Consensus, polarization, and nonlinear response in a two-layer voter model with intra-agent cross-layer reconciliation

We study a coupled two-layer opinion dynamics model in which agents simultaneously hold binary opinions on an online and an offline layer. The dynamics combine same-layer social imitation with a symmetric intra-agent cross-layer reconciliation process: with probability p, one layer of an agent is updated to match the other layer. Thus p quantifies the rate of internal cross-layer coordination, rather than a directed preference for either the online or offline state. Monte Carlo simulations reveal an observation-window-dependent crossover from a rapid-consensus regime to a long-lived polarized regime as p increases, with effective crossover centers pc close to unity for both network types. The crossover depends on network topology: Erdős-Rényi networks show a mean-field-like connectivity dependence, whereas Barabási-Albert scale-free networks exhibit a much weaker and more scattered dependence, consistent with stabilization by topological heterogeneity. Under periodic external driving with frequency f, both topologies display hysteresis loops whose enclosed area A scales as A∼f1/2 over the simulated range. The central analytical result is that the leading mean-field driven dynamics admits an exact cancellation of the internal coupling p from the total-magnetization equation. This cancellation leads to a square-root hysteresis scaling that is independent of p at leading order and only weakly dependent on topology, thereby separating topology-sensitive finite-time consensus stability from a comparatively topology-insensitive nonlinear response.