  Coupled oscillator networks show a complex interrelations between topological
characteristics of the network and the nonlinear stability of single nodes with
respect to large but realistic perturbations. We extend previous results on
these relations by incorporating sampling-based measures of the transient
behaviour of the system, its survivability, as well as its asymptotic
behaviour, its basin stability. By combining basin stability and survivability
we uncover novel, previously unknown asymptotic states with solitary,
desynchronized oscillators which are rotating with a frequency different from
their natural one. They occur almost exclusively after perturbations at nodes
with specific topological properties.
  More generally we confirm and significantly refine the results on the
distinguished role tree-shaped appendices play for nonlinear stability. We find
a topological classification scheme for nodes located in such appendices, that
exactly separates them according to their stability properties, thus
establishing a strong link between topology and dynamics. Hence, the results
can be used for the identification of vulnerable nodes in power grids or other
coupled oscillator networks. From this classification we can derive general
design principles for resilient power grids. We find that striving for
homogeneous network topologies facilitates a better performance in terms of
nonlinear dynamical network stability. While the employed second-order
Kuramoto-like model is parametrized to be representative for power grids, we
expect these insights to transfer to other critical infrastructure systems or
complex network dynamics appearing in various other fields.
