  The study of feedback has been mostly limited to single-hop communication
settings. In this paper, we consider Gaussian networks where sources and
destinations can communicate with the help of intermediate relays over multiple
hops. We assume that links in the network can be bidirected providing
opportunities for feedback. We ask the following question: can the information
transfer in both directions of a link be critical to maximizing the end-to-end
communication rates in the network? Equivalently, could one of the directions
in each bidirected link (and more generally at least one of the links forming a
cycle) be shut down and the capacity of the network still be approximately
maintained? We show that in any arbitrary Gaussian network with bidirected
edges and cycles and unicast traffic, we can always identify a directed acyclic
subnetwork that approximately maintains the capacity of the original network.
For Gaussian networks with multiple-access and broadcast traffic, an acyclic
subnetwork is sufficient to achieve every rate point in the capacity region of
the original network, however, there may not be a single acyclic subnetwork
that maintains the whole capacity region. For networks with multicast and
multiple unicast traffic, on the other hand, bidirected information flow across
certain links can be critically needed to maximize the end-to-end capacity
region. These results can be regarded as generalizations of the conclusions
regarding the usefulness of feedback in various single-hop Gaussian settings
and can provide opportunities for simplifying operation in Gaussian multi-hop
networks.
