We study a critical exponent of the flocking behavior to the one-dimensional 1D Cucker–Smale (C–S) model with a regular inverse power law communication on a general network with a spanning tree. For this, we propose a new nonlinear functional which can control the velocity diameter and decays exponentially fast as time goes on. As an application of the time-evolution of the nonlinear functional, we show that the C–S model on a line exhibits a unique critical exponent for unconditional flocking on a general network so that this improves an earlier result [S.-Y. Ha and J.-G. Liu, A simple proof of Cucker–Smale flocking dynamics and mean field limit, Commun. Math. Sci. 7 (2009) 297–325.] on the all-to-all network. Our result also resolves the critical exponent conjecture posed in Cucker–Dong’s work [On the critical exponent for flocks under hierarchical leadership, Math. Models Methods Appl. Sci. 19 (2009) 1391–1404] for 1D setting. Emergent behavior of the C–S model is independent of the special structure of the underlying network, as long as it contains a spanning tree.

中文翻译：

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一般图上一维 Cucker-Smale 模型的临界指数

我们研究了一维 1D Cucker-Smale (C-S) 模型的群集行为的关键指数，该模型在具有生成树的一般网络上具有规则的逆幂律通信。为此，我们提出了一种新的非线性函数，它可以控制速度直径并随着时间的推移呈指数快速衰减。作为非线性泛函的时间演化的一个应用，我们展示了一条线上的 C-S 模型对于一般网络上的无条件植绒表现出独特的临界指数，从而改进了早期的结果 [S.-Y. 哈和 J.-G。刘，Cucker-Smale 群集动力学和平均场极限的简单证明，Commun。数学。科学。7 (2009) 297–325.] 在全对全网络上。我们的结果也解决了 Cucker–Dong 的工作中提出的临界指数猜想 [论等级领导下的羊群的临界指数，Math. 模型方法应用程序。科学。19 (2009) 1391–1404] 用于一维设置。C-S 模型的涌现行为与底层网络的特殊结构无关，只要它包含生成树即可。