$$\begin{aligned}
Y^E=B_G x^V \oplus Z^E, \label{lin}\end{aligned}$$ where $B_G$ is the incidence matrix of the graph, i.e., the $m\times n$ matrix, with $m=|E|$ and $n=|V|$, such that $B_G(e,v)=1$ if and only if edge $e$ is incident to vertex $v$, and $Z$ is a random vector of dimension $|E|$ representing the noise.

In the above setting, the forward problem of recovering the most likely edge-variables given the vertex-variables is trivial and amounts to maximizing $Q$ for each edge. The inverse problem, however, is more challenging: the most likely vertex-variables (say with a uniform prior) given the edge-variables cannot be found by local maximization.

This problem can be interpreted as a community detection problem with censored edges: Consider a population with $n$ vertices and two communities, the blues and the reds. The colors of the vertices, encoded by the binary variables $\{X_i\}_{i \in [n]}$, are unknown and the goal is to recover them by observing pairwise interactions of these nodes. However, not all ${n \choose 2}$ interactions are observed, only the ones encoded by the graph $G$. In the noiseless case, the observation is perfect and allows to determine whether $X_i$ and $X_j$ are in the same community or not, i.e., $Y_{ij}=X_i \oplus 