  We reveal large fluctuations in the response of real multiplex networks to
random damage of nodes. These results indicate that the average response to
random damage, traditionally considered in mean-field approaches to
percolation, is a poor metric of system robustness. We show instead that a
large deviation approach to percolation provides a more accurate
characterization of system robustness. We identify an effective percolation
threshold at which we observe a clear abrupt transition separating two distinct
regimes in which the most likely response to damage is either a functional or a
dismantled multiplex network. We leverage our findings to propose a new metric,
named safeguard centrality, able to single out the nodes that control the
response of the entire multiplex network to random damage. We show that
safeguarding the function of top-scoring nodes is sufficient to prevent system
collapse.
