Entropy Vectors, Network Information Theory and Wireless Networks

Abstract

Information theory is well poised to have an impact on the manner in which future networks are designed and maintained, both because wired networks are ripe for applications such as network coding and also because wireless networks cannot be satisfactorily dealt with using conventional networking tools. The challenge is that most network information theory problems are notoriously difficult and so the barriers that must be overcome are often quite high. In particular, there are only a limited number of tools available and so fresh approaches are quite welcome. We describe an approach based on the definition of the space of "normalized" entropic vectors. In this framework, for a large class of acyclic memoryless networks, the capacity region for an arbitrary set of sources and destinations can be found by maximization of a linear function over the set of channel-constrained normalized entropic vectors and some linear constraints. The key point is that the closure of this set is convex and compact. While this may not necessarily make the problem simpler, it certainly circumvents the "infinite-letter characterization" issue, as well as the nonconvexity of earlier formulations. It also exposes the core of the problem as that of determining the space of normalized entropic vectors. The approach has several interesting consequences: it allows one to obtain the classical cutset bounds via a duality argument; for wired networks, it shows one need only consider the space of unconstrained normalized entropic vectors, thus separating channel and network coding---a result very recently recognized in the community. Outer bounds to the space of normalized entropic vectors are known to be related to non-Shannon inequalities. We develop inner bounds on this space using lattice-generated distributions and show how they can be used to compute inner bounds on the capacity region of networks using linear programming.

Files

Additional details