Approximate estimation in a class of directed networks

Recent advances in computing and measurement technologies have led to an explosion in the amount of increasing availability of network data in many different fields. To capture the bi-degree heterogeneity of directed networks nodes, the logistic-linear model and the implicit log-linear model have been proposed in the literature. However, computation of the MLEs is complicated and practical choice of these two models can be confusing. In this article, we reveal that these models can be viewed as instances of a broader class of null models, and we derived an approximate estimation for the MLEs under a sparse graph regime. Simulation studies and real data examples are conducted to further demonstrate our theoretical results.