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.

翻译：随机图的$\boldsymbol{\beta}$模型通常用于表示具有度异质性的网络中的成对交互。为了超越成对交互，Stasi等人（2014）引入了超图$\boldsymbol{\beta}$模型，以捕捉具有高阶（多方）交互的网络中的度异质性。在本文中，我们首次对具有多层结构的超图$\boldsymbol{\beta}$模型展开严格研究，该模型允许不同层中存在不同大小的超边。首先，我们推导了最大似然（ML）估计的收敛速率，并证明了其极小极大速率最优性。我们还推导了ML估计的极限分布，并为模型参数构建了渐近有效的置信区间。其次，我们考虑了超图$\boldsymbol{\beta}$模型中的拟合优度检验问题。具体而言，我们建立了零假设下似然比（LR）检验的渐近正态性，推导了其检测阈值，以及在该阈值处的极限功效。有趣的是，LR检验的检测阈值被证明是极小极大最优的，即所有检验在该阈值以下都是渐近无效的。数值实验进一步验证了这些理论结果。除了为超图$\boldsymbol{\beta}$模型建立估计与推断的理论框架外，上述结果还填补了图$\boldsymbol{\beta}$模型文献中的若干空白，例如ML估计的极小极大最优性以及LR检验的非零假设性质，据我们所知，这些内容此前尚未被研究过。