  A network of 3 nodes mutually communicating with each other is studied. This
multi-way network is a suitable model for 3-user device-to-device
communications. The main goal of this paper is to characterize the capacity
region of the underlying Gaussian 3-way channel (3WC) within a constant gap. To
this end, a capacity outer bound is derived using cut-set bounds and
genie-aided bounds. For achievability, the 3WC is first transformed into an
equivalent star-channel. This latter is then decomposed into a set of
`successive' sub-channels, leading to a sub-channel allocation problem. Using
backward decoding, interference neutralization, and known results on the
capacity of the star-channel relying of physical-layer network coding, an
achievable rate region for the 3WC is obtained. It is then shown that the
achievable rate region is within a constant gap of the developed outer bound,
leading to the desired capacity approximation. Interestingly, in contrast to
the Gaussian two-way channel (TWC), adaptation is necessary in the 3WC.
Furthermore, message splitting is another ingredient of the developed scheme
for the 3WC which is not required in the TWC. The two setups are, however,
similar in terms of their sum-capacity pre-log which is equal to 2. Finally,
some interesting networks and their approximate capacities are recovered as
special cases of the 3WC, such as the cooperative BC and MAC.
