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 it is of prime importance to understand the latent structure of network. In this paper, we model the network using the Degree-Corrected Mixed Membership (DCMM) model. In DCMM model, for each node $i$, there exists a membership vector $\boldsymbol{\pi}_ i = (\boldsymbol{\pi}_i(1), \boldsymbol{\pi}_i(2),\ldots, \boldsymbol{\pi}_i(K))$, where $\boldsymbol{\pi}_i(k)$ denotes the weight that node $i$ puts in community $k$. We derive novel finite-sample expansion for the $\boldsymbol{\pi}_i(k)$s which allows us to obtain asymptotic distributions and confidence interval of the membership mixing probabilities and other related population quantities. This fills an important gap on uncertainty quantification on the membership 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 member's profile with respect to a given community. The validity of our theoretical results is further demonstrated by via numerical experiments in both real and synthetic data examples.