  This paper considers the problem of reconstructing $n$ independent uniform
spins $X_1,\dots,X_n$ living on the vertices of an $n$-vertex graph $G$, by
observing their interactions on the edges of the graph. This captures instances
of models such as (i) broadcasting on trees, (ii) block models, (iii)
synchronization on grids, (iv) spiked Wigner models. The paper gives an
upper-bound on the mutual information between two vertices in terms of a bond
percolation estimate. Namely, the information between two vertices' spins is
bounded by the probability that these vertices are connected in a bond
percolation model, where edges are opened with a probability that "emulates"
the edge-information. Both the information and the open-probability are based
on the Chi-squared mutual information. The main results allow us to re-derive
known results for information-theoretic non-reconstruction in models (i)-(iv),
with more direct or improved bounds in some cases, and to obtain new results,
such as for a spiked Wigner model on grids. The main result also implies a new
subadditivity property for the Chi-squared mutual information for symmetric
channels and general graphs, extending the subadditivity property obtained by
Evans-Kenyon-Peres-Schulman [EKPS00] for trees.
