一般加权有向图上Cucker-Smale和Motsch-Tadmor模型群集的概率方法
Flocking of the Cucker-Smale and Motsch-Tadmor models on general weighted digraphs via a probabilistic method
摘要：
本文讨论了Cucker-Smale和Motsch-Tadmor模型在具有一般通信函数的一般有向加权图上的群集现象。我们提出了一种新的方法来研究这个问题的基础上的概率解释的解决方案。我们提供植绒的相互作用矩阵的四个假设下的结果，我们强调他们如何涉及到一定的马尔可夫跳跃过程的全变差的收敛。事实上，我们完善以前的结果的最小情况下，图承认一个独特的封闭通信类。考虑到邻接矩阵是置乱的和存在正可逆测度的两种特殊情况，我们改进了最小情况下的群集条件。在最后一种情况下，我们证明了渐近速度。我们还研究了层次领导的情况下，我们给出了一个新的一般群集条件，允许处理的情况下$ψ(r)\propto(1+r^2)^{-β/2}$和$β\geq1$。对于层次领导假设下的Motsch-Tadmor模型，我们展示了一种情况下，无论初始条件和通信功能，特别是即使$β\geq1$的群集现象发生。
英文摘要：
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 $β\geq1$.
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