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Abstract

A complex system’s structural-dynamical interplay plays a profound role in determining its collective
behavior. Irregular behavior in the form of macroscopic chaos, for instance, can be potentially
exhibited by the Kuramoto model of coupled phase oscillators at intermediate coupling strength with
frequency assortativity and this behavior is theoretically interesting. In practice, however, such irregular
behavior is often not under control and is undesired for the system’s functioning. How the
underlying structural and oscillators’ dynamical interplay affects a collective phenomenon (and its
corresponding stability) after being subjected to disturbances, attracts great attention. Here, we
exploit the concept of a coherency performance metric, as a sum of phase differences and frequency
displacements, to evaluate the response to perturbations on network-coupled oscillators. We derive
the performance metric as a quadratic form of the eigenvalues and eigenmodes corresponding to
the unperturbed system and the perturbation vector, and analyze the influences of perturbation
direction as well as strength on the metric. We further apply a computational approach to obtain
the performance metric’s derivative with respect to the oscillators’ inertia. We finally extend the
metric to a local definition which reflects the pairwise casual effects between any two oscillators.
These results deepen the understanding of the combined effects of the structural (eigenmodes) and
dynamical (inertia) effects on the system stability.