For statistical analysis of network data, the $\beta$-model has emerged as a useful tool, thanks to its flexibility in incorporating nodewise heterogeneity and theoretical tractability. To generalize the $\beta$-model, this paper proposes the Sparse $\beta$-Regression Model (S$\beta$RM) that unites two research themes developed recently in modelling homophily and sparsity. In particular, we employ differential heterogeneity that assigns weights only to important nodes and propose penalized likelihood with an $\ell_1$ penalty for parameter estimation. While our estimation method is closely related to the LASSO method for logistic regression, we develop new theory emphasizing the use of our model for dealing with a parameter regime that can handle sparse networks usually seen in practice. More interestingly, the resulting inference on the homophily parameter demands no debiasing normally employed in LASSO type estimation. We provide extensive simulation and data analysis to illustrate the use of the model. As a special case of our model, we extend the Erd\H{o}s-R\'{e}nyi model by including covariates and develop the associated statistical inference for sparse networks, which may be of independent interest.

翻译：在网络数据的统计分析中，$\beta$模型因其能够灵活纳入节点异质性且具有理论可处理性，已成为一种实用工具。为推广$\beta$模型，本文提出稀疏$\beta$回归模型（S$\beta$RM），该模型融合了近期在建模同质性与稀疏性方面发展的两个研究主题。具体而言，我们采用差异化异质性方法，仅对重要节点赋予权重，并提出带有$\ell_1$惩罚项的惩罚似然函数进行参数估计。虽然我们的估计方法与逻辑回归的LASSO方法密切相关，但我们发展了新的理论，强调该模型适用于处理实际中常见稀疏网络的参数体系。更有趣的是，对同质性参数的推断无需采用LASSO类估计中通常所需的去偏处理。我们通过大量仿真与数据分析展示了该模型的应用。作为本模型的特例，我们通过纳入协变量扩展了Erd\H{o}s-R\'{e}nyi模型，并建立了适用于稀疏网络的相关统计推断方法，该扩展可能具有独立的研究价值。