Maximum likelihood degree of the $\beta$-stochastic blockmodel

TL;DR

This paper derives a closed-form formula for the maximum likelihood degree of the $eta$-stochastic blockmodel, revealing it factors into a product of Eulerian numbers, thus advancing understanding of the model's algebraic complexity.

Contribution

It provides the first explicit formula for the maximum likelihood degree of the $eta$-stochastic blockmodel, connecting algebraic properties with combinatorial numbers.

Findings

Maximum likelihood degree factors into Eulerian numbers.

Closed-form formula for likelihood equations solutions.

Enhances understanding of algebraic complexity of the model.

Abstract

Log-linear exponential random graph models are a specific class of statistical network models that have a log-linear representation. This class includes many stochastic blockmodel variants. In this paper, we focus on $\beta$-stochastic blockmodels, which combine the $\beta$-model with a stochastic blockmodel. Here, using recent results by Almendra-Hern\'{a}ndez, De Loera, and Petrovi\'{c}, which describe a Markov basis for $\beta$-stochastic block model, we give a closed form formula for the maximum likelihood degree of a $\beta$-stochastic blockmodel. The maximum likelihood degree is the number of complex solutions to the likelihood equations. In the case of the $\beta$-stochastic blockmodel, the maximum likelihood degree factors into a product of Eulerian numbers.