Coupling dynamics of fluctuating-damping oscillators in Erdős-Rényi random networks

The intrinsic randomness of network topology and individual-level environmental perturbations are fundamental determinants of the diversity of collective dynamics in coupled systems, particularly in producing complex behaviors that elude deterministic frameworks. In this study, we investigate the coupling dynamics of fluctuating-damping oscillators on Erdős-Rényi (ER) random networks, wherein randomness arises from both the network structure and time-varying local dynamics. We begin by employing random matrix theory (RMT) to characterize structural randomness and examine how the interplay between network characteristics and damping fluctuations influences synchronization behavior. Specifically, we derive the condition for stochastic asymptotic synchronization and quantify its probability based on the spectral distribution of random Laplacian matrix. Our analysis demonstrates that greater noise intensity or lower switching rates hinder synchronization, whereas stronger coupling strength, larger network scale, and higher wiring probability promote it. Within the synchronization regime, we further establish conditions for stochastic asymptotic stability and analytically characterize the steady-state response of collective behavior, providing explicit expressions for the first moment of the mean field. Numerical simulations validate the theoretical findings and reveal non-monotonic dependencies of the synchronization process on key structural parameters, including coupling strength, network scale, and wiring probability. These results elucidate nonlinear, parameter-sensitive mechanisms by which topological characteristics and damping fluctuations jointly govern collective dynamics. Moreover, this study offers a potential methodological framework for investigating coupling dynamics in broader classes of complex networks through the integration of RMT.