A Unified Theory of Edge Weights: Stability of General Laplacian Networks from Matrix Phases and Asymmetry Rayleigh Ratios
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866911607899029504 |
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| author | Kastendiek, Nina Niehues, Jakob Hellmann, Frank |
| author_facet | Kastendiek, Nina Niehues, Jakob Hellmann, Frank |
| contents | We study the properties and stability of networks with arbitrary Laplacian coupling. Classic approaches to studying networked systems require unrealistic assumptions, including homogeneous node dynamics, one-dimensional and undirected edges, or constant edge weights. We develop a unified formulation of Laplacian-style couplings that drops these assumptions, providing a unified notion for the edge weights of adaptive, directed, and multi-dimensional edges. We show that the recently developed theory of matrix phases can capture essential stability properties of the network and its edges. We quantify the impact of the asymmetry of the higher-dimensional edge dynamics on the system's phase properties by introducing the Asymmetry Rayleigh Ratio. These theoretical advances allow us to derive new sufficient stability conditions for AC power grids, directed diffusion, and the Kuramoto-Sakaguchi model. The resulting conditions are less conservative than the specific results known for these systems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_18017 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Unified Theory of Edge Weights: Stability of General Laplacian Networks from Matrix Phases and Asymmetry Rayleigh Ratios Kastendiek, Nina Niehues, Jakob Hellmann, Frank Adaptation and Self-Organizing Systems Dynamical Systems We study the properties and stability of networks with arbitrary Laplacian coupling. Classic approaches to studying networked systems require unrealistic assumptions, including homogeneous node dynamics, one-dimensional and undirected edges, or constant edge weights. We develop a unified formulation of Laplacian-style couplings that drops these assumptions, providing a unified notion for the edge weights of adaptive, directed, and multi-dimensional edges. We show that the recently developed theory of matrix phases can capture essential stability properties of the network and its edges. We quantify the impact of the asymmetry of the higher-dimensional edge dynamics on the system's phase properties by introducing the Asymmetry Rayleigh Ratio. These theoretical advances allow us to derive new sufficient stability conditions for AC power grids, directed diffusion, and the Kuramoto-Sakaguchi model. The resulting conditions are less conservative than the specific results known for these systems. |
| title | A Unified Theory of Edge Weights: Stability of General Laplacian Networks from Matrix Phases and Asymmetry Rayleigh Ratios |
| topic | Adaptation and Self-Organizing Systems Dynamical Systems |
| url | https://arxiv.org/abs/2604.18017 |