Analysis and Consensus Control of Emergent Dynamic Polarization in Minimally-Nonlinear Opinion Dynamics
2026-08-10 • Social and Information Networks
Social and Information Networks
AI summaryⓘ
The authors study how opinions in social networks change over time and why groups sometimes get stuck in ongoing disagreement, called dynamic polarization. They create a simple mathematical model that shows this behavior can happen naturally from how individuals update their opinions locally. They prove that when people react strongly enough, the network shifts from agreement to stable back-and-forth opinion clusters. The authors also find ways to stop this oscillation by controlling just one person's opinion, helping the whole group reach steady agreement again. Their work is backed by detailed math and computer simulations.
Opinion dynamicsSocial networksDynamic polarizationNonlinear systemsBifurcation theoryConsensus equilibriumPeriodic orbitDirected graphsLocal controlAgent-based modeling
Authors
Rajul Kumar, Ningshi Yao
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.