A Sparse Beta Regression Model for Network Analysis

For statistical analysis of network data, the β-model has emerged as a useful tool, thanks to its flexibility in incorporating nodewise heterogeneity and theoretical tractability. To generalize the β-model, this article proposes the Sparse β-Regression Model (S β RM) that unites two research themes developed recently in modeling homophily and sparsity. In particular, we employ differential heterogeneity that assigns weights only to important nodes and propose penalized likelihood with an l1 penalty for parameter estimation. While our estimation method is closely related to the LASSO method for logistic regression, we develop a 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 Erdős-Rényi model by including covariates and develop the associated statistical inference for sparse networks, which may be of independent interest. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.

针对网络数据的统计分析，β模型（β-model）凭借其可灵活纳入节点异质性且理论上易于处理的特性，已成为一项实用工具。为推广β模型，本文提出稀疏β回归模型（Sparse β-Regression Model，简称SβRM），该模型整合了近期在同质性与稀疏性建模领域提出的两大研究方向。具体而言，本文采用仅对重要节点赋予权重的差异化异质性结构，并提出基于L1惩罚的惩罚似然法用于参数估计。尽管本文的估计方法与逻辑回归的LASSO方法密切相关，但我们构建了一套全新的理论，重点阐述如何利用所提模型处理实际中常见的稀疏网络所对应的参数场景。更具意义的是，针对同质性参数的推断无需像常规LASSO类估计那样进行去偏处理。本文通过大量模拟实验与数据分析案例，展示了该模型的应用路径。作为所提模型的一个特例，本文通过引入协变量拓展了埃尔德什-莱利模型（Erdős-Rényi model），并针对稀疏网络构建了对应的统计推断方法，该方法本身或具备独立研究价值。本文的补充材料可在线获取，其中包含可用于复现研究成果的标准化材料说明。