  We study the community detection problem on a Gaussian mixture model, in
which (1) vertices are divided into $k\geq 2$ distinct communities that are not
necessarily equally-sized; (2) the Gaussian perturbations for different entries
in the observation matrix are not necessarily independent or identically
distributed. We prove necessary and sufficient conditions for the exact
recovery of the maximum likelihood estimation (MLE), and discuss the cases when
these necessary and sufficient conditions give sharp threshold. Applications
include the community detection on a graph where the Gaussian perturbations of
observations on each edge is the sum of i.i.d.~Gaussian random variables on its
end vertices, in which we explicitly obtain the threshold for the exact
recovery of the MLE.
