Network risk and forecasting power in phase-flipping dynamical networks.

To model volatile real-world network behavior, we analyze a 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-flipping dynamics, and we 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, fn(t) and fℓ(t), and we define two estimators to quantify the level of risk in a network. We find hysteresis in the correlations of fn(t) due to failures at the node level, and we derive conditional probabilities for phase-flipping in networks. We apply our model to economic and traffic networks.