Early warning signals for phase transitions in networks

TL;DR

This work defines a real-field–like susceptibility for site percolation on arbitrary networks by introducing random observers that monitor local connectivity. The susceptibility χ(h) = dS(h)/dh, where S(h) is the fraction of nodes reachable from observers, is shown to be governed by the distribution of finite clusters and diverges near criticality, serving as an early warning signal for both continuous and mixed-order transitions. The authors derive explicit equations for χ in undirected, directed, and dependency-embedded networks, obtain exact results for tree-like and regular cases, and unify the critical behavior with a Landau mean-field framework, demonstrating universality across network types. This approach provides a practical, model-agnostic tool for detecting proximity to percolation thresholds in complex networks and offers insights into the role of finite clusters and higher-order network structures in critical phenomena.

Abstract

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.

Figures (3)

• Figure 1: The size of the giant cluster $S$ vs. $p$ in a $3$-regular random graph with the probability $r$ for nodes to have a dependency link. The curves from left to right: $r = 0$, $0.1$, $0.18$, $0.25$, $0.35$, $0.5$, $0.7$, $0.85$. For $r=1$, the giant connected percolation exists only at $p=1$, where $S=1$. The phase transition is of second order for $r < 1/4$, when $S \propto (p-p_c(r))$ near the critical parameter $p_c(r)$. The curve $r=0.25$ corresponds to the tricritical point. The phase transition is hybrid for $r > 1/4$. The jump of $S$ is shown by the dashed line. At $p=p_c(r)$, the singularity of $S$ is square-root, including the case of $r=1/4$. The insert shows the derivative $(dS/dp)|_{p=p_c(r)}$ at the critical point of the continuous percolation transition.
• Figure 2: Phase diagram on the $(r,p)$-plane for a $3$-regular random graph with dependency links. The solid curve represents the critical points of the continuous percolation transitions with the critical exponent $\beta =1$, see Eq. (\ref{['50']}). The black dot shows the tricritical point with $\beta =1/2$. The dashed curve shows the critical curve of the mixed order phase transition. The region above the critical curve is the region with the giant connected component, i.e., $S>0$. The region below the curve is the region with $S=0$.
• Figure 3: (a) The susceptibility $\chi$ vs. $p$ in a $3$-regular random graph with dependency links for different values of the probability $r$ to have a dependency link: (i) $r = 0.1$, the ordinary percolation transition with the critical exponent $\beta=1$; (ii) $r=r^\ast=0.25$, the tricritical point with $\beta=1/2$; (iii) $r=0.5$, the mixed order phase transition. The insert represents the zoom of $\chi$ at $r=0.5$. (b) The corresponding $1/\chi$ vs. $p$.