Asymptotics in directed exponential random graph models with an increasing bi-degree sequence

Although asymptotic analyses of undirected network models based on degree sequences have started to appear in recent literature, it remains an open problem to study the statistical properties of directed network models. In this paper, we provide for the first time a rigorous analysis of directed exponential random graph models using the in-degrees and out-degrees as sufficient statistics with binary and non-binary weighted edges. We establish the uniform consistency and the asymptotic normality of the maximum likelihood estimator, when the number of parameters grows and only one realized observation of the graph is available. One key technique in the proofs is to approximate the inverse of the Fisher information matrix using a simple matrix with high accuracy. Along the way, we also establish a geometrically fast rate of convergence for the Newton iterative algorithm, which is used to obtain the maximum likelihood estimate. Numerical studies confirm our theoretical findings.