  We study the community detection problem on a Gaussian mixture model, in
which vertices are divided into $k\geq 2$ distinct communities. The major
difference in our model is that the intensities for Gaussian perturbations are
different for different entries in the observation matrix, and we do not assume
that every community has the same number of vertices. We explicitly find the
threshold for the exact recovery of the maximum likelihood estimation.
Applications include the community detection on hypergraphs.
