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Degree Heterogeneity in Higher-Order Networks: Inference in the Hypergraph $\boldsymbolβ$-Model

Sagnik Nandy, Bhaswar B. Bhattacharya

TL;DR

The paper develops a rigorous inference framework for the hypergraph β-model with multiple layers, generalizing the graph β-model to higher-order interactions. It obtains precise ML convergence rates in both $L_2$ and $L_\infty$ norms, proves central limit theorems for finite-dimensional ML estimates, and constructs asymptotically valid confidence sets. It also analyzes goodness-of-fit via likelihood-ratio tests, deriving the null distribution and minimax detection thresholds in both $L_2$ and $L_\infty$ settings, with tests shown to be optimal up to logarithmic factors. Numerical experiments corroborate the theory and illustrate practical applicability to multi-way networks.

Abstract

The $\boldsymbolβ$-model for random graphs is commonly used for representing pairwise interactions in a network with degree heterogeneity. Going beyond pairwise interactions, Stasi et al. (2014) introduced the hypergraph $\boldsymbolβ$-model for capturing degree heterogeneity in networks with higher-order (multi-way) interactions. In this paper we initiate the rigorous study of the hypergraph $\boldsymbolβ$-model with multiple layers, which allows for hyperedges of different sizes across the layers. To begin with, we derive the rates of convergence of the maximum likelihood (ML) estimate and establish their minimax rate optimality. We also derive the limiting distribution of the ML estimate and construct asymptotically valid confidence intervals for the model parameters. Next, we consider the goodness-of-fit problem in the hypergraph $\boldsymbolβ$-model. Specifically, we establish the asymptotic normality of the likelihood ratio (LR) test under the null hypothesis, derive its detection threshold, and also its limiting power at the threshold. Interestingly, the detection threshold of the LR test turns out to be minimax optimal, that is, all tests are asymptotically powerless below this threshold. The theoretical results are further validated in numerical experiments. In addition to developing the theoretical framework for estimation and inference for hypergraph $\boldsymbolβ$-models, the above results fill a number of gaps in the graph $\boldsymbolβ$-model literature, such as the minimax optimality of the ML estimates and the non-null properties of the LR test, which, to the best of our knowledge, have not been studied before.

Degree Heterogeneity in Higher-Order Networks: Inference in the Hypergraph $\boldsymbolβ$-Model

TL;DR

The paper develops a rigorous inference framework for the hypergraph β-model with multiple layers, generalizing the graph β-model to higher-order interactions. It obtains precise ML convergence rates in both and norms, proves central limit theorems for finite-dimensional ML estimates, and constructs asymptotically valid confidence sets. It also analyzes goodness-of-fit via likelihood-ratio tests, deriving the null distribution and minimax detection thresholds in both and settings, with tests shown to be optimal up to logarithmic factors. Numerical experiments corroborate the theory and illustrate practical applicability to multi-way networks.

Abstract

The -model for random graphs is commonly used for representing pairwise interactions in a network with degree heterogeneity. Going beyond pairwise interactions, Stasi et al. (2014) introduced the hypergraph -model for capturing degree heterogeneity in networks with higher-order (multi-way) interactions. In this paper we initiate the rigorous study of the hypergraph -model with multiple layers, which allows for hyperedges of different sizes across the layers. To begin with, we derive the rates of convergence of the maximum likelihood (ML) estimate and establish their minimax rate optimality. We also derive the limiting distribution of the ML estimate and construct asymptotically valid confidence intervals for the model parameters. Next, we consider the goodness-of-fit problem in the hypergraph -model. Specifically, we establish the asymptotic normality of the likelihood ratio (LR) test under the null hypothesis, derive its detection threshold, and also its limiting power at the threshold. Interestingly, the detection threshold of the LR test turns out to be minimax optimal, that is, all tests are asymptotically powerless below this threshold. The theoretical results are further validated in numerical experiments. In addition to developing the theoretical framework for estimation and inference for hypergraph -models, the above results fill a number of gaps in the graph -model literature, such as the minimax optimality of the ML estimates and the non-null properties of the LR test, which, to the best of our knowledge, have not been studied before.
Paper Structure (37 sections, 24 theorems, 249 equations, 1 figure)

This paper contains 37 sections, 24 theorems, 249 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Summary of Results
  3. Related Work on the Graph $\bm \beta$-Model
  4. Asymptotic Notation
  5. Maximum Likelihood Estimation in Hypergraph $\bm \beta$-Models
  6. Rates of Convergence
  7. Minimax Rates
  8. Central Limit Theorems and Confidence Intervals
  9. Goodness-of-Fit: Asymptotics of the Likelihood Ratio Test and Minimax Detection Rates
  10. Asymptotics of the Likelihood Ratio Test
  11. Minimax Detection Rate in the $L_2$ Norm
  12. Testing in the $L_\infty$ Norm
  13. Numerical Experiments
  14. Proof of Theorem \ref{['thm:layermle']}
  15. Convergence Rate in the $L_2$ Norm
  16. ...and 22 more sections

Key Result

Theorem 2.1

Suppose $H_n \sim \mathsf{H}_{n, [r]}(n, \bm B)$ is a sample from the $r$-layered hypergraph $\bm \beta$-model as defined in eq:Hlayeredbeta. Then with probability $1-o(1)$ the likelihood equations eq:def_k_unif have a unique solution $\hat{\bm B} = ( \hat{\bm \beta}_{2}, \ldots, \hat{\bm \beta}_{r} for $2 \leq s \leq r$.

Figures (1)

  • Figure 1: (a) QQ plot of the ML estimate $\hat{\beta}_1$, (b) confidence intervals for $\beta_1$, and (c) power of the LR test for the goodness of fit problem \ref{['eq:H01s']}. The first two plots correspond to the 3-uniform hypergraph $\bm \beta$-model whereas plot (c) corresponds to $s$-uniform hypergraph $\bm \beta$-model for $s=2,3$.

Theorems & Definitions (37)

  • Definition 1.1
  • Theorem 2.1
  • Corollary 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Remark 2.1
  • Corollary 2.2
  • Theorem 2.4
  • Theorem 3.1
  • Theorem 3.2
  • ...and 27 more