The Annals of Statistics

Estimating and understanding exponential random graph models

Sourav Chatterjee and Persi Diaconis

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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 Erdős–Rényi model. We also find classes of models where the limiting graphs differ from Erdős–Rényi graphs. A limitation of our approach, inherited from the limitation of graph limit theory, is that it works only for dense graphs.

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Ann. Statist., Volume 41, Number 5 (2013), 2428-2461.

First available in Project Euclid: 5 November 2013

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Zentralblatt MATH identifier

Primary: 62F10: Point estimation 05C80: Random graphs [See also 60B20]
Secondary: 62P25: Applications to social sciences 60F10: Large deviations

Random graph Erdős–Rényi graph limit exponential random graph models parameter estimation


Chatterjee, Sourav; Diaconis, Persi. Estimating and understanding exponential random graph models. Ann. Statist. 41 (2013), no. 5, 2428--2461. doi:10.1214/13-AOS1155.

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