Decoding binary node labels from censored edge measurements: Phase transition and efficient recovery

We consider the problem of clustering a graph G into two communities by observing a subset of the vertex correlations. Specifically, we consider the inverse problem with observed variables Y=BG x oplus Z, where BG is the incidence matrix of a graph G, x is the vector of unknown vertex variables (with a uniform prior) and Z is a noise vector with Bernoulli(varepsilon) i.i.d. entries. All variables and operations are Boolean. This model is motivated by coding, synchronization, and community detection problems. In particular, it corresponds to a stochastic block model or a correlation clustering problem with two communities and censored edges. Without noise, exact recovery (up to global flip) of x is possible if and only the graph G is connected, with a sharp threshold at the edge probability log(n)/n for ErdHos-R'enyi random graphs. The first goal of this paper is to determine how the edge probability p needs to scale to allow exact recovery in the presence of noise. Defining the degree (oversampling) rate of the graph by α =np/log(n), it is shown that exact recovery is possible if and only if α >2/(1-2varepsilon)2+ o(1/(1-2varepsilon)2). In other words, 2/(1-2varepsilon)2 is the information theoretic threshold for exact recovery at low-SNR. In addition, an efficient recovery algorithm based on semidefinite programming is proposed and shown to succeed in the threshold regime up to twice the optimal rate. For a deterministic graph G, defining the degree rate as α=d/log(n), where d is the minimum degree of the graph, it is shown that the proposed method achieves the rate α> 4((1+λ)/(1-λ)2)/(1-2varepsilon)2+ o(1/(1-2varepsilon)2), where 1-λ is the spectral gap of the graph G.