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Non-asymptotic goodness-of-fit tests and model selection in valued stochastic blockmodels

Félix Almendra-Hernández, Miles Bakenhus, Vishesh Karwa, Mitsunori Ogawa, Sonja Petrović

TL;DR

This work extends finite-sample goodness-of-fit testing to valued stochastic blockmodels, unifying Bernoulli, Poisson, and labeled edge-type SBMs within an exponential-family framework. It derives explicit Markov-basis moves to sample from conditional reference distributions on dyad-sufficient statistic fibers, enabling exact conditional GoF tests and enabling both plug-in and partial Bayes approaches for settings with unknown block structure. The authors introduce GoF statistics tailored to Poisson and labeled SBMs, prove consistency of MLEs under varying known/unknown block assignments, and study the asymptotic power under block-merging to support a minimal-k model selection rule. Empirical results on simulations and two host-parasite networks show good Type I error control, substantial power when appropriate blocks are specified, and block-number recommendations that differ from prior literature, underscoring practical impact for network modeling with non-Bernoulli dyads.

Abstract

A valued stochastic blockmodel (SBM) is a general way to view networked data in which nodes are grouped into blocks and links between them are measured by counts or labels. This family allows for varying dyad sampling schemes, thereby including the classical, Poisson, and labeled SBMs, as well as those in which some edge observations are censored. This paper addresses the question of testing goodness-of-fit of such non-Bernoulli SBMs, focusing in particular on finite-sample tests. We derive explicit Markov bases moves necessary to generate samples from reference distributions and define goodness-of-fit statistics for determining model fit, comparable to those in the literature for related model families. For the labeled SBM, which includes in particular the censored-edge model, we study the asymptotic behavior of said statistics. One of the main purposes of testing goodness-of-fit of an SBM is to determine whether block membership of the nodes influences network formation. Power and Type 1 error rates are verified on simulated data. Additionally, we discuss the use of asymptotic results in selecting the number of blocks under the latent-block modeling assumption. The method derived for Poisson SBM is applied to ecological networks of host-parasite interactions. Our data analysis conclusions differ in selecting the number of blocks for the species from previous results in the literature.

Non-asymptotic goodness-of-fit tests and model selection in valued stochastic blockmodels

TL;DR

This work extends finite-sample goodness-of-fit testing to valued stochastic blockmodels, unifying Bernoulli, Poisson, and labeled edge-type SBMs within an exponential-family framework. It derives explicit Markov-basis moves to sample from conditional reference distributions on dyad-sufficient statistic fibers, enabling exact conditional GoF tests and enabling both plug-in and partial Bayes approaches for settings with unknown block structure. The authors introduce GoF statistics tailored to Poisson and labeled SBMs, prove consistency of MLEs under varying known/unknown block assignments, and study the asymptotic power under block-merging to support a minimal-k model selection rule. Empirical results on simulations and two host-parasite networks show good Type I error control, substantial power when appropriate blocks are specified, and block-number recommendations that differ from prior literature, underscoring practical impact for network modeling with non-Bernoulli dyads.

Abstract

A valued stochastic blockmodel (SBM) is a general way to view networked data in which nodes are grouped into blocks and links between them are measured by counts or labels. This family allows for varying dyad sampling schemes, thereby including the classical, Poisson, and labeled SBMs, as well as those in which some edge observations are censored. This paper addresses the question of testing goodness-of-fit of such non-Bernoulli SBMs, focusing in particular on finite-sample tests. We derive explicit Markov bases moves necessary to generate samples from reference distributions and define goodness-of-fit statistics for determining model fit, comparable to those in the literature for related model families. For the labeled SBM, which includes in particular the censored-edge model, we study the asymptotic behavior of said statistics. One of the main purposes of testing goodness-of-fit of an SBM is to determine whether block membership of the nodes influences network formation. Power and Type 1 error rates are verified on simulated data. Additionally, we discuss the use of asymptotic results in selecting the number of blocks under the latent-block modeling assumption. The method derived for Poisson SBM is applied to ecological networks of host-parasite interactions. Our data analysis conclusions differ in selecting the number of blocks for the species from previous results in the literature.
Paper Structure (19 sections, 10 theorems, 72 equations, 1 figure, 4 tables, 2 algorithms)

This paper contains 19 sections, 10 theorems, 72 equations, 1 figure, 4 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Blockmodels for valued networks
  3. Classical SBM
  4. Poisson SBM
  5. Labeled SBM
  6. Testing goodness-of-fit of valued SBMs
  7. Conditional goodness-of-fit test under fixed block assignment
  8. Plug-in conditional test with estimated block assignment
  9. Goodness-of-fit test for unknown block valued SBM: partial Bayes
  10. A goodness-of-fit statistic
  11. Poisson SBM.
  12. Labeled SBM.
  13. Consistency of the MLE for valued SBM with unknown $z$
  14. Asymptotic power of the Gof statistic under a merged block assignment
  15. Simulations
  16. ...and 4 more sections

Key Result

Lemma 3.2

Let $g\in \mathbb G$ denote a graph, and let $\mathbb P_\theta(G=g)$ be defined as in eqn:exponential_family, defining an exponential family model on $g$ where $h(g)$ is the base measure, $\theta$ is a vector of parameters, $T_z(g)$ is the vector of sufficient statistics. Then where $\mathcal{F}_{z,t}$, defined in eq:fiber, is the set of graphs $g\in\mathbb G$ whose sufficient statistics are equa

Figures (1)

  • Figure 1: Since block assignments partition $[n]$, the rows and columns of the adjacency matrix $G$ for a graph may be grouped according to this block assignment. Assuming this layout, the figure shows how the merge operation on blocks $B_{k-1}$ and $B_{k}$, relabels nodes in the adjacency matrix to be in block $\tilde{B}_{k-1} = B_{k-1} \cup B_k$.

Theorems & Definitions (34)

  • Definition 2.1: Valued SBM
  • Remark 2.2
  • Example 2.3: Modeling censored network data
  • Definition 3.1
  • Lemma 3.2: Cf. Lemma 1 in newDirections2024
  • Definition 3.3: Markov basis
  • Proposition 3.4
  • proof : Proof of \ref{['prop:Markov_Basis_poissonSBM']}
  • Proposition 3.5
  • proof : Proof of \ref{['prop:Markov_Basis_labeledSBM']}
  • ...and 24 more