## Expander graphs are globally synchronising

### With Victor Souza, Cambridge

# Expander graphs are globally synchronising

The Kuramoto model is a prototypical model used for rigorous mathematical analysis in the field of synchronisation and nonlinear dynamics. A realisation of this model consists of a collection of identical oscillators with interactions given by a network, which we identify respectively with vertices and edges of a graph. We show that a graph with sufficient expansion must be globally

synchronising, meaning that the Kuramoto model on such a graph will converge to the fully synchronised state with all the oscillators with same phase, for every initial state up to a set of measure zero. In particular, we show that for p ≥ (1 + eps)(log n)/n, the Kuramoto

model on the Erdős—Rényi graph G(n,p) is globally synchronising with high probability, settling a conjecture of Ling, Xu and Bandeira. We also show the global synchrony of any d-regular Ramanujan graph with d ≥ 600.

Joint work with P. Abdalla, A. Bandeira, M. Kassabov, S. Strogatz and A. Townsend.

- Speaker: Victor Souza, Cambridge
- Monday 21 November 2022, 16:00–17:00
- Venue: MR14, Centre for Mathematical Sciences.
- Series: Probability; organiser: Roland Bauerschmidt.