Mobile Geometric Scale-Free Random Graphs

Summary

In recent years scale-free geometric random graphs have been used extensively to study real world networks, such as telecommunication or social (media) networks. These models and their applications tend to be static in nature, with the nodes of the network (and their respective power) fixed in space and the connections between them not updating. Recent work has been done in order to introduce dynamics into the system, first by making the connections update over time (in non-geometric/non-spatial versions of such models) and more recently by making the nodes of the network move in space. Such dynamics break many of the standing assumptions for the static case and introduce correlations that make standard random graph theory techniques difficult to apply. Luckily, techniques developed for interacting particle systems provide a solid foundation on which random graph theory can be of use again. This project aims to continue the work in bridging the two worlds in order to explore many of the interesting and natural questions that arise from the setup, such as the following.

- The existence of connections (hereupon edges) between the nodes (hereupon vertices) is often more then just a deterministic function of the location of the vertices and their respective weights; it is instead random itself, with the probability of an edge existing increasing with proximity and the weight/influence of the two respective vertices. If vertices are mobile, the question of when and how edges update becomes relevant and warrants study.
- In the static case topological properties of the graph are of interest, such as the emergence of an infinite connected component and typical distances in the graph. If one looks at snapshots in the mobile case these properties are retained. It is however not clear whether like in the case of dynamical percolation, exceptional times with different properties can exist.
- The contact process has been studied on static scale-free geometric random graphs. For non-geometric models, results about the contact process are also known when edges update at various rates. First steps have been made for the mobile scale-free geometric random graphs in studying instantaneous propagation of information which indicate that many of the known techniques should be applicable here as well, but it is unclear whether the motion of the particles helps or hinders the contact process in its survival.

Literature

• P. Gracar and A. Grauer. Geometric scale-free random graphs on mobile vertices: broadcast and percolation times, 2024. (ArXiv preprint)
• P. Gracar and A. Grauer. The contact process on scale-free geometric random graphs. Stochastic Processes and their Applications, 173:104360, 2024.
• E. Jacob, A. Linker, and P. Mörters. The contact process on dynamical scale-free networks, 2022. (ArXiv preprint)
• Y. Peres, A. Sinclair, P. Sousi, and A. Stauffer. Mobile geometric graphs: Detection, coverage and percolation. In Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms. Society for Industrial and Applied Mathematics, 2011.

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