Synchronization of heterogeneous oscillators under network modifications: Perturbation and optimization of the synchrony alignment function

Synchronization is central to many complex systems in engineering physics (e.g., the power grid, Josephson junction circuits, and electrochemical oscillators) and biology (e.g., neuronal, circadian, and cardiac rhythms). Despite these widespread applications - for which proper functionality depends sensitively on the extent of synchronization - there remains a lack of understanding for how systems can best evolve and adapt to enhance or inhibit synchronization. We study how network modifications affect the synchronization properties of network-coupled dynamical systems that have heterogeneous node dynamics (e.g., phase oscillators with nonidentical frequencies), which is often the case for real-world systems. Our approach relies on a synchrony alignment function (SAF) that quantifies the interplay between heterogeneity of the network and of the oscillators and provides an objective measure for a system's ability to synchronize. We conduct a spectral perturbation analysis of the SAF for structural network modifications including the addition and removal of edges, which subsequently ranks the edges according to their importance to synchronization. Based on this analysis, we develop gradient-descent algorithms to efficiently solve optimization problems that aim to maximize phase synchronization via network modifications. We support these and other results with numerical experiments.