A Unified Framework for Structural Early Warning Signals: Decoupling Compression and Adaptive Capacity in Complex Systems

This work introduces a unified framework for analyzing early warning signals (EWS) in complex systems by explicitly separating structural compression (Φ) from adaptive capacity (R) as independent dimensions of system stability.   Classical early warning signals are primarily based on dynamical indicators such as variance and lag-1 autocorrelation, reflecting the phenomenon of critical slowing down. While effective in certain regimes, these indicators implicitly assume that instability is driven by dynamical amplification. This framework addresses a complementary failure mode: the gradual reduction of independently accessible degrees of freedom in the system’s state space.   Structural compression (Φ) is defined via the spectral entropy of the covariance matrix, capturing changes in the effective dimensionality of system behavior. Adaptive capacity (R) is operationalized as a recovery-rate-based measure, defined as R = 1 - \mathrm{AR}(1), reflecting the system’s responsiveness to perturbations. The resulting viability index T = R / \Phi provides a combined diagnostic perspective on stability.   A central contribution of this work is the identification of mechanism-dependent regimes governing the relationship between structural and dynamical indicators. In Relaxation–Coupling Failure Mode (RCFM), structural and dynamical properties are intrinsically coupled, limiting the additional value of structural indicators. In contrast, Structural–Dynamic Separability (SDS) describes systems in which structural compression and dynamical behavior evolve independently. In these regimes, structural compression can decline prior to observable increases in variance, providing earlier warning signals of instability.   This framework does not aim to replace classical early warning signals but rather to define the conditions under which structural indicators provide complementary and potentially earlier insight into system collapse. The approach is applicable to a wide range of domains, including ecological, socio-technical, and engineered systems, where structural constraints may precede observable dynamical instability.   Early Warning Signals Critical Transitions Complex Systems Structural Compression Adaptive Capacity   Covariance Structure Spectral Entropy Effective Rank AR(1) Stability Analysis   Mechanism-Dependent Regimes Structural–Dynamic Separability Relaxation–Coupling Failure Mode System Collapse Multivariate Dynamics