Minimizing Polarization in Geometric Opinion Dynamics on Social Networks

We propose a multidimensional geometric opinion dynamics model on social networks, in which opinions evolve through alignment-based interactions. We rigorously analyze the dynamics to identify its characteristics and show that polarization can emerge endogenously. To mitigate the effect of polarization, we introduce recommendation-based interventions with exposure as a controllable parameter, yielding a nonlinear optimization problem. We show that the dynamics have a Lipschitz-continuous Jacobian and that a broad class of resulting objectives has Lipschitz-continuous gradients, establishing their tractability for optimization. We further apply a Zeroth-order projected optimization method to minimize polarization using only function evaluations, with guarantees of convergence to first-order stationary points up to smoothing bias. Experiments on synthetic and real-world network topologies demonstrate that recommendations effectively reduce polarization, and that the zeroth-order method achieves performance comparable to first-order approaches.