Asymptotic theory in bipartite graph models with a growing number of parameters

Affiliation networks contain a set of actors and a set of events, where edges denote the affiliation relationships between actors and events. Here, we introduce a class of affiliation network models for modelling the degree heterogeneity, where two sets of degree parameters are used to measure the activeness of actors and the popularity of events, respectively. We develop the moment method to infer these degree parameters. We establish a unified theoretical framework in which the consistency and asymptotic normality of the moment estimator hold as the numbers of actors and events both go to infinity. We apply our results to several popular models with weighted edges, including generalized $\beta$-, Poisson and Rayleigh models. Simulation studies and a realistic example that involves the Poisson model provide concrete evidence that supports our theoretical findings.