Analytical Prediction of Critical Transitions in Oscillator Networks With Non-Local Links

A wide range of dynamical systems can be modeled as oscillators coupled through an interaction network. Such frameworks have been widely used to explore the emergence of collective phenomena across diverse applications. Here, we develop an analytical framework for oscillator-based Ising systems with Kuramoto-type pairwise coupling and a second harmonic term. Spectral dimension reduction identifies a stability boundary governed by the balance between global coupling and second-harmonic strength, while a mean-field Fokker-Planck analysis predicts the transition between single-cluster and two-cluster stationary states and clarifies the role of noise. Low-energy Ising configurations are predominantly obtained when the second harmonic term is strong relative to noise. Partition-function analysis further shows that, for the ring-based topology, links to third neighbors yield more desired solutions than the other added non-local connections. Applications to graph coloring and MaxCut demonstrate the relevance of these predictions to oscillator-based Ising optimization. These results provide a predictive framework for understanding how dynamical parameters, noise, and network structure jointly shape collective states and low-energy configurations.