Open Access
November 2020 Exponential-Family Models of Random Graphs: Inference in Finite, Super and Infinite Population Scenarios
Michael Schweinberger, Pavel N. Krivitsky, Carter T. Butts, Jonathan R. Stewart
Statist. Sci. 35(4): 627-662 (November 2020). DOI: 10.1214/19-STS743


Exponential-family Random Graph Models (ERGMs) constitute a large statistical framework for modeling dense and sparse random graphs with short- or long-tailed degree distributions, covariate effects and a wide range of complex dependencies. Special cases of ERGMs include network equivalents of generalized linear models (GLMs), Bernoulli random graphs, $\beta $-models, $p_{1}$-models and models related to Markov random fields in spatial statistics and image processing. While ERGMs are widely used in practice, questions have been raised about their theoretical properties. These include concerns that some ERGMs are near-degenerate and that many ERGMs are non-projective. To address such questions, careful attention must be paid to model specifications and their underlying assumptions, and to the inferential settings in which models are employed. As we discuss, near-degeneracy can affect simplistic ERGMs lacking structure, but well-posed ERGMs with additional structure can be well-behaved. Likewise, lack of projectivity can affect non-likelihood-based inference, but likelihood-based inference does not require projectivity. Here, we review well-posed ERGMs along with likelihood-based inference. We first clarify the core statistical notions of “sample” and “population” in the ERGM framework, separating the process that generates the population graph from the observation process. We then review likelihood-based inference in finite, super and infinite population scenarios. We conclude with consistency results, and an application to human brain networks.


Download Citation

Michael Schweinberger. Pavel N. Krivitsky. Carter T. Butts. Jonathan R. Stewart. "Exponential-Family Models of Random Graphs: Inference in Finite, Super and Infinite Population Scenarios." Statist. Sci. 35 (4) 627 - 662, November 2020.


Published: November 2020
First available in Project Euclid: 17 November 2020

MathSciNet: MR4175389
Digital Object Identifier: 10.1214/19-STS743

Keywords: ERGM , exponential-family random graph model , model degeneracy , projectivity , Social network

Rights: Copyright © 2020 Institute of Mathematical Statistics

Vol.35 • No. 4 • November 2020
Back to Top