# Geometry of weighted recursive and affine preferential attachment trees

## TL;DR

This paper analyzes the geometry of weighted recursive and affine preferential attachment trees, establishing their equivalence and deriving asymptotic properties of various tree statistics.

## Contribution

It shows that affine preferential attachment trees can be represented as weighted recursive trees with random weights and proves convergence results for key tree metrics.

## Key findings

- Affine preferential attachment trees are distributionally equivalent to weighted recursive trees with random weights.
- Established almost sure scaling limits for degree sequences, height, and profile of the trees.
- Proved weak convergence of measures associated with the tree structure.

## Abstract

We study two models of growing recursive trees. For both models, initially the tree only contains one vertex $u_1$ and at each time $n\geq 2$ a new vertex $u_n$ is added to the tree and its parent is chosen randomly according to some rule. In the \emph{weighted recursive tree}, we choose the parent $u_k$ of $u_n$ among $\{u_1,u_2,\dots, u_{n-1}\}$ with probability proportional to $w_k$, where $(w_n)_{n\geq1}$ is some deterministic sequence that we fix beforehand. In the \emph{affine preferential attachment tree with fitnesses}, the probability of choosing any $u_k$ is proportional to $a_k+\mathrm{deg}^{+}(u_k)$, where $\mathrm{deg}^{+}(u_k)$ denotes its current number of children, and the sequence of \emph{fitnesses} $(a_n)_{n\geq 1}$ is deterministic and chosen as a parameter of the model. We show that for any sequence $(a_n)_{n\geq 1}$, the corresponding preferential attachment tree has the same distribution as some weighted recursive tree with a \emph{random} sequence of weights (with some explicit distribution). We then prove almost sure scaling limit convergences for some statistics associated with weighted recursive trees as time goes to infinity, such as degree sequence, height, profile and also the weak convergence of some measures carried on the tree. Thanks to the connection between the two models, these results also apply to affine preferential attachment trees.