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PreprintCC BY 4.0arXiv

Simple Cuts, Simple Landscapes: Global Synchronization and Rank-Width-One Matrices

  • Hongjin Wu
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Abstract

We study how the structure of the coupling matrix A shapes the energy landscape EA(θ)=−1/2∑i,jAijcos(θi−θj) of the Kuramoto model. We prove that, for an entrywise nonnegative coupling matrix A with connected support, if A has rank-width one over ℝ, then the landscape is "simple", i.e., every local minimum is synchronized. The rank-width bound is sharp, and the result has many interesting implications for global synchronization. The key contribution of this work is to uncover a structural synchronization mechanism beyond the rank-one regime, which has served as a starting point for earlier analyses of global synchronization.

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Cite

BibTeXRIS
BibTeX
@misc{wu2026simple,
  title = {{Simple Cuts, Simple Landscapes: Global Synchronization and Rank-Width-One Matrices}},
  author = {Hongjin Wu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.38670},
  eprint = {2610.38670},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.38670},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Simple Cuts, Simple Landscapes: Global Synchronization and Rank-Width-One Matrices
AU  - Hongjin Wu
PY  - 2026
DA  - 2026/10/28
PB  - arXiv
DO  - 10.48550/arxiv.2610.38670
UR  - https://arxiv.org/abs/2610.38670
AB  - We study how the structure of the coupling matrix A shapes the energy landscape E_A(θ)=−1/2∑_i,jA_ijcos(θ_i−θ_j) of the Kuramoto model. We prove that, for an entrywise nonnegative coupling matrix A with connected support, if A has rank-width one over ℝ, then the landscape is "simple", i.e., every local minimum is synchronized. The rank-width bound is sharp, and the result has many interesting implications for global synchronization. The key contribution of this work is to uncover a structural synchronization mechanism beyond the rank-one regime, which has served as a starting point for earlier analyses of global synchronization.
ER  -