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Synchronization is central to many complex systems in engineering physics(e.g., the power-grid, Josephson junction circuits, and electro-chemicaloscillators) and biology (e.g., neuronal, circadian, and cardiac rhythms).Despite these widespread applications---for which proper functionality dependssensitively on the extent of synchronization---there remains a lack ofunderstanding for how systems evolve and adapt to enhance or inhibitsynchronization. We study how network modifications affect the synchronizationproperties of network-coupled dynamical systems that have heterogeneous nodedynamics (e.g., phase oscillators with non-identical frequencies), which isoften the case for real-world systems. Our approach relies on a synchronyalignment function (SAF) that quantifies the interplay between heterogeneity ofthe network and of the oscillators and provides an objective measure for asystem's ability to synchronize. We conduct a spectral perturbation analysis ofthe SAF for structural network modifications including the addition and removalof edges, which subsequently ranks the edges according to their importance tosynchronization. Based on this analysis, we develop gradient-descent algorithmsto efficiently solve optimization problems that aim to maximize phasesynchronization via network modifications. We support these and other resultswith numerical experiments.

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