  In this paper, we make use of channel symmetry properties to determine the
capacity region of three types of two-way networks: (a) two-user memoryless
two-way channels (TWCs), (b) two-user TWCs with memory, and (c) three-user
multiaccess/degraded broadcast (MA/DB) TWCs. For each network, symmetry
conditions under which a Shannon-type random coding inner bound (under
independent non-adaptive inputs) is tight are given. For two-user memoryless
TWCs, prior results are substantially generalized by viewing a TWC as two
interacting state-dependent one-way channels. The capacity of symmetric TWCs
with memory, whose outputs are functions of the inputs and independent
stationary and ergodic noise processes, is also obtained. Moreover, various
channel symmetry properties under which the Shannon-type inner bound is tight
are identified for three-user MA/DB TWCs. The results not only enlarge the
class of symmetric TWCs whose capacity region can be exactly determined but
also imply that interactive adaptive coding, not improving capacity, is
unnecessary for such channels.
