Vertices of high degree in the preferential attachment tree

TL;DR

This paper analyzes the degree distribution of vertices in a preferential attachment tree, showing concentration results for the number of vertices of each degree, especially for high-degree vertices as the tree grows.

Contribution

It provides new concentration results for the number of vertices of each degree in the preferential attachment process, especially for high degrees growing with the tree size.

Findings

Number of vertices of degree t/\u03bb^3 concentrates around its mean.

Concentration holds for degrees up to (t/log t)^{-1/3}.

Results are optimal up to a logarithmic factor.

Abstract

We study the basic preferential attachment process, which generates a sequence of random trees, each obtained from the previous one by introducing a new vertex and joining it to one existing vertex, chosen with probability proportional to its degree. We investigate the number $D_t(\ell)$ of vertices of each degree $\ell$ at each time $t$, focussing particularly on the case where $\ell$ is a growing function of $t$. We show that $D_t(\ell)$ is concentrated around its mean, which is approximately $4t/\ell^3$, for all $\ell \le (t/\log t)^{-1/3}$; this is best possible up to a logarithmic factor.