  Transition to amplitude death in scale-free networks of nonlinear oscillators
is investigated. As the coupling strength increases, the network will undergo
three stages in approaching to the state of complete amplitude death. The first
stage is featured by a \emph{"stair-like"} distribution of the node amplitude,
and the transition is accomplished by a \emph{hierarchical death} of the
amplitude stairs. The second and third stages are characterized by,
respectively, a continuing elimination of the synchronous clusters and a fast
death of the non-synchronized nodes.
