Abstract:We consider a variant of the Hegselmann-Krause model of consensus formation where information between agents propagates with a finite speed $\mathfrak {c}>0$. This leads to a system of ordinary differential equations (ODE) with state-dependent delay. Observing that the classical well-posedness theory for ODE systems does not apply, we provide a proof of global existence and uniqueness of solutions of the model. We prove that asymptotic consensus is always reached in the spatially one-dimensional setting of the model, as long as agents travel slower than $\mathfrak {c}$. We also provide sufficient conditions for asymptotic consensus in the spatially multidimensional setting.

---

中文翻译：

---

具有有限信息传播速度的 Hegselmann-Krause 模型中的适定性和渐近一致性

摘要：我们考虑了 Hegselmann-Krause 共识形成模型的一种变体，其中代理之间的信息以有限速度 $\mathfrak {c}>0$ 传播。这导致了具有状态相关延迟的常微分方程 (ODE) 系统。观察到 ODE 系统的经典适定性理论不适用，我们提供了模型解的全局存在性和唯一性的证明。我们证明了在模型的空间一维设置中总能达成渐近共识，只要代理的移动速度比 $\mathfrak {c}$ 慢。我们还为空间多维环境中的渐近共识提供了充分条件。

---