Statistical Inference in a Directed Network Model With Covariates

Networks are often characterized by node heterogeneity for which nodes exhibit different degrees of interaction and link homophily for which nodes sharing common features tend to associate with each other. In this article, we rigorously study a directed network model that captures the former via node-specific parameterization and the latter by incorporating covariates. In particular, this model quantifies the extent of heterogeneity in terms of outgoingness and incomingness of each node by different parameters, thus allowing the number of heterogeneity parameters to be twice the number of nodes. We study the maximum likelihood estimation of the model and establish the uniform consistency and asymptotic normality of the resulting estimators. Numerical studies demonstrate our theoretical findings and two data analyses confirm the usefulness of our model. Supplementary materials for this article are available online.

网络通常具有节点异质性（node heterogeneity）与边同配性（link homophily）两类核心特征：前者指节点间的交互程度存在差异，后者则指具有共同特征的节点更倾向于彼此建立连接。本文针对一类有向网络模型展开严谨研究，该模型通过节点专属参数化方式刻画节点异质性，并通过引入协变量来捕捉边同配性特征。具体而言，该模型通过不同参数分别量化每个节点的外向交互强度与内向交互强度层面的异质性程度，因此异质性参数的数量可达节点总数的两倍。本文对该模型的极大似然估计（maximum likelihood estimation）展开研究，并证明了所得估计量的一致相合性（uniform consistency）与渐近正态性（asymptotic normality）。数值实验验证了本文的理论结论，两项实证数据分析则证实了所提模型的实用性。本文的补充材料可在线获取。