Inferences on Mixing Probabilities and Ranking in Mixed-Membership Models

Network data is prevalent in numerous big data applications, including economics and health networks, where understanding the latent structure of the network is of prime importance. In this paper, we model the network using the Degree-Corrected Mixed Membership (DCMM) model. In the DCMM model, for each node i, there exists a membership vector πi=(πi(1),πi(2),…,πi(K)), where πi(k) denotes the weight that node i puts in community k. We derive a novel finite-sample expansion for the πi(k) s, which allows us to obtain asymptotic distributions and confidence intervals of the membership mixing probabilities and other related population quantities. This fills an important gap in uncertainty quantification on the member’s profile. We further develop a ranking scheme of the vertices based on the membership mixing probabilities on certain communities and perform relevant statistical inferences. A multiplier bootstrap method is proposed for ranking inference of individual membership profiles with respect to a given community. The validity of our theoretical results is further demonstrated via numerical experiments in both real and synthetic data examples.