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The bi-degree sequence characterizes many useful information in directed networks. Although asymptotic theory has been derived in an exponential random graph model with the bi-degree sequence as the sufficient statistic, asymptotic theory for general graph distributions parameterized by the bi-degrees is still missing in existing literature. In this article, we introduce a general class of random graph models parameterized by a set of out-degree parameters and in-degree parameters to model the degree heterogeneity of bi-degrees. In particular, we left the likelihood function unspecified, and used a moment estimation approach to infer the degree parameters. In this class of models, the number of parameters increases as the size of network grows and therefore, asymptotic inference is nonstandard. We establish a unified framework in which the consistency and asymptotic normality of the moment estimator hold when the number of nodes goes to infinity. We illustrate our unified results by two applications including the Probit model and the Poisson model. Simulations and a real data application under the Poisson model are carried out to further demonstrate the theoretical results.

Inference in a class of directed random graph models with an increasing number of parameters