A central limit theorem in the $β$-model for undirected random graphs with a diverging number of vertices

The $\beta$-model provides a convenient tool for analyzing network data and Chatterjee, Diaconis and Sly (2011) recently established the consistency of the maximum likelihood estimate (MLE) in the $\beta$-model when the number of vertices goes to infinity. In this note, by effectively approximating the inverse of the Fisher information matrix, we further obtain its asymptotic normality under mild conditions. Simulation studies and a data example are also provided to illustrate the theoretical results.