Semidefinite programs for exact recovery of a hidden community

We study a semidefinite programming (SDP) relaxation of the maximum likelihood estimation for exactly recovering a hidden community of cardinality K from an n × n symmetric data matrix A, where for distinct indices i, j, A_ij ∼ P if i, j are both in the community and A_ij ∼ Q otherwise, for two known probability distributions P and Q. We identify a sufficient condition and a necessary condition for the success of SDP for the general model. For both the Bernoulli case (P = Bern(p) and Q = Bern(q) with p > q) and the Gaussian case (P = N(µ, 1) and Q = N(0, 1) with µ > 0), which correspond to the problem of planted dense subgraph recovery and submatrix localization respectively, the general results lead to the following findings: (1) If K = ω(n/log n), SDP attains the information-theoretic recovery limits with sharp constants; (2) If K = Θ(n/log n), SDP is order-wise optimal, but strictly suboptimal by a constant factor; (3) If K = o(n/log n) and K → ∞, SDP is order-wise suboptimal. The same critical scaling for K is found to hold, up to constant factors, for the performance of SDP on the stochastic block model of n vertices partitioned into multiple communities of equal size K. A key ingredient in the proof of the necessary condition is a construction of a primal feasible solution based on random perturbation of the true cluster matrix.