Maximum lilkelihood estimation in the $\beta$-model

TL;DR

This paper investigates the conditions for the existence of maximum likelihood estimators in the $eta$-model for undirected random graphs, providing combinatorial characterizations and asymptotic guarantees.

Contribution

It derives necessary and sufficient conditions for MLE existence in the $eta$-model using polytope analysis and characterizes cases of nonestimability.

Findings

Characterization of MLE existence via degree sequence polytopes

Conditions for MLE existence with increasing nodes

Identification of sample points leading to nonestimability

Abstract

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.