Abrupt transitions in complex systems, from ecology to climate, are hard to predict in advance. Existing early warning methods rely on slowing dynamics and growing temporal correlations, but they often give the signal too late. A research direction using functional networks has been developing around this work: neighboring studies show how neural networks predict the unseen or how climate gives an earlier sign. However, these approaches either require knowledge of the model or provide limited lead time. Our work fits into this line of research but shifts the focus from global characteristics to the mesoscopic level: we do not look at the entire network as a whole, but at the probabilities that a node belongs to clusters of different sizes.
From time series at each point in space, pairwise Pearson correlations are computed. Pairs of points with correlation above a threshold γ are connected by links, forming a functional network. Then the percolation transition in this network is analyzed: the size of the largest connected component S1 and the average size of the other clusters ⟨s²⟩. The key idea is to use the probabilities $c_s$ that a randomly chosen node belongs to a cluster of size s. In random graphs, the maxima of $c_s$ occur at average degrees $p_s$ = (s−1)/s, which converge to the percolation point, but for small s they occur earlier. This sequence of peaks in $c_s$ serves as an early warning.
The method is tested on three models: lake eutrophication with a saddle-node bifurcation, the Ginzburg–Landau equation with a pitchfork bifurcation, and the Lorenz-96 system with a Hopf bifurcation and transition to chaos. In all cases, the percolation transition in the functional network occurs before the dynamical transition, and the peaks in $c_s$ give an even earlier warning. For example, for the lake model, the peak of c₂ occurs at p=0.635, whereas the dynamical jump occurs at $p_d$≈0.658, which is twice as early as the signal from the percolation transition. For the Ginzburg–Landau equation, two percolation transitions are observed on both sides of the bifurcation. In the Lorenz-96 system, the percolation phase exists in an intermediate range of parameters, and the peaks of c₂ approach the transition points as the threshold γ increases. On real sea surface temperature data in the NINO3.4 region, the method detected precursors of El Niño and La Niña events 115–240 days before their official onset.
This work differs from neighboring studies by introducing a mesoscopic view: instead of global metrics (variance, clustering) or local ones (node degree), it uses probabilities of belonging to small clusters. This allows for earlier warning than standard percolation indicators and does not require knowledge of the system's dynamics. The method is universal for systems with growing spatial correlations as they approach a bifurcation, which is typical for diffusively coupled systems with critical slowing down.
The method can be extended to systems with more complex topology, not necessarily spatially embedded. It is also promising to investigate the optimal choice of threshold γ and cluster size s in real time, possibly with adaptive algorithms. Application to other climate indices or biological systems may reveal new patterns.
The method could improve early warning for climate events such as El Niño and help monitor ecological or physiological systems where abrupt transitions are dangerous.
Test the method on other datasets, such as ice cores or financial time series. Develop an algorithm for automatic selection of threshold γ and cluster size s in real time.
In the field of predicting critical transitions, an open question remains: how to distinguish a true signal from noise in short time series. This method offers a way to amplify the signal through spatial aggregation, but it requires further testing on systems with strong non-stationarity.
For the lake model, the peak of probability c₂ occurs at p=0.635, while the transition itself occurs at p=0.658: a difference of 0.023 in the parameter, which corresponds to a noticeable lead time.
The idea of predicting a catastrophe from growing correlations echoes the concept of the 'psychohistory' from Isaac Asimov's 'Foundation' series, where Hari Seldon predicts the fall of the Empire through statistical patterns.