Abstract: To capture the heterozygosity of vertex degrees of networks and understand their distributions, a class of random graph models parameterized by the strengths of vertices is proposed. These models have a framework of mutually independent edges, where the number of parameters matches the size of the network. The asymptotic properties of the maximum likelihood estimator have been derived in such models as the β-model, but general results are lacking. In these models, the likelihood equations are identical to the moment equations. Here, we establish a unified asymptotic result that includes the consistency and asymptotic normality of the moment estimator instead of the maximum likelihood estimator, when the number of parameters goes to infinity. We apply it to the generalized β-model, maximum entropy models, and Poisson models.

Key words and phrases: Asymptotical normality, consistency, increasing number of parameters, moment estimators, undirected network models.