Thread by @stevenstrogatz

I’m thrilled about this preprint w Martin Kassabov and Alex Townsend. We prove that a network of identical Kuramoto oscillators synchronizes —regardless of the details of its wiring diagram — if every oscillator is connected to at least 75% of the others. arxiv.org/abs/2105.11406

This puzzle has fascinated a lot of us in my little corner of nonlinear dynamics since 2012. What’s the smallest level of connectivity that guarantees that a homogeneous Kuramoto model (the simplest kind of oscillator system) will always fall into sync?

We still don’t know the answer. The magic level of connectivity was previously proven to lie between 68.38 and 78.89%, and conjectured to be exactly 75%. In this preprint we’ve now reduced the upper bound to 75%. But that’s still far away from the best known lower bound.

The gap between the upper and lower bound is tantalizing. We argue in the paper that the gap may be unbridgeable by linear stability analysis; there may be a minefield of states in the gap whose linear stability is marginal. If so, they’ll require a delicate nonlinear analysis.

In any case, we know for certain that the upper bound of 0.75 is the best one can achieve by linear arguments. If the actual value of the critical connectivity turns out to be < 0.75, that will require a nonlinear argument for its proof. (We explain why in the paper.)

Caveat: our results are only for identical Kuramoto oscillators. If the oscillators are not identical, or if they are other types of dynamical systems, our analysis does not apply. Almost nothing is known about networks of generic dynamical systems. Beware of generalizing here!