  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.
