Broadcasting on trees near criticality

TL;DR

This paper investigates the broadcasting problem on $d$-ary trees near the critical noise threshold, using innovative information-theoretic methods to analyze reconstructability and phase transition behavior.

Contribution

It introduces the application of less-noisy channel comparison techniques to study the phase transition in broadcasting on trees, providing new insights into the reconstructability near criticality.

Findings

Reconstructability is established for $\delta<\delta_c$ using information-theoretic methods.

The phase transition at the critical noise level is characterized.

The approach offers a new perspective compared to traditional combinatorial analyses.

Abstract

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 $\mathrm{BSC}_\delta$ to $d$ descendants. The goal is to reconstruct $X_0$ given the vector $X_{L_h}$ of values of all variables at depth $h$. It is well known that reconstruction (better than a random guess) is possible as $h\to \infty$ if and only if $\delta < \delta_c(d)$. In this paper, we study the behavior of the mutual information and the probability of error when $\delta$ 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 $\delta<\delta_c$) using purely information-theoretic ideas. This is in contrast with previous derivations, which explicitly analyze distribution of the Hamming weight of $X_{L_h}$ (a so-called Kesten-Stigum bound).