Estimating and understanding exponential random graph models

TL;DR

This paper introduces a theoretical analysis method for exponential random graph models using large-deviations approximation, explaining degeneracy issues and showing that many models resemble Erdős-Rényi graphs, with limitations to dense graphs.

Contribution

It provides the first rigorous proofs of degeneracy in exponential random graph models and offers a large-deviations based analysis to understand model behavior.

Findings

Most graphs are either almost empty or complete.

Many models are statistically indistinguishable from Erdős-Rényi graphs.

The approach is limited to dense graphs.

Abstract

We introduce a method for the theoretical analysis of exponential random graph models. The method is based on a large-deviations approximation to the normalizing constant shown to be consistent using theory developed by Chatterjee and Varadhan [European J. Combin. 32 (2011) 1000-1017]. The theory explains a host of difficulties encountered by applied workers: many distinct models have essentially the same MLE, rendering the problems ``practically'' ill-posed. We give the first rigorous proofs of ``degeneracy'' observed in these models. Here, almost all graphs have essentially no edges or are essentially complete. We supplement recent work of Bhamidi, Bresler and Sly [2008 IEEE 49th Annual IEEE Symposium on Foundations of Computer Science (FOCS) (2008) 803-812 IEEE] showing that for many models, the extra sufficient statistics are useless: most realizations look like the results of a simple Erd\H{o}s-R\'{e}nyi model. We also find classes of models where the limiting graphs differ from Erd\H{o}s-R\'{e}nyi graphs. A limitation of our approach, inherited from the limitation of graph limit theory, is that it works only for dense graphs.