Geometry of weighted recursive and affine preferential attachment trees

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.