Optimal design of higher-order interactions for mutual synchronization among three collectively oscillating networks

Many networked dynamical systems consist of interacting subnetworks, each of which can exhibit collective oscillations. Mutual synchronization among such collectively oscillating networks is ubiquitously observed, and its efficient control is an important engineering problem. While control of synchronization has been extensively studied for pairwise interactions, many systems also involve higher-order interactions among three or more subnetworks that cannot be decomposed into independent pairwise couplings. In this study, we investigate the optimization of higher-order internetwork coupling for mutual synchronization among three collectively oscillating networks. Using phase reduction theory for collectively oscillating networks, we reduce the full network dynamics to a two-dimensional system of phase-difference equations, in which the linear stability of the phase-locked state and the rotational property of the phase differences around it are characterized by a higher-order coupling tensor shared by the three networks. We then derive the optimal coupling tensors that maximize either the linear stability or the rotational property for a fixed overall strength of the internetwork coupling. As a numerical example, we consider three collectively oscillating networks, each consisting of four FitzHugh-Nagumo elements coupled in a tree structure, and demonstrate that, compared with a random coupling tensor of the same overall strength, the two types of optimal coupling yield faster convergence to the phase-locked state and stronger rotational motion of the phase differences around it, respectively.