  This paper is concerned with jointly recovering $n$ node-variables $\left\{
x_{i}\right\}_{1\leq i\leq n}$ from a collection of pairwise difference
measurements. Imagine we acquire a few observations taking the form of
$x_{i}-x_{j}$; the observation pattern is represented by a measurement graph
$\mathcal{G}$ with an edge set $\mathcal{E}$ such that $x_{i}-x_{j}$ is
observed if and only if $(i,j)\in\mathcal{E}$. To account for noisy
measurements in a general manner, we model the data acquisition process by a
set of channels with given input/output transition measures. Employing
information-theoretic tools applied to channel decoding problems, we develop a
\emph{unified} framework to characterize the fundamental recovery criterion,
which accommodates general graph structures, alphabet sizes, and channel
transition measures. In particular, our results isolate a family of
\emph{minimum} \emph{channel divergence measures} to characterize the degree of
measurement corruption, which together with the size of the minimum cut of
$\mathcal{G}$ dictates the feasibility of exact information recovery. For
various homogeneous graphs, the recovery condition depends almost only on the
edge sparsity of the measurement graph irrespective of other graphical metrics;
alternatively, the minimum sample complexity required for these graphs scales
like \[ \text{minimum sample complexity }\asymp\frac{n\log
n}{\mathsf{Hel}_{1/2}^{\min}} \] for certain information metric
$\mathsf{Hel}_{1/2}^{\min}$ defined in the main text, as long as the alphabet
size is not super-polynomial in $n$. We apply our general theory to three
concrete applications, including the stochastic block model, the outlier model,
and the haplotype assembly problem. Our theory leads to order-wise tight
recovery conditions for all these scenarios.
