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 $Y=B_G x \oplus Z$, where $B_G$ 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 Erdős-Rényi 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 $α>2/(1-2\varepsilon)^2+ o(1/(1-2\varepsilon)^2)$. In other words, $2/(1-2\varepsilon)^2$ 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 $α> 4((1+λ)/(1-λ)^2)/(1-2\varepsilon)^2+ o(1/(1-2\varepsilon)^2)$, where $1-λ$ is the spectral gap of the graph $G$.

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