Max-Flow Min-Cut Theorems for Multi-User Communication Networks

TL;DR

This paper introduces new theoretical frameworks and results for multi-user communication networks, including a max-flow min-cut theorem for Rényi entropy and insights into the limitations of network coding in relay networks.

Contribution

It presents novel ideas such as the model concept, variants of information dispersion theorems, and establishes a max-flow min-cut theorem for Rényi entropy, expanding understanding of network capacity and coding limitations.

Findings

Max-flow min-cut theorem holds for Rényi entropy with order less than one.

The theorem fails for Rényi entropy with order greater than one.

Simple packet-switching achieves capacity in static single-receiver networks, but not in relay 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. 2) We introduce {\em the term model}, which is a simple, graph-free symbolic approach to communication networks. 3) We state and prove variants of a theorem concerning the dispersion of information in single-receiver communications. 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. One technical result is a max-flow min-cut theorem for the R\'enyi 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 the R\'enyi entropy with order greater than one. We leave the status of the theorem with regards to the ordinary Shannon Entropy measure (R\'enyi entropy of order one and the limit case between validity or failure of the theorem) as an open question. In non-dynamic static communication networks with a single receiver, a simple application of Menger's theorem shows that the optimal throughput can be achieved without proper use of network coding i.e. just by using ordinary packet-switching. This fails dramatically in relay networks with a single receiver. We show that even a powerful method like linear network coding fails miserably for relay networks. With that in mind, it is noticeable that our rather weak form of network coding (routing with dynamic headers) is asymptotically sufficient to reach capacity.