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

TL;DR

This paper investigates how network modifications influence synchronization in heterogeneous oscillator systems, introducing a spectral perturbation approach and optimization algorithms to enhance synchronization in complex networks.

Contribution

It develops a spectral perturbation method for the synchrony alignment function and creates gradient-based algorithms to optimize network structure for better synchronization.

Findings

Spectral perturbation analysis ranks edges by importance to synchronization.

Optimization algorithms effectively enhance synchronization through network modifications.

Numerical experiments validate the proposed methods.

Abstract

Synchronization is central to many complex systems in engineering physics (e.g., the power-grid, Josephson junction circuits, and electro-chemical 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 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 non-identical 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.