From trees to seeds: on the inference of the seed from large trees in the uniform attachment model

TL;DR

This paper investigates how the initial seed influences the structure of large trees in the uniform attachment model, demonstrating that different seeds result in distinguishable limiting tree distributions.

Contribution

It introduces new statistics to measure global properties of trees, showing seed influence extends beyond preferential attachment to uniform attachment models.

Findings

Different seeds lead to distinct limiting tree distributions.

Constructed statistics effectively distinguish seed influence.

Results extend understanding of seed effects in random tree models.

Abstract

We study the influence of the seed in random trees grown according to the uniform attachment model, also known as uniform random recursive trees. We show that different seeds lead to different distributions of limiting trees from a total variation point of view. To do this, we construct statistics that measure, in a certain well-defined sense, global "balancedness" properties of such trees. Our paper follows recent results on the same question for the preferential attachment model.