Anticipating the onset of collective synchronization is important in many networked systems, yet observing every oscillator is often impractical. We investigate whether synchronization transitions can be detected from a small set of monitored, or sentinel, nodes. Using a stochastic Kuramoto model on networks, we numerically compare three early warning signals: the local order parameter, its temporal variance, and the variance of individual oscillator phases after removing their mean rotational trends. We also compare sentinel-selection strategies based on node dynamics, degree, and random sampling. We show that, under partial observation, the local order parameter and the variance of detrended phases provide substantially stronger warning signals than the variance of the local order parameter. Selecting nodes according to their dynamical behavior near the transition consistently improves performance over other sentinel-selection methods. With only $\lfloor\ln N\rfloor$ dynamically selected sentinels, two success indicators approach the performance obtained by observing all $N$ nodes. These results demonstrate that synchronization transitions can be anticipated from sparse observations when the warning signal and monitored nodes are chosen appropriately.
Spatial early warning signals (EWSs) seek evidence of an approaching tipping point from a single snapshot of many interacting elements. Existing theory largely assumes spatial homogeneity, whereas networks introduce systematic differences among nodes that may obscure fluctuation-based warning signals. We develop a mathematical framework for spatial EWSs in stochastic dynamical systems on networks. We find that the expected spatial variance, a popular spatial EWS, decomposes exactly into a structural contribution from heterogeneity in the equilibrium state and a fluctuation contribution determined by the stationary covariance. Near a simple steady-state bifurcation, the potentially divergent covariance concentrates along the critical eigendirection: the left eigenvector determines how strongly noise excites the critical fluctuation, while the right eigenvector determines its spatial pattern. Consequently, the spatial variance has a divergent fluctuation contribution when the limiting critical eigendirection is noise-excited and spatially nonuniform after centering. In contrast, the spatial coefficient of variation generally saturates, while skewness, kurtosis, and Moran's $I$ approach network-dependent limits without a universal warning direction. We also derive results for homogeneous networks, node-wise baseline subtraction as preprocessing, and Hopf bifurcations, for which the limiting distributions are qualitatively different. These results clarify when spatial EWSs provide reliable warnings and why their performance depends on network structure, noise, and preprocessing.
Naoki Masuda· 0 citations
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