  We study the vertex classification problem on a graph whose vertices are in
$k\ (k\geq 2)$ different communities, edges are only allowed between distinct
communities, and the number of vertices in different communities are not
necessarily equal. The observation is a weighted adjacency matrix, perturbed by
a scalar multiple of the Gaussian Orthogonal Ensemble (GOE), or Gaussian
Unitary Ensemble (GUE) matrix. For the exact recovery of the maximum likelihood
estimation (MLE) with various weighted adjacency matrices, we prove sharp
thresholds of the intensity $\sigma$ of the Gaussian perturbation. These
weighted adjacency matrices may be considered as natural models for the
electric network. Surprisingly, these thresholds of $\sigma$ do not depend on
whether the sample space for MLE is restricted to such classifications that the
number of vertices in each group is equal to the true value. In contrast to the
$\ZZ_2$-synchronization, a new complex version of the semi-definite programming
(SDP) is designed to efficiently implement the community detection problem when
the number of communities $k$ is greater than 2, and a common region
(independent of $k$) for $\sigma$ such that SDP exactly recovers the true
classification is obtained.
