Directed networks are conveniently represented as graphs in which ordered edges encode interactions between vertices. Despite their wide availability, there is a shortage of statistical models amenable for inference, specially when contextual information and degree heterogeneity are present. This paper presents an annotated graph model with parameters explicitly accounting for these features. To overcome the curse of dimensionality due to modelling degree heterogeneity, we introduce a sparsity assumption and propose a penalized likelihood approach with $\ell_1$-regularization for parameter estimation. We study the estimation and selection consistency of this approach under a sparse network assumption, and show that inference on the covariate parameter is straightforward, thus bypassing the need for the kind of debiasing commonly employed in $\ell_1$-penalized likelihood estimation. Simulation and data analysis corroborate our theoretical findings.

翻译：有向网络通常以图的形式表示，其中有序边编码了顶点之间的交互关系。尽管这类网络广泛存在，但适用于推断的统计模型仍然匮乏，尤其是在存在上下文信息和度异质性时。本文提出一种带注释图模型，其参数显式考虑了这些特征。为克服建模度异质性导致的维数灾难，我们引入稀疏性假设，并提出带$\ell_1$正则化的惩罚似然方法进行参数估计。在稀疏网络假设下，我们研究了该方法的估计与选择一致性，并证明协变量参数的推断可直接进行，从而无需采用$\ell_1$惩罚似然估计中常用的去偏技术。仿真实验与数据分析验证了我们的理论结果。