On the Geometry of Discrete Exponential Families with Application to Exponential Random Graph Models

TL;DR

This paper explores the geometric structure of discrete exponential families, especially exponential random graph models, to better understand their properties and address issues like degeneracy and maximum likelihood estimation.

Contribution

It characterizes the geometry of discrete exponential families using the normal fan of the convex support and applies these insights to analyze ERG models and their degeneracy.

Findings

Normal fan of the convex support is key to understanding exponential family properties

Geometric analysis helps explain ERG model degeneracy

Provides a framework for improved maximum likelihood estimation in ERG models

Abstract

There has been an explosion of interest in statistical models for analyzing network data, and considerable interest in the class of exponential random graph (ERG) models, especially in connection with difficulties in computing maximum likelihood estimates. The issues associated with these difficulties relate to the broader structure of discrete exponential families. This paper re-examines the issues in two parts. First we consider the closure of $k$-dimensional exponential families of distribution with discrete base measure and polyhedral convex support $\mathrm{P}$. We show that the normal fan of $\mathrm{P}$ is a geometric object that plays a fundamental role in deriving the statistical and geometric properties of the corresponding extended exponential families. We discuss its relevance to maximum likelihood estimation, both from a theoretical and computational standpoint. Second, we apply our results to the analysis of ERG models. In particular, by means of a detailed example, we provide some characterization of the properties of ERG models, and, in particular, of certain behaviors of ERG models known as degeneracy.