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Maximum likelihood degree of the $β$-stochastic blockmodel

Cashous Bortner, Jennifer Garbett, Elizabeth Gross, Christopher McClain, Naomi Krawzik, Derek Young

TL;DR

This paper computes the maximum likelihood degree (MLdeg) for β-stochastic blockmodels (β-SBMs), a toric log-linear ERGM that combines the β-model with stochastic block structure. Leveraging a quadratic Markov-basis result for β-SBMs, it derives a closed-form multiplicative formula for MLdeg, showing it factors as a product of Eulerian-number terms corresponding to blocks of size greater than two. Specifically, MLdeg$(n_1,\

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.

Maximum likelihood degree of the $β$-stochastic blockmodel

TL;DR

This paper computes the maximum likelihood degree (MLdeg) for β-stochastic blockmodels (β-SBMs), a toric log-linear ERGM that combines the β-model with stochastic block structure. Leveraging a quadratic Markov-basis result for β-SBMs, it derives a closed-form multiplicative formula for MLdeg, showing it factors as a product of Eulerian-number terms corresponding to blocks of size greater than two. Specifically, MLdeg$(n_1,\

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 -stochastic blockmodels, which combine the -model with a stochastic blockmodel. Here, using recent results by Almendra-Hernández, De Loera, and Petrović, which describe a Markov basis for -stochastic block model, we give a closed form formula for the maximum likelihood degree of a -stochastic blockmodel. The maximum likelihood degree is the number of complex solutions to the likelihood equations. In the case of the -stochastic blockmodel, the maximum likelihood degree factors into a product of Eulerian numbers.
Paper Structure (10 sections, 9 theorems, 38 equations, 2 figures, 1 table)

This paper contains 10 sections, 9 theorems, 38 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background
  3. The beta-Stochastic Blockmodel
  4. The Design Matrix and Likelihood Equations
  5. The Main Theorem
  6. Proof of the Main Theorem
  7. Proof of the Factoring Lemma
  8. Constructing new models by contracting blocks
  9. Proof of Factoring Lemma
  10. Acknowledgements

Key Result

Lemma 2.4

Let $k$ be a positive integer and $\tau:[k]\rightarrow[k]$ be a permutation of $[k]$. If $M_1=\mathcal{M}(n_1,n_2,\ldots,n_k)$ and $M_2=\mathcal{M}(n_{\tau(1)},n_{\tau(2)},\ldots,n_{\tau(k)})$ are $\beta$-SBMs then $\mathop{\mathrm{MLdeg}}\nolimits(M_1)=\mathop{\mathrm{MLdeg}}\nolimits(M_2)$.

Figures (2)

  • Figure 1: A graph with three blocks of sizes $3,4,$ and $3$.
  • Figure 2: The blocks of $M_1$ and $M_2$. This figure exhibits the blocks of $M$ together with the newly defined block $V_{*}$ while also indicating which blocks are members of $M_1$ and which blocks are members of $M_2$.

Theorems & Definitions (27)

  • Example 2.1
  • Example 2.2
  • Example 2.3
  • Lemma 2.4
  • proof
  • Theorem 3.1: The Maximum Likelihood Degree of $\beta$-SBMs
  • Example 3.2
  • Remark 3.3
  • Corollary 3.4
  • Corollary 3.5
  • ...and 17 more