Maximum likelihood degree of the ${\hat{I}}^2$-stochastic blockmodel

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 ${\hat{I}}^2$-stochastic blockmodels, which combine the ${\hat{I}}^2$-model with a stochastic blockmodel. Here, using recent results by Almendra-Hernández, De Loera, and Petrović, which describe a Markov basis for ${\hat{I}}^2$-stochastic block model, we give a closed form formula for the maximum likelihood degree of a ${\hat{I}}^2$-stochastic blockmodel. The maximum likelihood degree is the number of complex solutions to the likelihood equations. In the case of the ${\hat{I}}^2$-stochastic blockmodel, the maximum likelihood degree factors into a product of Eulerian numbers.