最小非线性观点动力学中涌现动态极化的分析与共识控制
Analysis and Consensus Control of Emergent Dynamic Polarization in Minimally-Nonlinear Opinion Dynamics
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中文总结 AI 辅助
本文提出M-NOD框架解析社交网络观点演化产生的动态极化，证明其稳定性条件，开发局部控制策略仅锚定单个智能体即可消除振荡分歧、恢复共识。
中文摘要 AI 辅助
社交网络中的集体观点通过局部交互规则演化，但这类局部更新如何产生动态极化——即网络层面对立观点集群间持续的振荡分歧——仍是未解之谜。本文提出最小非线性观点动力学（Minimally-Nonlinear Opinion Dynamics，简称M-NOD）框架，将动态极化解析为仅由智能体层面局部观点更新规则产生、无外部施加机制的真正涌现集体行为。通过引入最小三次非线性项，我们严格证明：当反应率超过临界阈值时，网络的共识平衡点会通过超临界翻转分岔失去稳定性；在分岔后区域，这种不稳定性会产生唯一的局部渐近稳定周期轨道，从而表征具有均衡二分观点集群的对称动态极化。我们进一步通过证明在具有非均匀影响权重的有向图下，非对称动态极化的存在性与局部渐近稳定性，确立了该行为的结构鲁棒性。最后，为解决这种不良的循环僵局，我们开发了智能体层面的局部控制策略，证明仅锚定单个智能体的观点就足以消除全网范围的振荡分歧，恢复渐近稳定的共识。数值模拟验证了理论分析，包括对称与非对称动态极化的涌现，并证明了所提控制干预的有效性。
英文摘要
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.
