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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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Table of 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 $λ>0$.