Broadcasting on random recursive trees

TL;DR

This paper investigates the limits of reconstructing the root's bit value in random recursive trees and preferential attachment trees, analyzing error probabilities and simple reconstruction methods like majority vote and centroid-based rules.

Contribution

It characterizes the conditions under which accurate reconstruction is possible in random recursive and preferential attachment trees, introducing bounds on error probabilities and analyzing simple algorithms.

Findings

Reconstruction is feasible when q is below a certain threshold.

Error probability can be bounded by a constant times q.

Simple rules like majority vote and centroid perform effectively.

Abstract

We study the broadcasting problem when the underlying tree is a random recursive tree. The root of the tree has a random bit value assigned. Every other vertex has the same bit value as its parent with probability $1-q$ and the opposite value with probability $q$, where $q \in [0,1]$. The broadcasting problem consists in estimating the value of the root bit upon observing the unlabeled tree, together with the bit value associated with every vertex. In a more difficult version of the problem, the unlabeled tree is observed but only the bit values of the leaves are observed. When the underlying tree is a uniform random recursive tree, in both variants of the problem we characterize the values of $q$ for which the optimal reconstruction method has a probability of error bounded away from $1/2$. We also show that the probability of error is bounded by a constant times $q$. Two simple reconstruction rules are analyzed in detail. One of them is the simple majority vote, the other is the bit value of the centroid of the tree. Most results are extended to linear preferential attachment trees as well.