Multiscale Synchronization Hierarchy and Universal Thresholds of Systemic Tau in Modular and High-Dimensional Chaotic Systems

Abstract Synchronization is a fundamental phenomenon in complex systems, yet reliably detecting transitions between synchronized and desynchronized regimes remains challenging, especially in high-dimensional and noisy environments. Here we introduce Systemic Tau (τs), a robust, nonparametric measure based on rank correlations, and demonstrate that it exhibits universal thresholds that consistently distinguish stable synchronization from chaotic dynamics. Using extensive numerical simulations on Coupled Map Lattices and four canonical complex network topologies (Barabási–Albert, Watts–Strogatz, Stochastic Block Model, and hierarchical networks), we show that the thresholds τs > +0.50 (stable) and |τs| < 0.41 (chaotic) remain consistent across system sizes, coupling strengths, and network architectures. A central finding of this work is the discovery of a temporal multiscale synchronization hierarchy in modular networks: local communities reach stable synchronization significantly earlier than global coherence emerges, acting as “synchronization nuclei.” We further demonstrate that additive noise functions as a sharp and tunable control parameter capable of rapidly destroying stable synchronization above a critical threshold. Comparative analyses reveal that Systemic Tau offers superior sharpness, lower variance, and better regime separation than the classical order parameter, particularly in chaotic regimes. These results establish Systemic Tau as a universal, scalable, and computationally efficient framework for characterizing synchronization transitions, with potential applications in neuroscience, biology, engineering, and the social sciences.