  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.
