The capacity of network coding for a single source has the nice and sweet max-flow min-cut characterization.  Once we move beyond single-source, the problem becomes much more difficult.

The capacity region for multisource network coding on acyclic networks was completely characterized by Yan, Yeung, and Zhang (2007) in terms of $\Gamma^*$, the region of all entropy functions.  This characterization is strongly tied to the study of non-Shannon-type inequalities.  In this tutorial, we will first explain this characterization on an intuitive basis.  This characterization, however, works only for block network codes with arbitrarily small probability of decoding error but does not work with variable-rate network codes that can possibly achieve zero error.  In the rest of the tutorial, we will explain the hurdles involved and report some of our latest efforts toward overcoming these hurdles.