Uniform temporal trees

Abstract

Motivated by the study of random temporal networks, we introduce a class of random trees that we coin 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 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 T 𝑛,𝑝 which states that ∣T 𝑛,𝑝∣ 𝑒np converges in distribution to an Exponential(1) random variable as 𝑛 → ∞. For the height 𝐻𝑛,𝑝, we prove that 𝐻𝑛,𝑝 np converges to 𝑒 in probability. Uniform temporal trees show some remarkable similarities to uniform random recursive trees.

This work was supported by the Natural Sciences and Engineering Research Council of Canada (RGPIN-2024-04164); Spanish Ministry of Economy and Competitiveness (PID2022-138268NB-I00); NSERC Postgraduate Scholarship; Fundación BBVA a Proyectos de Investigación Científica 2021.