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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2404.15124 |
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| _version_ | 1866929325552435200 |
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| author | Gracar, Peter Grauer, Arne |
| author_facet | Gracar, Peter Grauer, Arne |
| contents | We study the phenomenon of information propagation on mobile geometric scale-free random graphs, where vertices instantaneously pass on information to all other vertices in the same connected component. The graphs we consider are constructed on a Poisson point process of intensity $λ>0$, and the vertices move over time as simple Brownian motions on either $\mathbb{R}^d$ or the $d$-dimensional torus of volume $n$, while edges are randomly drawn depending on the locations of the vertices, as well as their a priori assigned marks. This includes mobile versions of the age-dependent random connection model and the soft Boolean model. We show that in the ultrasmall regime of these random graphs, information is broadcast to all vertices on a torus of volume $n$ in poly-logarithmic time and that on $\mathbb{R}^d$, the information will reach the infinite component before time $t$ with stretched exponentially high probability, for any $λ>0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_15124 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Geometric scale-free random graphs on mobile vertices: broadcast and percolation times Gracar, Peter Grauer, Arne Probability 05C80 (Primary), 82C22 (Secondary) We study the phenomenon of information propagation on mobile geometric scale-free random graphs, where vertices instantaneously pass on information to all other vertices in the same connected component. The graphs we consider are constructed on a Poisson point process of intensity $λ>0$, and the vertices move over time as simple Brownian motions on either $\mathbb{R}^d$ or the $d$-dimensional torus of volume $n$, while edges are randomly drawn depending on the locations of the vertices, as well as their a priori assigned marks. This includes mobile versions of the age-dependent random connection model and the soft Boolean model. We show that in the ultrasmall regime of these random graphs, information is broadcast to all vertices on a torus of volume $n$ in poly-logarithmic time and that on $\mathbb{R}^d$, the information will reach the infinite component before time $t$ with stretched exponentially high probability, for any $λ>0$. |
| title | Geometric scale-free random graphs on mobile vertices: broadcast and percolation times |
| topic | Probability 05C80 (Primary), 82C22 (Secondary) |
| url | https://arxiv.org/abs/2404.15124 |