Predicting Lifetime of Dynamical Networks Experiencing Persistent Random   Attacks
  Empirical estimation of critical points at which complex systems abruptly flip from one state to another is among the remaining challenges in network science. However, due to the stochastic nature of critical transitions it is widely believed that critical points are difficult to estimate, and it is even more difficult, if not impossible, to predict the time such transitions occur [1-4]. We analyze a class of decaying dynamical networks experiencing persistent attacks in which the magnitude of the attack is quantified by the probability of an internal failure, and there is some chance that an internal failure will be permanent. When the fraction of active neighbors declines to a critical threshold, cascading failures trigger a network breakdown. For this class of network we find both numerically and analytically that the time to the network breakdown, equivalent to the network lifetime, is inversely dependent upon the magnitude of the attack and logarithmically dependent on the threshold. We analyze how permanent attacks affect dynamical network robustness and use the network lifetime as a measure of dynamical network robustness offering new methodological insight into system dynamics. 
B. Podobnik, T. Lipic, D. Horvatic, A. Majdandzic, S. Bishop and H. E.   Stanley
cs.SI physics.soc-ph