Scaling limits and influence of the seed graph in preferential attachment trees

TL;DR

This paper investigates the asymptotic properties of preferential attachment trees, proving seed graph influence, and introduces the Brownian looptree as a scaling limit of large-degree node structures.

Contribution

It proves the influence of the seed graph on asymptotic behavior and introduces the Brownian looptree as a new scaling limit for large-degree node structures.

Findings

Seed graph significantly influences asymptotic tree behavior

Looptrees converge to the Brownian looptree in Gromov-Hausdorff sense

Brownian looptree has Hausdorff dimension 2

Abstract

We are interested in the asymptotics of random trees built by linear preferential attachment, also known in the literature as Barab\'asi-Albert trees or plane-oriented recursive trees. We first prove a conjecture of Bubeck, Mossel \& R\'acz concerning the influence of the seed graph on the asymptotic behavior of such trees. Separately we study the geometric structure of nodes of large degrees in a plane version of Barab\'asi-Albert trees via their associated looptrees. As the number of nodes grows, we show that these looptrees, appropriately rescaled, converge in the Gromov-Hausdorff sense towards a random compact metric space which we call the Brownian looptree. The latter is constructed as a quotient space of Aldous' Brownian Continuum Random Tree and is shown to have almost sure Hausdorff dimension $2$.