Strong Converse Theorems for Classes of Multimessage Multicast Networks: A R\'enyi Divergence Approach

TL;DR

This paper proves that for certain classes of discrete memoryless multicast networks, the probability of error approaches one if the rate exceeds the cut-set bound, using a Rényi divergence approach.

Contribution

It establishes the strong converse for specific DM-MMN classes where the cut-set bound is tight, using a novel Rényi divergence-based proof technique.

Findings

Strong converse holds for wireless erasure networks.

Strong converse applies to DM-MMNs with independent DMCs.

Error probability tends to one outside the cut-set bound.

Abstract

This paper establishes that the strong converse holds for some classes of discrete memoryless multimessage multicast networks (DM-MMNs) whose corresponding cut-set bounds are tight, i.e., coincide with the set of achievable rate tuples. The strong converse for these classes of DM-MMNs implies that all sequences of codes with rate tuples belonging to the exterior of the cut-set bound have average error probabilities that necessarily tend to one (and are not simply bounded away from zero). Examples in the classes of DM-MMNs include wireless erasure networks, DM-MMNs consisting of independent discrete memoryless channels (DMCs) as well as single-destination DM-MMNs consisting of independent DMCs with destination feedback. Our elementary proof technique leverages properties of the R\'enyi divergence.