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.

💡 Research Summary

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The paper addresses the long‑standing problem of detecting the percolation transition in finite‑size complex networks, where the emergence or disappearance of a giant connected component (GCC) is smoothed out by finite‑size effects. Traditional early‑warning indicators, such as the zero‑field susceptibility defined as the mean cluster size or the probability that two randomly chosen nodes belong to the same cluster, require full knowledge of the cluster structure and are therefore computationally demanding, especially for directed or multiplex networks.

To overcome these limitations, the authors introduce a novel, physically motivated observable: a set of randomly chosen “observers” that act as a real external field. A fraction (h) (with (0<h\le 1)) of the nodes is selected uniformly at random; these nodes constitute the observer set (\mathcal{C}_h). For each observer (i\in\mathcal{C}_h) the authors consider the set of nodes reachable from (i) by following edges (in undirected graphs) or by following edges forward and backward (in directed graphs). The union of all reachable sets is denoted (\mathcal{T}_h) (or (\mathcal{T}^{\text{tot}}_h) for directed graphs). The fraction of the whole network that can be reached from at least one observer is \