Collective dynamical behaviors of coupled Langevin oscillators with damping fluctuation in attractive–repulsive weighted random networks

The randomness that arises from both the network topology and individual-level environmental fluctuations plays a crucial role in shaping the collective dynamics of coupled systems. Since many real-world systems are random, weighted, and often incorporate both attractive and repulsive couplings, understanding how structural randomness interacts with local perturbations is essential for revealing mechanisms of collective dynamics. In this work, we investigate coupled Langevin oscillators with fluctuating damping on weighted random networks that incorporate both attractive and repulsive couplings, where structural and dynamical uncertainties coexist. To analyze the collective behavior, we combine random matrix theory (RMT) to derive stochastic asymptotic synchronization criteria and probabilistic estimates informed by the eigenvalue statistics of random Laplacian matrices. In the synchronized state, we additionally derive stochastic asymptotic stability conditions and provide analytical formulations for the steady-state collective response. Theoretical analysis shows that stronger weight expectation, weaker weight variance, and larger network scale facilitate synchronization, while higher noise intensity and lower switching rates suppress it. Numerical experiments not only confirm the theoretical results but also demonstrate that synchronization time exhibits non-monotonic dependence on network characteristics and noise parameters. Overall, the present work introduces a unified analytical framework for investigating coupled blue oscillators with damping fluctuations on random weighted networks, advancing the understanding of how network structure and perturbations jointly shape synchronization and stability in complex systems.