  In this paper, we study the complete synchronization of the Kuramoto model
with general network containing a spanning tree, when the initial phases are
distributed in an open half circle. As lack of uniform coercivity in general
digraph, in order to capture the dissipation structure on a general network, we
apply the node decomposition criteria in \cite{H-L-Z20} to yield a hierarchical
structure, which leads to the hypo-coercivity. This drives the phase diameter
into a small region after finite time in a large coupling regime, and the
uniform boundedness of the diameter eventually leads to the emergence of
exponentially fast synchronization.
