Degree Heterogeneity in Higher-Order Networks: Inference in the Hypergraph $\boldsymbol{\beta}$-Model

TL;DR

This paper develops a rigorous statistical framework for the hypergraph β-model, enabling accurate inference and hypothesis testing for complex higher-order network data with multiple layers and degree heterogeneity.

Contribution

It introduces the first comprehensive analysis of the hypergraph β-model, establishing convergence rates, asymptotic distributions, and optimal detection thresholds for inference in multi-layer higher-order networks.

Findings

ML estimators are minimax rate optimal.

Likelihood ratio test is asymptotically normal under the null.

Detection threshold is minimax optimal, with tests powerless below it.

Abstract

The $\boldsymbol{\beta}$-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 $\boldsymbol{\beta}$-model for capturing degree heterogeneity in networks with higher-order (multi-way) interactions. In this paper we initiate the rigorous study of the hypergraph $\boldsymbol{\beta}$-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 $\boldsymbol{\beta}$-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 $\boldsymbol{\beta}$-models, the above results fill a number of gaps in the graph $\boldsymbol{\beta}$-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.