Convergence to consensus for a Hegselmann-Krause-type model with distributed time delay

Consensus of the Hegselmann-Krause opinion formation model with time delay

In this paper, we study Hegselmann-Krause models with a time-variable time delay. Under appropriate assumptions, we show the exponential asymptotic consensus when the time delay satisfies a suitable smallness assumption. Our main strategies for this are based on Lyapunov functional approach and careful estimates on the trajectories. We then study the mean-field limit from the many-individual Hegselmann-Krause equation to the continuity-type partial differential equation as the number N of individuals goes to infinity. For the limiting equation, we prove global-in-time existence and uniqueness …

Optimal condition for asymptotic consensus in the Hegselmann-Krause model with finite speed of information propagation

We prove that asymptotic global consensus is always reached in the Hegselmann-Krause model with finite speed of information propagation $\mathfrak{c}>0$ under minimal (i.e., necessary) assumptions on the influence function. In particular, we assume that the influence function is globally positive, which is necessary for reaching global consensus, and such that the agents move with speeds strictly less than $\mathfrak{c}$, which is necessary for well-posedness of solutions. From this point of view, our result is optimal. The proof is based on the fact that the state-dependent delay, induced by …

带环境噪声的 Hegselmann--Krause 模型

Hegselmann--Krause model with environmental noise

本文研究带环境噪声的 Hegselmann-Krause 连续时间观点动力学，通过正则化和混沌传播推导出 McKean-Vlasov 随机微分方程极限，并证明其强解存在唯一性及相关随机 Fokker-Planck 方程的适定性。

Well-Posedness of the Limiting Equation of a Noisy Consensus Model in Opinion Dynamics

This paper establishes the global well-posedness of the nonlinear Fokker-Planck equation for a noisy version of the Hegselmann-Krause model. The equation captures the mean-field behavior of a classic multiagent system for opinion dynamics. We prove the global existence, uniqueness, nonnegativity and regularity of the weak solution. We also exhibit a global stability condition, which delineates a forbidden region for consensus formation. This is the first nonlinear stability result derived for the Hegselmann-Krause model.

具有领导者-追随者相互作用的时间延迟意见动力学：共识、稳定性与均场极限

Time-delayed opinion dynamics with leader-follower interactions: consensus, stability, and mean-field limits

本文研究了具有领导者和非领导者的时滞意见形成模型，分析了共识、稳定性及均场极限，证明了粒子系统的指数收敛，并在两种情形下讨论了均场极限的存在性与唯一性。