  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
Erd\H{o}s-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
$\alpha =np/\log(n)$, it is shown that exact recovery is possible if and only
if $\alpha >2/(1-2\varepsilon)^2+ o(1/(1-2\varepsilon)^2)$. 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 $\alpha=d/\log(n)$, where $d$ is the minimum degree of the
graph, it is shown that the proposed method achieves the rate $\alpha>
4((1+\lambda)/(1-\lambda)^2)/(1-2\varepsilon)^2+ o(1/(1-2\varepsilon)^2)$,
where $1-\lambda$ is the spectral gap of the graph $G$.
