Uniform temporal trees
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916577791705088 |
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| author | Atamanchuk, Caelan Devroye, Luc Lugosi, Gabor |
| author_facet | Atamanchuk, Caelan Devroye, Luc Lugosi, Gabor |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_13044 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Uniform temporal trees Atamanchuk, Caelan Devroye, Luc Lugosi, Gabor Probability 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. |
| title | Uniform temporal trees |
| topic | Probability |
| url | https://arxiv.org/abs/2501.13044 |