Maximum Likelihood Estimation in Network Models

Details Maximum Likelihood Estimation in Network Models Maximum Likelihood Estimation in Network Models

2011-05-31

arxiv.org/abs/1105.6145v2

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We study maximum likelihood estimation for the statistical model for both directed and undirected random graph models in which the degree sequences are minimal sufficient statistics. In the undirected case, the model is known as the beta model. We derive necessary and sufficient conditions for the existence of the MLE that are based on the polytope of degree sequences. We characterize in a combinatorial fashion sample points leading to a nonexistent MLE, and non-estimability 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 nodes increases. We illustrate our approach on other random graph models for networks, such as the Rasch model, the Bradley-Terry model and the more general p1 model of Holland and Leinhardt (1981).

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