Simultaneous first- and second-order percolation transitions in interdependent networks

In a system of interdependent networks, an initial failure of nodes invokes a cascade of iterative failures that may lead to a total collapse of the whole system in the form of an abrupt first-order transition. When the fraction of initial failed nodes $1-p$ reaches criticality $p=p_c$, the abrupt collapse occurs by spontaneous cascading failures. At this stage, the giant component decreases slowly in a plateau form and the number of iterations in the cascade $\tau$ diverges. The origin of this plateau and its increasing with the size of the system have been unclear. Here we find that, simultaneously with the abrupt first-order transition, a spontaneous second-order percolation occurs during the cascade of iterative failures. This sheds light on the origin of the plateau and how its length scales with the size of the system. Understanding the critical nature of the dynamical process of cascading failures may be useful for designing strategies for preventing and mitigating catastrophic collapses.