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

TL;DR

This paper provides the first rigorous asymptotic analysis of directed exponential random graph models using degree sequences, establishing consistency and normality of estimators for large networks with weighted edges.

Contribution

It introduces a novel theoretical framework for directed ERGMs with increasing parameters, including proofs of estimator properties and approximation techniques.

Findings

Maximum likelihood estimators are uniformly consistent.

Estimators are asymptotically normal under increasing parameters.

Numerical studies support theoretical results.

Abstract

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 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 as well as continuous weighted edges. We establish the uniform consistency and the asymptotic normality for the maximum likelihood estimate, 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. Numerical studies confirm our theoretical findings.