Network Coding Theorem for Dynamic 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. 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 and its Renyi entropies measure the amount of information that can be inferred about the input from the outputs. It was proved that the maximum dispersion and the maximum Renyi entropy of order less than one asymptotically reach the min-cut of the term set. In this paper, we first generalize the term set framework for multi-user communications and state a multi-user version of the dispersion (and Renyi entropy) theorem. We then model dynamic networks as a collection of term sets and the possible demands of users via a utility function. We apply the multi-user theorem to derive a general principle for many-to-many cast communications in dynamic multi-user networks. In general, we show that if each user´s demand can be satisfied locally, then all the demands can be satisfied globally.