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 cr
itical 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.
In order to model volatile real-world network behavior, we analyze phase-flipping dynamical scale-free network in which nodes and links fail and recover. We investigate how stochasticity in a parameter governing the recovery process affects phase-fli
pping dynamics, and find the probability that no more than q% of nodes and links fail. We derive higher moments of the fractions of active nodes and active links, $f_n(t)$ and $f_{ell}(t)$, and define two estimators to quantify the level of risk in a network. We find hysteresis in the correlations of $f_n(t)$ due to failures at the node level, and derive conditional probabilities for phase-flipping in networks. We apply our model to economic and traffic networks.
