  In this paper, we will study the emergent behavior of Kuramoto model with
frustration on a general digraph containing a spanning tree. We provide a
sufficient condition for the emergence of asymptotical synchronization if the
initial data is confined in half circle. As lack of uniform coercivity in
general digraph, we apply the node decomposition criteria in \cite{H-L-Z20} to
capture a clear hierarchical structure, which successfully yields the
dissipation mechanism of phase diameter and a small invariant set after finite
time. Then the dissipation of frequency diameter will be clear, which
eventually leads to the synchronization.
