Nonlinear Opinion Dynamics of Stubborn Individuals in Time-Varying Cooperation–Competition Networks

We study a nonlinear opinion dynamics model for stubborn individuals in time-varying signed cooperation–competition networks. The model incorporates nonlinear response functions, stochastic perturbations, and memory effects within a unified dynamical framework. We establish a uniform boundedness result under bounded external perturbations and derive a sufficient condition for contraction of trajectory differences based on the Lipschitz property of the nonlinear response and the interaction strengths. Polarization is characterized through a variance-based indicator together with the observed cluster structure. Numerical simulations illustrate the effects of nonlinear response strength, temporal network variability, stochastic perturbations, and memory mechanisms on convergence, clustering, and polarization-like behavior. An additional robustness test on the empirical Epinions signed social network shows that the qualitative dependence on nonlinear response strength persists under a large-scale heterogeneous topology. The proposed framework provides a mathematical and mechanism-based approach for analyzing nonlinear collective dynamics in heterogeneous signed networks.