Dense networks that do not synchronize and sparse ones that do

Consider any network of $n$ identical Kuramoto oscillators in which each oscillator is coupled bidirectionally with unit strength to at least $μ(n−1)$ other oscillators. Then, there is a critical value of $μ$ above which the system is guaranteed to converge to the in-phase synchronous state for almost all initial conditions. The precise value of $μ$ remains unknown. In 2018, Ling, Xu, and Bandeira proved that if each oscillator is coupled to at least 79.29% of all the others, global synchrony is ensured. In 2019, Lu and Steinerberger improved this bound to 78.89%. Here, we find clues that the critical connectivity may be exactly 75%. Our methods yield a slight improvement on the best known lower bound on the critical connectivity from $68.18%$ to $68.28%$⁠. We also consider the opposite end of the connectivity spectrum, where the networks are sparse rather than dense. In this regime, we ask how few edges one needs to add to a ring of $n$ oscillators to turn it into a globally synchronizing network. We prove a partial result: all the twisted states in a ring of size $n=2m$ can be destabilized by adding just $O(nlog2⁡n)$ edges. To finish the proof, one needs to rule out all other candidate attractors. We have done this for $n≤8$ but the problem remains open for larger $n$⁠. Thus, even for systems as simple as Kuramoto oscillators, much remains to be learned about dense networks that do not globally synchronize and sparse ones that do.