Circulant networks of identical Kuramoto oscillators: Seeking dense networks that do not globally synchronize and sparse ones that do
There is a critical connectivity $0<\mu_c<1$ for systems of identical Kuramoto oscillators. Any network of size $n$ in which each oscillator interacts with at least $\mu(n-1)$ others is globally synchronizing if $\mu\geq\mu_c$; otherwise, it may not be. The best known bounds on $\mu_c$ are $15/22 \approx 0.6818 \leq \mu_c\leq (3-\sqrt{2})/2 \approx 0.7929$. Focusing on circulant networks leads us to conjecture that $\mu_c = 0.75$. At the sparse end of the connectivity spectrum, we find that a ring of $n$ oscillators can be turned into a globally synchronizing network by adding just $\mathcal{O}(n \log_2 n)$ edges.