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The $\boldsymbol{\beta}$-model for random graphs is commonly used forrepresenting pairwise interactions in a network with degree heterogeneity.Going beyond pairwise interactions, Stasi et al. (2014) introduced thehypergraph $\boldsymbol{\beta}$-model for capturing degree heterogeneity innetworks with higher-order (multi-way) interactions. In this paper we initiatethe rigorous study of the hypergraph $\boldsymbol{\beta}$-model with multiplelayers, which allows for hyperedges of different sizes across the layers. Tobegin with, we derive the rates of convergence of the maximum likelihood (ML)estimate and establish their minimax rate optimality. We also derive thelimiting distribution of the ML estimate and construct asymptotically validconfidence intervals for the model parameters. Next, we consider thegoodness-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 alsoits limiting power at the threshold. Interestingly, the detection threshold ofthe LR test turns out to be minimax optimal, that is, all tests areasymptotically powerless below this threshold. The theoretical results arefurther validated in numerical experiments. In addition to developing thetheoretical framework for estimation and inference for hypergraph$\boldsymbol{\beta}$-models, the above results fill a number of gaps in thegraph $\boldsymbol{\beta}$-model literature, such as the minimax optimality ofthe ML estimates and the non-null properties of the LR test, which, to the bestof our knowledge, have not been studied before.

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