Exact Recovery of Community Detection in k-partite Graph Models
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We study the vertex classification problem on a graph in which the vertices are in k (k≥ 2) different groups, or communities, and edges are only allowed between vertices in distinct groups. The observation is the weighted adjacency matrix, perturbed by a Gaussian Orthogonal Ensemble (GOE), or Gaussian Unitary Ensemble (GUE) matrix. Different from the standard symmetric _2-synchronization in which there are 2 communities with equal number of vertices, we do not assume that the number of vertices in each group is the same. For the exact recovery of the maximum likelihood estimation (MLE) with various weighted adjacency matrix, we prove a sharp phase transition result with respect to the intensity σ of the Gaussian perturbation. These weighted adjacency matrices may be considered as natural models for the electric network. Surprisingly, the threshold, or critical value, of σ does not depend on whether or not the sample space for MLE is restricted to the classifications such that the number of vertices in each group is the same as the true value. In contrast to the well-studied _2-sychronization, 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 for σ such that SDP exactly recovers the true classification is obtained, which is independent of k.
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