  Dynamical entities interacting with each other on complex networks often
exhibit multistability. The stability of a desired steady regime (e.g., a
synchronized state) to large perturbations is critical in the operation of many
real-world networked dynamical systems such as ecosystems, power grids, the
human brain, etc. This necessitates the development of appropriate quantifiers
of stability of multiple stable states of such systems. Motivated by the
concept of basin stability (BS) (Menck et al., Nature Physics 9, 89 (2013)), we
propose here the general framework of \emph{multi-node basin stability} for
gauging global stability and robustness of networked dynamical systems in
response to non-local perturbations simultaneously affecting multiple nodes of
a system. The framework of multi-node BS provides an estimate of the critical
number of nodes which when simultaneously perturbed, significantly reduces the
capacity of the system to return to the desired stable state. Further, this
methodology can be applied to estimate the minimum number of nodes of the
network to be controlled or safeguarded from external perturbations to ensure
proper operation of the system. Multi-node BS can also be utilized for probing
the influence of spatially localised perturbations or targeted attacks to
specific parts of a network. We demonstrate the potential of multi-node BS in
assessing the stability of the synchronized state in a deterministic scale-free
network of R\"{o}ssler oscillators and a conceptual model of the power grid of
the United Kingdom with second-order Kuramoto-type nodal dynamics.
