  For any network of identical Kuramoto oscillators with identical positive
coupling, there is a critical connectivity above which the system is guaranteed
to converge to the in-phase synchronous state, for almost all initial
conditions. But the precise value of this critical connectivity remains
unknown. In 2018, Ling, Xu, and Bandeira proved that if each oscillator is
coupled to at least 79.29 percent of all the others, global synchrony is
ensured. In 2019, Lu and Steinerberger improved this bound to 78.89 percent.
Here, by focusing on circulant networks, we find clues that the critical
connectivity may be exactly 75 percent. 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 $n=2^m$ can be destabilized by adding just $\mathcal{O}(n \log_2
n)$ edges. To finish the proof, one also needs to rule out all other candidate
attractors. We have done this for $n=8$ with computational algebraic geometry,
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.
