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Monte Carlo goodness-of-fit tests for degree corrected and related stochastic blockmodels

Vishesh Karwa, Debdeep Pati, Sonja Petrović, Liam Solus, Nikita Alexeev, Mateja Raič, Dane Wilburne, Robert Williams, Bowei Yan

TL;DR

The general testing methodology developed here extends to any finite mixture of log-linear models on discrete data, and as such is the first application of the algebraic statistics machinery for latent-variable models.

Abstract

We construct Bayesian and frequentist finite-sample goodness-of-fit tests for three different variants of the stochastic blockmodel for network data. Since all of the stochastic blockmodel variants are log-linear in form when block assignments are known, the tests for the \emph{latent} block model versions combine a block membership estimator with the algebraic statistics machinery for testing goodness-of-fit in log-linear models. We describe Markov bases and marginal polytopes of the variants of the stochastic blockmodel, and discuss how both facilitate the development of goodness-of-fit tests and understanding of model behavior. The general testing methodology developed here extends to any finite mixture of log-linear models on discrete data, and as such is the first application of the algebraic statistics machinery for latent-variable models.

Monte Carlo goodness-of-fit tests for degree corrected and related stochastic blockmodels

TL;DR

The general testing methodology developed here extends to any finite mixture of log-linear models on discrete data, and as such is the first application of the algebraic statistics machinery for latent-variable models.

Abstract

We construct Bayesian and frequentist finite-sample goodness-of-fit tests for three different variants of the stochastic blockmodel for network data. Since all of the stochastic blockmodel variants are log-linear in form when block assignments are known, the tests for the \emph{latent} block model versions combine a block membership estimator with the algebraic statistics machinery for testing goodness-of-fit in log-linear models. We describe Markov bases and marginal polytopes of the variants of the stochastic blockmodel, and discuss how both facilitate the development of goodness-of-fit tests and understanding of model behavior. The general testing methodology developed here extends to any finite mixture of log-linear models on discrete data, and as such is the first application of the algebraic statistics machinery for latent-variable models.

Paper Structure

This paper contains 32 sections, 7 theorems, 35 equations, 10 figures, 7 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Outline of the paper.
  3. Three variants of the stochastic blockmodel
  4. A generic SBM with known and latent block assignments.
  5. The Erdös-Rényi-SBM
  6. The Additive SBM
  7. The $\beta$-SBM
  8. Testing goodness-of-fit of the SBMs
  9. A frequentist conditional goodness-of-fit test
  10. A Bayesian conditional goodness-of-fit test for latent blocks
  11. Choosing a goodness-of-fit statistic
  12. Estimating conditional $p$-values: algebraic statistics Monte Carlo
  13. Known block assignment.
  14. Latent block assignment.
  15. Algebra and Geometry of SBMs
  16. ...and 17 more sections

Key Result

Proposition 3.3

Consider an exponential family model as in Definition defn:fiber. Let $u$ denote a fixed value of the sufficient statistics. Then the conditional distribution on the fiber is

Figures (10)

  • Figure 1: Two graphs $g$ and $h$ in the same fiber of the ER-SBM with block structure $\mathbb{B}=\{B_1, B_2, B_3\}$ and a Markov move corresponding to the linear form $x_{23}-x_{15}\in\ker(\varphi_{\mathrm{ER}})$ that moves from $g$ to $h$. The blue, loosely dashed line indicates edge insertion and the red, densely dashed line indicates edge deletion.
  • Figure 2: Two graphs $g$ and $h$ in the same fiber of the additive SBM with block structure $\mathbb{B}=\{B_1,B_2,B_3\}$ and a quadratic Markov move corresponding to the binomial $x_{26}x_{45}-x_{24}x_{56}\in\ker(\varphi_{\mathrm{Add}})$ that moves from $g$ to $h$. The blue, loosely dashed lines indicate edge insertion and the red, densely dashed lines indicate edge deletion.
  • Figure 3: Two graphs $g$ and $h$ in the same fiber of the $\beta$-SBM with block structure $\mathbb{B}=\{B_1,B_2,B_3\}$ and a cubic Markov move corresponding to the binomial $x_{23}x_{45}x_{46}-x_{24}x_{34}x_{56}\in\ker(\varphi_\beta)$ that moves from $g$ to $h$. The blue, loosely dashed lines indicate edge insertion and the red, densely dashed lines indicate edge deletion.
  • Figure 4: Histograms of the GoF statistic for testing the fit of ER-SBM with $2$ blocks. Data generated from a $2$-block additive SBM.
  • Figure 5: Histograms of the GoF statistic for testing the fit of a $2$-block ER-SBM. Data generated from a $2$-block $\beta$-SBM.
  • ...and 5 more figures

Theorems & Definitions (23)

  • Definition 2.1: ER-SBM
  • Definition 2.2: Additive SBM
  • Remark 2.3
  • Definition 2.4: $\beta$-SBM
  • Remark 2.5
  • Definition 3.1: A fiber for a discrete exponential family
  • Remark 3.2
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • ...and 13 more