Characteristics for two kinds of cascading events

Avalanche or cascade failure is ubiquitous. We first classify the cascading phenomena into two categories: the cascading disasters which result in large-scale functional failures and the cascading events that do not lead to disasters. We elucidate that two important factors, the increasing amount of events and the acceleration of event frequency, can induce the crossover from the cascading phenomenon to the cascading disaster. Through a simplified sandpile model and a heuristic logistic map, we demonstrate that the dependence of the event number on the observation time behaves as a power-law and as an exponential for these two different cascading events, respectively. The analytic derivations are found to be consistent with several empirical observations. Our present findings contribute to the understanding of the transition between different cascading events, providing a basis for the further understanding of the transitions among more general critical events.