Uniform temporal trees

Abstract

Motivated by the study of random temporal networks, we introduce a class of random trees that we coin \emph{uniform temporal trees}. A uniform temporal tree is obtained by assigning independent uniform $[0,1]$ labels to the edges of a rooted complete infinite $n$-ary tree and keeping only those vertices for which the path from the root to the vertex has decreasing edge labels. The $p$-percolated uniform temporal tree, denoted by $\mathcal{T}_{n,p}$, is obtained similarly, with the additional constraint that the edge labels on each path are all below $p$. We study several properties of these trees, including their size, height, the typical depth of a vertex, and degree distribution. In particular, we establish a limit law for the size of $\mathcal{T}_{n,p}$ which states that $\frac{|\mathcal{T}_{n,p}|}{e^{np}}$ converges in distribution to an $\exponential(1)$ random variable as $n \to \infty$. For the height $H_{n,p}$, we prove that $\frac{H_{n,p}}{np}$ converges to $e$ in probability. Uniform temporal trees show some remarkable similarities to uniform random recursive trees.