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arXiv · 2410.06223

Maximum likelihood degree of the $β$-stochastic blockmodel

Cashous Bortner·Jennifer Garbett·Elizabeth Gross·Christopher McClain·Naomi Krawzik·Derek Young
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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 $β$-stochastic blockmodels, which combine the $β$-model with a stochastic blockmodel. Here, using recent results by Almendra-Hernández, De Loera, and Petrović, which describe a Markov basis for $β$-stochastic block model, we give a closed form formula for the maximum likelihood degree of a $β$-stochastic blockmodel. The maximum likelihood degree is the number of complex solutions to the likelihood equations. In the case of the $β$-stochastic blockmodel, the maximum likelihood degree factors into a product of Eulerian numbers.

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Cashous Bortner, Jennifer Garbett, Elizabeth Gross, Christopher McClain, Naomi Krawzik, Derek Young. 2024-10-08. Maximum likelihood degree of the $β$-stochastic blockmodel. https://doi.org/10.2140/astat.2025.16.77

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