Max-flow min-cut theorem for Rényi entropy in communication networks

A symbolic approach to communication networks, where the topology of the underlying network is contained in a set of formal terms, was recently introduced. Many communication problems can be recast as dispersion problems in this setup. The so-called min-cut of a term set represents its number of degrees of freedom. For any assignment of function symbols, its dispersion measures the amount of information sent to the destinations. It was proved that the maximum dispersion asymptotically reaches the min-cut of the term set. In this paper, we refine this result in two ways. First, we prove a max-flow min-cut theorem for the Rényi entropy with order less than one, given that the inputs are equiprobably distributed; conversely, there is no max-flow min-cut theorem for Rényi entropy with order greater than one. Second, although linear coding functions have the practical appeal of low complexity, we prove that they are insufficient in general to reach the min-cut. More specifically, there exist term sets which have an arbitrarily large dispersion for non-linear coding functions, yet limited dispersion when linear coding functions are considered. Conversely, we show that if there is a solution based on low degree polynomials, then there exists a linear solution.