  We consider the problem of estimating community memberships of nodes in a
network, where every node is associated with a vector determining its degree of
membership in each community. Existing provably consistent algorithms often
require strong assumptions about the population, are computationally expensive,
and only provide an overall error bound for the whole community membership
matrix. This paper provides uniform rates of convergence for the inferred
community membership vector of each node in a network generated from the Mixed
Membership Stochastic Blockmodel (MMSB); to our knowledge, this is the first
work to establish per-node rates for overlapping community detection in
networks. We achieve this by establishing sharp row-wise eigenvector deviation
bounds for MMSB. Based on the simplex structure inherent in the
eigen-decomposition of the population matrix, we build on established
corner-finding algorithms from the optimization community to infer the
community membership vectors. Our results hold over a broad parameter regime
where the average degree only grows poly-logarithmically with the number of
nodes. Using experiments with simulated and real datasets, we show that our
method achieves better error with lower variability over competing methods, and
processes real world networks of up to 100,000 nodes within tens of seconds.
