Emergence of synchronization in Kuramoto model with general digraph

In this paper, we study the complete synchronization of Kuramoto model with general network containing a spanning tree, when initial phases are distributed in an open half circle. As lack uniform coercivity digraph, order to capture dissipation structure on network, apply strongly connected graph decomposition criteria Ha et al. [Math. Models Methods Appl. Sci., 30 (2020)] yield hierarchical structure, which will induce estimates similar hypo-coercivity estimates. This drives phase diameter into invariant region confined quarter circle finite time under large coupling regime. Then uniformly small boundedness eventually leads emergence exponentially fast synchronization.