Dualities Between Entropy Functions and Network Codes

In communications networks, the capacity region of multisource network coding is given in terms of the set of entropy functions $Gamma ^{ast }$. More broadly, determination of $Gamma ^{ast }$ would have an impact on converse theorems for multi-terminal problems in information theory. This paper provides several new dualities between entropy functions and network codes. Given a function $ggeq 0$ defined on all subsets of $N$ random variables, we provide a construction for a network multicast problem which is “solvable” if and only if $g$ is the entropy function of a set of quasi-uniform random variables. The underlying network topology is fixed and the multicast problem depends on $g$ only through link capacities and source rates. A corresponding duality is developed for linear network codes, where the constructed multicast problem is linearly solvable if and only if $g$ is linear group characterizable. Relaxing the requirement that the domain of $g$ be subsets of random variables, we obtain a similar duality between polymatroids and the linear programming bound. These duality results provide an alternative proof of the insufficiency of linear (and abelian) network codes, and demonstrate the utility of non-Shannon inequalities to tighten outer bounds on network coding capacity regions.