Broadcasting on trees near criticality

Abstract

© 2020 IEEE. We revisit the problem of broadcasting on d-ary trees: starting from a Bernoulli(1/2) random variable X 0 at a root vertex, each vertex forwards its value across binary symmetric channels BSC δ to d descendants. The goal is to reconstruct X 0 given the vector X Lh of values of all variables at depth h. It is well known that reconstruction (better than a random guess) is possible as h →∞ if and only if δ < δ c (d). In this paper, we study the behavior of the mutual information and the probability of error when δ is slightly subcritical. The innovation of our work is application of the recently introduced less-noisy channel comparison techniques. For example, we are able to derive the positive part of the phase transition (reconstructability when δ < δ c ) using purely information-theoretic ideas. This is in contrast with previous derivations, which explicitly analyze distribution of the Hamming weight of X Lh (a so-called Kesten-Stigum bound).