Max-flow min-cut theorems on dispersion and entropy measures for communication networks

Abstract

The paper presents four distinct new ideas and results for communication networks:1) We show that relay-networks (i.e. communication networks where different nodes use the same coding functions) can be used to model dynamic networks, in a way, akin to Kripke's possible worlds. Changes in the network are modelled by considering a multiverse where different possible situations arise as worlds existing in parallel.2) We introduce the term model, which is a simple, graph-free symbolic approach to communication networks. This model yields an algorithm to calculate the capacity of a given communication network.3) We state and prove variants of a theorem concerning the dispersion of information in single-receiver communications. The dispersion theorem resembles the max-flow min-cut theorem for commodity networks. The proof uses a very weak kind of network coding, called routing with dynamic headers.4) We show that the solvability of an abstract multi-user communication problem is equivalent to the solvability of a single-target communication in a suitable relay network.In the paper, we develop a number of technical ramifications of these ideas and results. We prove a max-flow min-cut theorem for the Rényi entropy with order less than one, given that the sources are equiprobably distributed; conversely, we show that the max-flow min-cut theorem fails for order greater than one. We also show that linear network coding fails for relay networks, although routing with dynamic headers is asymptotically sufficient to reach capacity.