Estimating and Understanding Exponential Random Graph Models

Details Estimating and Understanding Exponential Random Graph Models Estimating and Understanding Exponential Random Graph Models

2011-02-14

arxiv.org/abs/1102.2650v3

Mathematics

Probability

57 pages, 8 figures. Corrections to the proof of Theorem 5.1

Scientific paper

We introduce a new method for estimating the parameters 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. 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 showing that for many models, the extra sufficient statistics are useless: most realizations look like the results of a simple Erdos-Renyi model. We also find classes of models where the limiting graphs differ from Erdos-Renyi graphs and begin to make the link to models where the natural parameters alternate in sign.

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