Existence of maximum likelihood estimates in exponential random graph models

TL;DR

This paper provides a simplified proof of the fundamental condition for the existence of maximum likelihood estimates in exponential random graph models, linking the estimate's existence to the position of the target statistic within the convex hull.

Contribution

It offers a streamlined proof of the core theoretical result in ERGMs, clarifying the conditions for MLE existence.

Findings

MLE exists iff target statistic is in the relative interior of the convex hull

Provides a simplified, rigorous proof of a key ERGM property

Clarifies the geometric conditions for ERGM parameter estimation

Abstract

We present a streamlined proof of the foundational result in the theory of exponential random graph models (ERGMs) that the maximum likelihood estimate exists if and only if the target statistic lies in the relative interior of the convex hull of the set of realizable statistics. .