Critical exponent for Cucker–Smale model under group-hierarchical multi-leadership

In this paper, we focus on the critical exponent for the Cucker–Smale model in R^d(d≥1) under group-hierarchical multi-leadership (GHML) topology. The GHML is an asymmetric topology with a group hierarchical structure and multiple leaders. The exponent β in communication weight function measures the decay rate with respect to the distance of particles. In literature, for d≥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.