Analysis and Consensus Control of Emergent Dynamic Polarization in Minimally-Nonlinear Opinion Dynamics

Abstract

Collective opinions in social networks evolve through local interaction rules, yet how such local updates give rise to dynamic polarization--persistent oscillatory disagreement between opposing opinion clusters at the network level--remains unexplained. This paper proposes a Minimally-Nonlinear Opinion Dynamics (or M-NOD) framework that analytically characterizes dynamic polarization as a truly emergent collective behavior arising solely from local, agent-level opinion-update rules without externally imposed mechanisms. By introducing a minimal cubic nonlinearity, we rigorously prove that, as the reactivity rate exceeds a critical threshold, the network's consensus equilibrium loses stability via a supercritical flip bifurcation. In the post-bifurcation regime, this instability gives rise to a unique, locally asymptotically stable periodic orbit, thereby characterizing symmetric dynamic polarization with balanced bipartite opinion clusters. We further establish the structural robustness of this behavior by proving the existence and local asymptotic stability of asymmetric dynamic polarization under directed graphs with nonuniform influence weights. Finally, to resolve this undesirable cyclic deadlock, we develop local agent-level control strategies. We prove that anchoring the opinion of only a single agent is sufficient to eliminate network-wide oscillatory disagreement and restore asymptotically stable consensus. Numerical simulations substantiate the theoretical analysis, including the emergence of symmetric and asymmetric dynamic polarization, and demonstrate the efficacy of the proposed control interventions.