论文标题
通过概率方法，在一般加权挖掘上的Cucker-Smale和Motsch-Tadmor模型的植入
Flocking of the Cucker-Smale and Motsch-Tadmor models on general weighted digraphs via a probabilistic method
论文作者
论文摘要
在本文中，我们讨论了与一般的通信功能的一般定向和加权图上的cucker-smale和motsch-tadmor模型的植被现象。我们提出了一种基于解决方案的概率解释来研究此问题的新方法。我们在相互作用矩阵的四个假设下提供了羊群结果，我们强调了它们与某个马尔可夫跳跃过程的总变化中的收敛方式。确实，我们在最小情况下完善了以前的结果，该案例允许该图允许独特的封闭通信类。考虑到邻接矩阵正在争夺或接受正面可逆度量的两种特殊情况，我们改善了最小情况下获得的植入条件。在最后一个情况下，我们表征了渐近速度。我们还研究了分层领导案例，我们提供了一种新的一般羊群条件，该条件允许处理$ψ（r）\ propto（1+r^2）^{ - β/2} $和$β\ geq1 $。对于在层次领导假设下的Motsch-Tadmor模型中，我们表现出一种情况，即使最初的条件和通信功能如何，羊群现象也会发生，即使$β\ geq1 $也是如此。
In this paper, we discuss the flocking phenomenon for the Cucker-Smale and Motsch-Tadmor models in continuous time on a general oriented and weighted graph with a general communication function. We present a new approach for studying this problem based on a probabilistic interpretation of the solutions. We provide flocking results under four assumptions on the interaction matrix and we highlight how they relate to the convergence in total variation of a certain Markov jump process. Indeed, we refine previous results on the minimal case where the graph admits a unique closed communication class. Considering the two particular cases where the adjacency matrix is scrambling or where it admits a positive reversible measure, we improve the flocking condition obtained for the minimal case. In the last case, we characterise the asymptotic speed. We also study the hierarchical leadership case where we give a new general flocking condition which allows to deal with the case $ψ(r)\propto(1+r^2)^{-β/2}$ and $β\geq1$. For the Motsch-Tadmor model under the hierarchical leadership assumption, we exhibit a case where the flocking phenomenon occurs regardless of the initial conditions and the communication function, in particular even if $β\geq1$.
