# Critical synchronization dynamics of the Kuramoto model on connectome and small world graphs

arXiv: https://arxiv.org/abs/1903.00385 · HTML: https://mlsanity.com/paper/1903.00385/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 Mar 2019.

## Built on by (most cited first, 2 of 2)

- [Enhancing deep neural networks through complex-valued representations and Kuramoto synchronization dynamics](https://mlsanity.com/paper/2502.21077/) (2025)
- [Connecting the geometry and dynamics of many-body complex systems with message passing neural operators](https://mlsanity.com/paper/2502.15913/) (2025)

## Abstract (arXiv)

The hypothesis, that cortical dynamics operates near criticality also suggests, that it exhibits universal critical exponents which marks the Kuramoto equation, a fundamental model for synchronization, as a prime candidate for an underlying universal model. Here, we determined the synchronization behavior of this model by solving it numerically on a large, weighted human connectome network, containing 804092 nodes, in an assumed homeostatic state. Since this graph has a topological dimension d < 4, a real synchronization phase transition is not possible in the thermodynamic limit, still we could locate a transition between partially synchronized and desynchronized states. At this crossover point we observe power-law--tailed synchronization durations, with τt simeq 1.2(1), away from experimental values for the brain. For comparison, on a large two-dimensional lattice, having additional random, long-range links, we obtain a mean-field value: τt simeq 1.6(1). However, below the transition of the connectome we found global coupling control-parameter dependent exponents 1 < τt ≤ 2, overlapping with the range of human brain experiments. We also studied the effects of random flipping of a small portion of link weights, mimicking a network with inhibitory interactions, and found similar results. The control-parameter dependent exponent suggests extended dynamical criticality below the transition point.