Fine asymptotics for the maximum degree in weighted recursive trees with bounded random weights

TL;DR

This paper investigates the asymptotic behavior of the maximum degree in weighted recursive trees with bounded random weights, providing higher-order corrections and detailed distributional results.

Contribution

It extends existing results for recursive trees to the weighted case, deriving precise asymptotics and distributional properties of maximum degrees and related quantities.

Findings

Higher-order corrections to maximum degree growth.

Asymptotic independence of degrees of randomly chosen vertices.

Distribution of the number of vertices with near-maximum degree.

Abstract

A weighted recursive tree is an evolving tree in which vertices are assigned random vertex-weights and new vertices connect to a predecessor with a probability proportional to its weight. Here, we study the maximum degree and near-maximum degrees in weighted recursive trees when the vertex-weights are almost surely bounded and their distribution function satisfies a mild regularity condition near zero. We are able to specify higher-order corrections to the first order growth of the maximum degree established in prior work. The accuracy of the results depends on the behaviour of the weight distribution near the largest possible value and in certain cases we manage to find the corrections up to random order. Additionally, we describe the tail distribution of the maximum degree, the distribution of the number of vertices attaining the maximum degree, and establish asymptotic normality of the number of vertices with near-maximum degree. Our analysis extends the results proved for random recursive trees (where the weights are constant) to the case of random weights. The main technical result shows that the degrees of several uniformly chosen vertices are asymptotically independent with explicit error corrections.