  In an increasingly connected world, the resilience of networked dynamical
systems is important in the fields of ecology, economics, critical
infrastructures, and organizational behaviour. Whilst we understand small-scale
resilience well, our understanding of large-scale networked resilience is
limited. Recent research in predicting the effective network-level resilience
pattern has advanced our understanding of the coupling relationship between
topology and dynamics. However, a method to estimate the resilience of an
individual node within an arbitrarily large complex network governed by
non-linear dynamics is still lacking. Here, we develop a sequential mean-field
approach and show that after 1-3 steps of estimation, the node-level resilience
function can be represented with up to 98\% accuracy. This new understanding
compresses the higher dimensional relationship into a one-dimensional dynamic
for tractable understanding, mapping the relationship between local dynamics
and the statistical properties of network topology. By applying this framework
to case studies in ecology and biology, we are able to not only understand the
general resilience pattern of the network, but also identify the nodes at the
greatest risk of failure and predict the impact of perturbations. These
findings not only shed new light on the causes of resilience loss from cascade
effects in networked systems, but the identification capability could also be
used to prioritize protection, quantify risk, and inform the design of new
system architectures.
