Weak local limit of preferential attachment random trees with additive fitness

TL;DR

This paper studies the local structure of preferential attachment trees with additive fitness, showing they converge to a mixed Poisson process and providing convergence rates for degree distributions.

Contribution

It introduces a weak local limit for these trees using mixed Poisson processes and quantifies the convergence rate in total variation distance.

Findings

Weak local limit constructed via mixed Poisson processes.

Explicit rate of convergence in total variation distance.

Limiting degree distributions for vertices and ancestors.

Abstract

We consider linear preferential attachment random trees with additive fitness, where fitness is defined as the random initial vertex attractiveness. We show that when the fitness distribution has positive bounded support, the weak local limit of this family can be constructed using a sequence of mixed Poisson point processes. We also provide a rate of convergence of the total variation distance between the r- neighbourhood of the uniformly chosen vertex in the preferential attachment tree and that of the root vertex of its weak local limit. We apply the theorem to obtain the limiting degree distributions of the uniformly chosen vertex and its ancestors, that is, the vertices that are on the path between the uniformly chosen vertex and the initial vertex. Rates of convergence in the total variation distance are established for these results.