In this paper, we focus on the critical exponent for the Cucker–Smale model in ${\mathbb{R}}^d(d\ge 1)$ under group-hierarchical multi-leadership (GHML) topology. The GHML is an asymmetric topology with a group hierarchical structure and multiple leaders. The exponent $\beta$ in communication weight function measures the decay rate with respect to the distance of particles. In literature, for $d\ge 2$, the critical exponent for unconditional flocking is proven to be $1/2$ only for symmetric topologies or hierarchical leadership. For general digraphs, the exponent below which the unconditional flocking occurs depends on the digraph and is less than $1/2$. In this paper, we prove that the critical exponent is $1/2$.

在本文中，我们关注 Cucker-Smale 模型的临界指数${\mathbb{R}}^d(d\ge 1)$在组层级多领导（GHML）拓扑下。GHML 是具有组层次结构和多个领导者的非对称拓扑。指数$\beta$在通信中，权重函数测量相对于粒子距离的衰减率。在文学中，对于$d\ge 2$，无条件植绒的临界指数被证明是$1/2$仅适用于对称拓扑或分层领导。对于一般有向图，无条件聚集发生的指数取决于有向图并且小于$1/2$. 在本文中，我们证明了临界指数是$1/2$.