Leaves of preferential attachment trees

TL;DR

This paper characterizes the joint distribution of degree and leafdegree in large preferential attachment trees, providing detailed probabilistic descriptions and extending the approach to other models like recursive trees and redirection models.

Contribution

It introduces a comprehensive probabilistic framework for leafdegree statistics in preferential attachment trees, applicable to various models and enabling new tractable analyses.

Findings

Derived the joint degree-leafdegree distribution $n_{k, au}$ from its generating function.

Obtained leafdegree distribution $m_{ au}$ and fraction of protected vertices $n_{k,0}$.

Analyzed fluctuations and concentration of empirical counts $N_{k, au}$.

Abstract

We provide a local probabilistic description of the limiting statistics of large preferential attachment trees in terms of the ordinary degree (number of neighbors) but augmented with information on leafdegree (number of neighbors that are leaves). The full description is the joint degree-leafdegree distribution $n_{k,\ell}$, which we derive from its associated multivariate generating function. From $n_{k,\ell}$ we obtain the leafdegree distribution, $m_{\ell}$, as well as the fraction of vertices that are protected (nonleaves with leafdegree zero) as a function of degree, $n_{k,0}$, among numerous other results. We also examine fluctuations and concentration of joint degree-leafdegree empirical counts $N_{k,\ell}$. Although our main findings pertain to the preferential attachment tree, the approach we present is highly generalizable and can characterize numerous existing models, in addition to facilitating the development of tractable new models. We further demonstrate the approach by analyzing $n_{k,\ell}$ in two other models: the random recursive tree, and a redirection-based model.