Dualities Between Entropy Functions and Network Codes

TL;DR

This paper establishes a duality between entropy functions and network codes, showing how entropy properties influence network solvability and capacity bounds, and highlighting the limitations of linear codes.

Contribution

It introduces a new duality framework linking entropy functions with network coding problems, and demonstrates the importance of non-Shannon inequalities for capacity bounds.

Findings

Network solvability is characterized by entropy functions.

Duality between polymatroids and linear programming bounds.

Linear and abelian network codes are insufficient for certain capacities.

Abstract

This paper provides a new duality between entropy functions and network codes. Given a function $g\geq 0$ defined on all proper subsets of $N$ random variables, we provide a construction for a network multicast problem which is solvable if and only if $g$ is entropic. The underlying network topology is fixed and the multicast problem depends on $g$ only through edge capacities and source rates. 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.