A central limit theorem for the number of isolated vertices in a preferential attachment random graph

TL;DR

This paper establishes a central limit theorem for the count of isolated vertices in a preferential attachment graph, demonstrating that the rescaled number converges to a Gaussian distribution using Stein's method.

Contribution

It introduces a novel application of Stein's method and size-bias coupling to analyze the distribution of isolated vertices in a preferential attachment model.

Findings

Bounds in Wasserstein distance to Gaussian distribution

Asymptotic normality of the number of isolated vertices

Quantitative convergence rates

Abstract

We study the number of isolated vertices in a preferential attachment random graph introduced by Dereich and M\"orters in 2009. In this graph model vertices are added over time and newly arriving vertices connect to older ones with probability proportional to a (sub-)linear function of the indegree of the older vertex at that time. Using Stein's method and a size-bias coupling, we deduce bounds in the Wasserstein distance between the law of the properly rescaled number of isolated vertices and a standard Gaussian distribution.