  We consider the problem of data collection in a two-layer network consisting
of (1) links between $N$ distributed agents and a remote sink node; (2) a
sparse network formed by these distributed agents. We study the effect of
inter-agent communications on the overall energy consumption. Despite the
sparse connections between agents, we provide an in-network coding scheme that
reduces the overall energy consumption by a factor of $\Theta(\log N)$ compared
to a naive scheme which neglects inter-agent communications. By providing lower
bounds on both the energy consumption and the sparseness (number of links) of
the network, we show that are energy-optimal except for a factor of
$\Theta(\log\log N)$. The proposed scheme extends a previous work of Gallager
on noisy broadcasting from a complete graph to a sparse graph, while bringing
in new techniques from error control coding and noisy circuits.
