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：有向网络可方便地表示为图形，其中有序边缘对顶点之间的相互作用进行编码。尽管它们广泛可用，但在存在上下文信息和度数异质性时，可进行推断的统计模型不足。本文提出了一种带有显式参数的注释图模型，该参数考虑了这些特征。为了克服由于建模度异质性而引起的维数灾难，我们引入了一种稀疏性假设，并提出了基于$\ell_1$-规范化的罚函数最大似然方法以进行参数估计。我们研究了这种方法在稀疏网络假设下的估计和选择一致性，并表明了在协变参数上的推断是直接的，从而避免了通常在$\ell_1$-规范化最大似然估计中使用的去偏置的需要。模拟和数据分析证实了我们的理论发现。