Rare events and discontinuous percolation transitions.

Percolation theory characterizing the robustness of a network has applications ranging from biology, to epidemic spreading, and complex infrastructures. Percolation theory, however, only concerns the average response of a network to random damage of its nodes while in real finite networks, fluctuations around this average behavior are observable. Consequently, for finite networks, there is an urgent need to evaluate the risk of collapse in response to rare configurations of the initial damage. 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.