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February 2023 Statistics of the two star ERGM
Sumit Mukherjee, Yuanzhe Xu
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Bernoulli 29(1): 24-51 (February 2023). DOI: 10.3150/21-BEJ1448


In this paper, we explore the two star Exponential Random Graph Model, which is a two parameter exponential family on the space of simple labeled graphs. We introduce auxiliary variables to express the two star model as a mixture of the β model on networks. Using this representation, we study asymptotic distribution of the number of edges, and the sampling variance of the degrees. In particular, the limiting distribution for the number of edges has similar phase transition behavior to that of the magnetization in the Curie-Weiss Ising model of Statistical Physics. Using this, we show existence of consistent estimates for both parameters. Finally, we prove that the centered partial sum of degrees converges as a process to a Brownian bridge in all parameter domains, irrespective of the phase transition.

Funding Statement

The first author was partially supported by NSF Grants DMS-1712037 and DMS-2113414.


We thank the associate editor and two anonymous referees, whose comments and suggestions have greatly improved the quality of presentation in the paper.


Download Citation

Sumit Mukherjee. Yuanzhe Xu. "Statistics of the two star ERGM." Bernoulli 29 (1) 24 - 51, February 2023.


Received: 1 February 2021; Published: February 2023
First available in Project Euclid: 13 October 2022

Digital Object Identifier: 10.3150/21-BEJ1448

Keywords: auxiliary variables , consistent estimation , ERGM , phase transition , two-star


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Vol.29 • No. 1 • February 2023
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