Open Access

Lorinc Márton, Stefanie Winkelmann, Mauricio del Razo, Natasa Djurdjevac Conrad

• Opinions play a crucial role in shaping collective phenomena such as political polarization, cultural integration and demographic change. By continuously changing social environments in which opinions evolve, human migration serves as an important driver of collective opinion formation. While migration and opinion dynamics have both been extensively studied, the few existing models that couple the two are primarily deterministic and therefore cannot capture demographic fluctuations, finite-size effects or stochastic transitions between emergent collective states. To address this limitation, we introduce a unifying stochastic framework for opinion dynamics over migration networks that couples local opinion transitions, demographic processes and migration between communities. The dynamics are formulated through a spatio--temporal master equation, which provides a probabilistic description of the underlying population process. From this microscopic representation, we derive deterministic mean-field equations governing the co-evolution of community sizes and opinion compositions, thereby linking agent-level interactions to macroscopic population behavior. Using two representative case studies, we demonstrate how stochasticity and migration can qualitatively change the emergent dynamics and collective outcomes, including the emergence of consensus, polarization and the stabilization of oscillatory opinion dynamics. These examples highlight the rich interplay between social interactions, demographic change and migration in deterministic and stochastic settings, and they demonstrate that migration should be viewed as an integral component of collective opinion formation rather than only an external demographic process.