The number of descendants in a preferential attachment graph

TL;DR

This paper analyzes the asymptotic distribution of the number of descendants of the last added vertex in a preferential attachment graph, revealing a convergence to a scaled Gamma distribution influenced by model parameters.

Contribution

It establishes the limiting distribution of descendants in preferential attachment graphs using a Pólya urn approach, extending previous uniform attachment results.

Findings

The scaled number of descendants converges in distribution to a product involving a Gamma distribution.

The limiting distribution depends on the parameters m and ρ of the model.

Results include convergence of all moments and extensions to graphs with self-loops.

Abstract

We study the number $X^{(n)}$ of vertices that can be reached from the last added vertex $n$ via a directed path (the descendants) in the standard preferential attachment graph. In this model, vertices are sequentially added, each born with outdegree $m\ge 2$; the endpoint of each outgoing edge is chosen among previously added vertices with probability proportional to the current degree of the vertex plus some number $\rho$. We show that $X^{(n)}/n^\nu$ converges in distribution as $n\to\infty$, where $\nu$ depends on both $m$ and $\rho$, and the limiting distribution is given by a product of a constant factor and the $(1-\nu)$-th power of a Gamma(m/(m-1),1) variable. The proof uses a P\'olya urn representation of preferential attachment graphs, and the arguments of Janson (2024) where the same problem was studied in uniform attachment graphs. Further results, including convergence of all moments and analogues for the version with possible self-loops are provided.