The lower bound of the network connectivity guaranteeing in-phase synchronization

In-phase synchronization is a stable state of identical Kuramoto oscillators coupled on a network with identical positive connections, regardless of network topology. However, this fact does not mean that the networks always synchronize in-phase because other attractors besides the stable state may exist. The critical connectivity $\mu_{\mathrm{c}}$ is defined as the network connectivity above which only the in-phase state is stable for all the networks. In other words, below $\mu_{\mathrm{c}}$, one can find at least one network which has a stable state besides the in-phase sync. The best known evaluation of the value so far is $0.6828\cdots\leq\mu_{\mathrm{c}}\leq0.75$. In this paper, focusing on the twisted states of the circulant networks, we provide a method to systematically analyze the linear stability of all possible twisted states on all possible circulant networks. This method using integer programming enables us to find the densest circulant network having a stable twisted state besides the in-phase sync, which breaks a record of the lower bound of the $\mu_{\mathrm{c}}$ from $0.6828\cdots$ to $0.6838\cdots$. We confirm the validity of the theory by numerical simulations of the networks not converging to the in-phase state.