Fluctuations in a general preferential attachment model via Stein's method

TL;DR

This paper analyzes a general preferential attachment model with sublinear connection probabilities, providing convergence rates to the limiting distribution using Stein's method and Markov chain techniques.

Contribution

It introduces a novel application of Stein's method to quantify convergence rates in a flexible preferential attachment model with diverse behaviors.

Findings

Convergence rates in total variation distance are established.

The limiting distribution is characterized as the stationary distribution of a Markov chain.

The model exhibits power law and stretched exponential behaviors depending on parameters.

Abstract

We consider a general preferential attachment model, where the probability that a newly arriving vertex connects to an older vertex is proportional to a sublinear function of the indegree of the older vertex at that time. It is well known that the distribution of a uniformly chosen vertex converges to a limiting distribution. Depending on the parameters, this model can show power law, but also stretched exponential behaviour. Using Stein's method we provide rates of convergence for the total variation distance. Our proof uses the fact that the limiting distribution is the stationary distribution of a Markov chain together with the generator method of Barbour.