We study time-asymptotic interplay between time-delayed communication and Cucker-Smale (C-S) velocity alignment. For this, we present two sufficient frameworks for the asymptotic flocking to the continuous and discrete C-S models with $ q $-closest neighbors in the presence of time-delayed communications. Communication time-delays result from the finite-propagation speed of information and they are often ignored in the first place modeling of collective dynamics. In the absence of time-delays in communication, Cucker and Dong showed that the C-S model with $ q $-closest neighbors can exhibit a phase-transition like phenomenon for unconditional and conditional flockings depending on the size $ q $ relative to system size. In this paper, we investigate whether Cucker and Dong's result is robust with respect to the time-delayed communications or not. In fact, our flocking estimates show that the critical number of $ q $ for unconditional flocking is the same as in the case for zero time-delay, which shows the robustness of the Cucker and Dong's result with respect to small time-delay.

中文翻译：

---

关于可卡犬-男合奏 \ begin {document} $ q $ \ end {document}延迟通信下的最近邻居

我们研究时间传播和Cucker-Smale（CS）速度对准之间的时间渐近相互作用。为此，我们提出了两个足够的框架，用于在存在延时通信的情况下，以$ q $-最近邻居的连续和离散CS模型渐近植群。通信时间延迟是由信息的有限传播速度引起的，在集体动力学的首要建模中通常会忽略它们。在通信中没有时间延迟的情况下，Cucker和Dong证明，具有$ q $-最接近邻居的CS模型可以根据相对于系统大小的$ q $大小，表现出无条件和有条件植绒的相变现象。在本文中，我们调查了Cucker和Dong' 相对于延时通信，结果是否可靠。实际上，我们的植绒估计表明，无条件植绒的关键数量q q $与零时延的情况相同，这表明了Cucker和Dong的结果对于小时延的鲁棒性。