Early warning signals for phase 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. For this end, we introduce the susceptibility of arbitrary random undirected and directed networks and show 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.