highlighting differences from dense models.

Contribution

It offers new asymptotic results for sparse ERGMs, including exact estimates for mean, variance, and the log partition function, extending understanding beyond dense models.

Findings

Exact estimates for mean and variance of the limiting distribution

Sharp contrast with dense ERGM asymptotics

Analysis of directed sparse ERGMs with multiple outward stars

Abstract

We study the asymptotics for sparse exponential random graph models where the parameters may depend on the number of vertices of the graph. We obtain exact estimates for the mean and variance of the limiting probability distribution and the limiting log partition function of the edge-(single)-star model. They are in sharp contrast to the corresponding asymptotics in dense exponential random graph models. Similar analysis is done for directed sparse exponential random graph models parametrized by edges and multiple outward stars.