Uniform temporal trees

TL;DR

This work introduces uniform temporal trees, random rooted trees formed by marking edges with independent Uniform[0,1] labels and retaining only root-to-vertex paths that decrease along the labels, with a p-percolated variant T_{n,p}. The authors develop a uniform spacings coupling to translate label evolution into a branching-random-walk framework, enabling precise limit laws for key quantities. They show that the size, height, typical depth, and degree distribution of T_{n,p} exhibit sharp asymptotics: the size normalized by e^{np} converges to an Exponential(1) variable, the height scales linearly with np with constant e, typical depths concentrate around np on the same scale, and the asymptotic degree distribution is geometric with parameter 1/2. These results reveal close qualitative and quantitative parallels between uniform temporal trees and uniform random recursive trees, with explicit limiting distributions and moment bounds that enhance understanding of random temporal networks.

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.

Figures (4)

• Figure 1: A uniform temporal tree with $n=10$.
• Figure 2: The rotation of the uniform spacings around a vertex $x$. The blue section above $x$ is moved from above $x$ to below $0$. After the segment is moved the points are distributed uniformly over $[x-1,x]$.
• Figure 3: The evolution of labels in a $\mathcal{T}_{n,p}$ according to the spacings coupling. The random variables $S_1,S_2,S_1',S_2'$ are all uniform spacings. The label of a vertex is the label of its next lower-rank sibling (or parent if its rank is $1$) minus a spacing.
• Figure 4: The mapping $\phi$ up to index $L=3$. The left tree is $\mathcal{T}_{n,p}$ and the right is the binary tree $\mathcal{T}^*$ with the labelling obtained from $\phi$. The vertices are ordered from left to right in order of increasing index in $\mathcal{T}_{n,p}$. A left child (blue edge) in $\mathcal{T}^*$ corresponds to moving down to the vertex's child of the smallest index in $\mathcal{T}_{n,p}$, and a right child (red edge) corresponds to moving to a vertex's sibling of the smallest index in $\mathcal{T}_{n,p}$.

Theorems & Definitions (17)

• Theorem 2.1
• Remark
• Theorem 2.2
• Theorem 2.3
• Theorem 2.4
• Lemma 3.1
• proof
• Lemma 4.1
• Lemma 4.2
• proof : Proof of Lemma \ref{['branchrandwalk']}