  We establish a finite-sample Berry-Esseen theorem for the entrywise limits of
the eigenvectors for a broad collection of signal-plus-noise random matrix
models under challenging weak signal regimes. The signal strength is
characterized by a scaling factor $\rho_n$ through $n\rho_n$, where $n$ is the
dimension of the random matrix, and we allow $n\rho_n$ to grow at the rate of
$\log n$. The key technical contribution is a sharp finite-sample entrywise
eigenvector perturbation bound. The existing error bounds on the
two-to-infinity norms of the higher-order remainders are not sufficient when
$n\rho_n$ is proportional to $\log n$. We apply the general entrywise
eigenvector analysis results to the symmetric noisy matrix completion problem,
random dot product graphs, and two subsequent inference tasks for random
graphs: the estimation of pure nodes in mixed membership stochastic block
models and the hypothesis testing of the equality of latent positions in random
graphs.
