Percolation theory plays a pivotal role in network science as it sheds light on the fundamental structural properties of a network that determine its robustness when a fraction of node is initially damaged.
Despite the fact that the percolation transition is second order, cascade of failure events that abruptly dismantle a network are actually occurring in real systems, with major examples ranging from large electric blackouts to the sudden collapse of ecological systems.
Here we build a large deviation theory of percolation characterizing the response of a sparse network to rare events. This general theory includes the second order phase transition observed typically for random configurations of the initial damage but reveals also discontinuous transitions corresponding to rare configurations of the initial damage for which the size of the giant component is suppressed.
Moreover we will discuss recent results regarding cascade of failure and discontinuous transitions on multilayer networks and theoretical principles to design multilayer networks which boost their robustness.