The Annals of Statistics

Maximum lilkelihood estimation in the $\beta$-model

Alessandro Rinaldo, Sonja Petrović, and Stephen E. Fienberg

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We study maximum likelihood estimation for the statistical model for undirected random graphs, known as the $\beta$-model, in which the degree sequences are minimal sufficient statistics. We derive necessary and sufficient conditions, based on the polytope of degree sequences, for the existence of the maximum likelihood estimator (MLE) of the model parameters. We characterize in a combinatorial fashion sample points leading to a nonexistent MLE, and nonestimability of the probability parameters under a nonexistent MLE. We formulate conditions that guarantee that the MLE exists with probability tending to one as the number of nodes increases.

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Ann. Statist., Volume 41, Number 3 (2013), 1085-1110.

First available in Project Euclid: 13 June 2013

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Primary: 62F99: None of the above, but in this section

$\beta$-model polytope of degree sequences random graphs maximum likelihood estimator


Rinaldo, Alessandro; Petrović, Sonja; Fienberg, Stephen E. Maximum lilkelihood estimation in the $\beta$-model. Ann. Statist. 41 (2013), no. 3, 1085--1110. doi:10.1214/12-AOS1078.

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Supplemental materials

  • Supplementary material: Supplement to “Maximum lilkelihood estimation in the $\beta$-model”. In the supplementary material we extend our analysis to other models for network data: the Rasch model, the $\beta$-model with no sampling constraints on the number of observed edges per dyad, the Bradley–Terry model and the $p_{1}$ model of Holland and Leinhardt (1981). We also provide details on how to determine whether a given degree sequence belongs to the interior of the polytope of degree sequences $P_{n}$ and on how to compute the facial set corresponding to a degree sequence on the boundary of $P_{n}$.