Phase Synchronization in Random Geometric Graphs on the 2D Sphere
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866918151960133632 |
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| author | De Vita, Cecilia Groisman, Pablo Huang, Ruojun |
| author_facet | De Vita, Cecilia Groisman, Pablo Huang, Ruojun |
| contents | The Kuramoto model is a classical nonlinear ODE system designed to study synchronization phenomena. Each equation represents the phase of an oscillator and the coupling between them is determined by a graph. There is an increasing interest in understanding the relation between the graph topology and the spontaneous synchronization of the oscillators. Abdalla, Bandeira and Invernizzi considered random geometric graphs on the $d$-dimensional sphere and proved that the system synchronizes with high probability as long as the mean number of neighbors and the dimension $d$ go to infinity. They posed the question about the behavior when $d$ is small. In this paper, we prove that synchronization holds for random geometric graphs on the two-dimensional sphere, with high probability as the number of nodes goes to infinity, as long as the initial conditions converge to a smooth function. We conjecture a similar behavior for more general simply-connected closed Riemannian manifolds but we expect global synchronization to fail if the manifold is not simply-connected, as was shown in [11] and suggested in [9]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_01151 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Phase Synchronization in Random Geometric Graphs on the 2D Sphere De Vita, Cecilia Groisman, Pablo Huang, Ruojun Probability The Kuramoto model is a classical nonlinear ODE system designed to study synchronization phenomena. Each equation represents the phase of an oscillator and the coupling between them is determined by a graph. There is an increasing interest in understanding the relation between the graph topology and the spontaneous synchronization of the oscillators. Abdalla, Bandeira and Invernizzi considered random geometric graphs on the $d$-dimensional sphere and proved that the system synchronizes with high probability as long as the mean number of neighbors and the dimension $d$ go to infinity. They posed the question about the behavior when $d$ is small. In this paper, we prove that synchronization holds for random geometric graphs on the two-dimensional sphere, with high probability as the number of nodes goes to infinity, as long as the initial conditions converge to a smooth function. We conjecture a similar behavior for more general simply-connected closed Riemannian manifolds but we expect global synchronization to fail if the manifold is not simply-connected, as was shown in [11] and suggested in [9]. |
| title | Phase Synchronization in Random Geometric Graphs on the 2D Sphere |
| topic | Probability |
| url | https://arxiv.org/abs/2504.01151 |