  The problem of network coding for multicasting a single source to multiple
sinks has first been studied by Ahlswede, Cai, Li and Yeung in 2000, in which
they have established the celebrated max-flow mini-cut theorem on non-physical
information flow over a network of independent channels. On the other hand, in
1980, Han has studied the case with correlated multiple sources and a single
sink from the viewpoint of polymatroidal functions in which a necessary and
sufficient condition has been demonstrated for reliable transmission over the
network. This paper presents an attempt to unify both cases, which leads to
establish a necessary and sufficient condition for reliable transmission over a
noisy network for multicasting all the correlated multiple sources to all the
multiple sinks. Furthermore, we address also the problem of transmitting
"independent" sources over a multiple-access-type of network as well as over a
broadcast-type of network, which reveals that the (co-) polymatroidal
structures are intrinsically involved in these types of network coding.
