Statistical Inference in a Directed Network Model With Covariates

ABSTRACT

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.

KEYWORDS:

• Asymptotic normality
• Consistency
• Degree heterogeneity
• Homophily
• Increasing number of parameters
• Maximum likelihood estimator

Acknowledgments

The authors are very grateful to three referees, an associate editor, and the Editor Regina Liu for their valuable comments that have greatly improved the article. Our simulation code is available on request. The authors thank Wei Cai at Northeast China Normal University for sharing the Sina Weibo data.

Additional information

Funding

Yan’s research is partially supported by the National Natural Science Foundation of China (No. 11771171). Jiang’s research is partially supported by the Hong Kong RGC grant (PolyU 253023/16P). Leng’s research is partially supported by a Turing Fellowship under the EPSRC grant EP/N510129/1.