We present a direct proof of asymptotic consensus in the non-linear Hegselmann–Krause model with transmission-type delay, where the communication weights depend on the particle distance in phase space. Our approach is based on an explicit estimate of the shrinkage of the group diameter on finite time intervals and avoids the usage of Lyapunov-type functionals or results from non-negative matrix theory. It works for both the original formulation of the model with communication weights scaled by the number of agents, and the modification with weights normalized a'la Motsch–Tadmor. We pose only minimal assumptions on the model parameters. In particular, we only assume global positivity of the influence function, without imposing any conditions on its decay rate or monotonicity. Moreover, our result holds for any length of the delay.

中文翻译：

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具有传输类型延迟的 Hegselmann-Krause 模型中无条件渐近共识的直接证明

我们在具有传输类型延迟的非线性 Hegselmann-Krause 模型中提出了渐近一致性的直接证明，其中通信权重取决于相空间中的粒子距离。我们的方法基于对有限时间间隔内群直径收缩的显式估计，并避免使用 Lyapunov 型泛函或非负矩阵理论的结果。它适用于具有按代理数量缩放的通信权重的模型的原始公式，以及具有归一化 a'la Motsch-Tadmor 的权重的修改。我们只对模型参数做出最小的假设。特别是，我们只假设影响函数的全局正性，而不对其衰减率或单调性强加任何条件。此外，我们的结果适用于任何长度的延迟。