Statistics of the two-star ERGM

Abstract

In this paper, we explore the two-star Exponential Random Graph Model, which is a two parameter exponential family on the space of simple labeled graphs. We introduce auxiliary variables to express the two-star model as a mixture of the $\beta$ model on networks. Using this representation, we study asymptotic distribution of the number of edges, and the sampling variance of the degrees. In particular, the limiting distribution for the number of edges has similar phase transition behavior to that of the magnetization in the Curie-Weiss Ising model of Statistical Physics. Using this, we show existence of consistent estimates for both parameters in all parameter domains. Finally, we prove that the centered partial sum of degrees converges as a process to a Brownian bridge in all parameter domains, irrespective of the phase transition.

Significant changes. New author added. All main theorems now cover critical point. Theorem 1.4 new. This article also supersedes arxiv article 1310.4164 by the first author

References in corpus (2)

• Solution for the properties of a clustered network
• Phase transition in the two star Exponential Random Graph Model

Cited by in corpus (4)

• Concentration and consistency results for canonical and curved exponential-family models of random graphs
• Optimal Single Sample Tests for Structured versus Unstructured Network Data
• Efficient Sampling for Ising and Potts Models using Auxiliary Gaussian Variables
• The GHS and other correlation inequalities for the two-star model