A numerical study found that a small set of monitored network nodes could flag simulated synchronization transitions. The strongest candidates were the time-averaged local order parameter and the variance of oscillator phases after their mean rotational trend had been removed.
The finding is limited to the model tested. The work used numerical simulations on nine network structures, so it does not establish that the same signals will anticipate transitions in real systems.
The useful signals stood out
The researchers compared three early-warning measures: the time-averaged local order parameter, which summarizes how closely connected oscillators move together; the temporal variance of that measure; and the variance of each oscillator’s phase after its mean rotational trend had been removed. They also compared dynamics-based, degree-based and random sentinel selection. A sentinel is simply a monitored node in the network.
With the smaller, logarithmically sized sentinel set, the average local order parameter and detrended-phase variance generally rose as the simulated system approached its transition, although both patterns fluctuated. The local-order-parameter variance was nearly flat or fell, which is why the authors treated it as less useful under partial observation.
The study measured trend strength with Kendall’s tau, a rank-based measure of whether a signal tends to rise or fall as coupling increases. For the smaller set, the dynamics-based choice S2 produced mean tau values of 0.76 for the local-order-parameter average, -0.27 for its temporal variance and 0.76 for detrended-phase variance. Monitoring every node, S1, produced 0.87, 0.79 and 0.89 for those measures. The reported standard deviations were 0.09, 0.24 and 0.05 for S2, and 0.03, 0.04 and 0.03 for S1.
With twice the logarithmic sentinel count, S2’s mean tau was 0.83 for the local-order-parameter average, 0.09 for its temporal variance and 0.81 for detrended-phase variance. The corresponding standard deviations were 0.05, 0.29 and 0.04.
Detection depended on the signal and the number of nodes
To test early detection, the researchers defined the critical coupling, Kc, as the smallest coupling at which the normalized increase in the temporal mean global order parameter above its value at the first grid point exceeded 0.2. That first coupling value was 0.01. TIPMOC then compared a power-law-divergence model with a linear model and declared a transition when the difference in corrected AICc scores was -10 or lower for three consecutive windows.
With the smaller sentinel set, S2 detected the transition in 92% of simulations using the local-order-parameter average and in 100% using detrended-phase variance. It detected the transition in only 2% of simulations when using local-order-parameter variance. Full observation detected all three signals in 100% of simulations.
With twice the logarithmic sentinel count, S2 reached 98% detection for the local-order-parameter average and 100% for detrended-phase variance. For detrended-phase variance, detection was 96% with S3, which ranked nodes at 0.5 times the critical coupling, and 97% with S4, which selected the largest-degree nodes.
A tightly defined numerical test
Three model networks with 100 nodes each and six empirical networks were included. For every network, the researchers used ten independent frequency assignments and ten random-seed simulations for each assignment, producing 100 runs per network and 900 pooled runs across the nine networks.
Each run used Euler-Maruyama integration with a time step of 0.01, noise strength of 0.15 and a total duration of 300 time units. The first 150 time units were burn-in, followed by observations every 0.1 time units during another 150 time units.
The six sentinel strategies were labeled S1 through S6. S1 used all nodes. S2 ranked nodes by their early-warning behavior at 0.9 times the critical coupling, while S3 used the same ranking at 0.5 times the critical coupling. S4 selected the largest-degree nodes, S5 the smallest-degree nodes and S6 uniformly random nodes.
What remains to be shown
The authors interpret dynamics-based S2 selection as outperforming the other tested heuristics. They judge detrended-phase variance better for detecting the transition, while its Kendall’s tau and that of the local-order-parameter average were comparably high.
That result has an important practical qualification. S2 and S3 are defined using rankings made at 0.9 and 0.5 times Kc, so using those strategies depends on knowing the critical coupling. The reported detection rates were pooled across the nine networks, and confidence intervals were not reported.
The paper is therefore a computational test of warning metrics, not a validated monitoring rule for real systems. It is an arXiv version-1 preprint dated 28 August 2026.
Paper data and sources
Original title: Early warning signals for synchronization transitions from partial observations
Authors: Yusuke Kato, Naoki Masuda
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-28
DOI: Not available
Original paper · Full text