Early warning signals for percolation transitions in networks

The percolation phase transition in complex network systems attracts much attention and has numerous applications in various research fields. Finite-size effects smooth the transition and make it difficult to predict the critical point of appearance or disappearance of the giant connected component. To this end, we introduce a new method for computing and measuring susceptibility of arbitrary random undirected and directed networks. The importance of the characteristics is that a strong increase of the susceptibility is the early warning signal of approaching the transition point. Our method is based on the introduction of "observers," which are randomly chosen nodes monitoring the local connectivity of a network. To demonstrate efficiency of the method, we derive explicit equations determining the susceptibility and study its critical behavior near continuous and mixed-order phase transitions in uncorrelated random undirected and directed networks, networks with dependency links, and k-cores of networks. The universality of the critical behavior is supported by the phenomenological Landau theory of phase transitions.