Beyond 11/16: dense Kuramoto networks that resist synchrony
Abstract
How densely must a network of identical Kuramoto oscillators be wired before synchrony is its only stable state? The answer is a critical connectivity μc (the minimum degree divided by n−1), known to satisfy 11/16≤μc<3/4. We raise the lower bound from 11/16=0.6875 to μc≥(4+2q*)/(7+q*)=0.6929077…, where q*≈0.6506 is a root of an explicit polynomial of degree 8. The networks approaching this bound carry stable states that hug the four-cluster state: three neighboring clusters each split into two slightly separated halves, and the four crossed links between neighboring halves (those whose phase gap deviates most from a quarter turn) have densities approaching q*. We build these networks from a coarse-grained cluster model, verify stability with interval arithmetic, and blow the model up into large networks with an exact minimum degree. The states are fragile and easily missed by numerical searches that impose stability margins.
Related
- Perturbation responses on topological synchrony in simplicial Kuramoto model
- Phase-delays shape multistability and basin sizes in Kuramoto networks: analytical estimates from network structure
- Existence and Stability of Dancing Equilibria in Asymmetric Kuramoto Networks
- Boltzmann-like occupation of nonequilibrium steady states on dense disordered networks
- Deep Learning-Based Multi-User Communication Design for Dense IoT Networks: Interference-Aware Finite-Blocklength Communication and Preliminary MIMO Extensions
BibTeX
@misc{okeeffe2026beyond,
title = {{Beyond 11/16: dense Kuramoto networks that resist synchrony}},
author = {Kevin P. O'Keeffe},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.21271},
eprint = {2610.21271},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.21271},
}
RIS
TY - UNPB
TI - Beyond 11/16: dense Kuramoto networks that resist synchrony
AU - Kevin P. O'Keeffe
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.21271
UR - https://arxiv.org/abs/2610.21271
AB - How densely must a network of identical Kuramoto oscillators be wired before synchrony is its only stable state? The answer is a critical connectivity μ_c (the minimum degree divided by n−1), known to satisfy 11/16≤μ_c<3/4. We raise the lower bound from 11/16=0.6875 to μ_c≥(4+2q^*)/(7+q^*)=0.6929077…, where q^*≈0.6506 is a root of an explicit polynomial of degree 8. The networks approaching this bound carry stable states that hug the four-cluster state: three neighboring clusters each split into two slightly separated halves, and the four crossed links between neighboring halves (those whose phase gap deviates most from a quarter turn) have densities approaching q^*. We build these networks from a coarse-grained cluster model, verify stability with interval arithmetic, and blow the model up into large networks with an exact minimum degree. The states are fragile and easily missed by numerical searches that impose stability margins.
ER -