Broadcasting on random recursive trees

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 ∈ [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.

Broadcasting induced colourings of random recursive trees and preferential attachment trees

“…For a rooted tree T = (V, E) with root ρ and p ∈ [0, 1], we define a broadcasting induced colouring σ T,p of T to be a random 2-colouring σ T,p : V (T ) → {red, blue} of the vertices of T such that (i) P(σ T,p (ρ) = red) = P(σ T,p (ρ) = blue) = 1 2 and…”

“…Applications of the reconstruction problem in trees include its connection to stochastic block models [27,14], a random graph model with applications in machine learning. Of particular interest to this work, Addario-Berry, Devroye, Lugosi, and Velona studied the reconstruction problem in random recursive trees and preferential attachment trees [1].…”

“…. , − 1 4 , − 1 3 , − 1 2 }, then there may be vertices v in T α,n for which (1) is negative. This problem is avoided when α = − 1 d , since (1) is positive when deg + (u) < d and is zero when deg + (u) = d, resulting in a tree T −1/d,n whose vertices all have outdegree less than or equal to d.…”

“…As an example of the growth process we describe, consider the trees with broadcasting induced colourings in Figure 1. The tree T 8 is grown from T 7 by choosing v according to the probability (1) and adding a child u (notice that deg + (v) = 0). The probability that u takes a different colour from v is 1 − p, and so…”