The random graph process is globally synchronizing
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909462921478144 |
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| author | Jain, Vishesh Mizgerd, Clayton Sawhney, Mehtaab |
| author_facet | Jain, Vishesh Mizgerd, Clayton Sawhney, Mehtaab |
| contents | The homogeneous Kuramoto model on a graph $G = (V,E)$ is a network of $|V|$ identical oscillators, one at each vertex, where every oscillator is coupled bidirectionally (with unit strength) to its neighbors in the graph. A graph $G$ is said to be globally synchronizing if, for almost every initial condition, the homogeneous Kuramoto model converges to the all-in-phase synchronous state. Confirming a conjecture of Abdalla, Bandeira, Kassabov, Souza, Strogatz, and Townsend, we show that with high probability, the random graph process becomes globally synchronizing as soon as it is connected. This is best possible, since connectivity is a necessary condition for global synchronization. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_12205 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The random graph process is globally synchronizing Jain, Vishesh Mizgerd, Clayton Sawhney, Mehtaab Combinatorics Probability The homogeneous Kuramoto model on a graph $G = (V,E)$ is a network of $|V|$ identical oscillators, one at each vertex, where every oscillator is coupled bidirectionally (with unit strength) to its neighbors in the graph. A graph $G$ is said to be globally synchronizing if, for almost every initial condition, the homogeneous Kuramoto model converges to the all-in-phase synchronous state. Confirming a conjecture of Abdalla, Bandeira, Kassabov, Souza, Strogatz, and Townsend, we show that with high probability, the random graph process becomes globally synchronizing as soon as it is connected. This is best possible, since connectivity is a necessary condition for global synchronization. |
| title | The random graph process is globally synchronizing |
| topic | Combinatorics Probability |
| url | https://arxiv.org/abs/2501.12205 |