Fundamental Limits of Stability Inference in High-Dimensional Complex Systems

Abstract: Many complex systems, including ecosystems, neural circuits, and financial markets, are inferred to operate close to a threshold of instability, at which a small perturbation can propagate across the entire system. This proximity is often interpreted as functionally advantageous, yet it poses a question common to all these fields: from a finite, noisy recording, how precisely can the distance of a system from that threshold be estimated? Using the multivariate Ornstein-Uhlenbeck process as the canonical linear model of relaxation near a stable fixed point, we show that the attainable precision is governed by three factors: an effective measurement budget, set by the number of samples relative to the system dimension and the sampling interval; the signal-to-noise ratio, given by the magnitude of deterministic interactions relative to stochastic forcing; and the distance to criticality, which simultaneously sets the system's correlation times and degrades both of the preceding factors. As the slowest dynamical mode softens near the threshold, the curvature of the log-likelihood flattens along the direction that determines stability, so that the relative uncertainty on the estimated distance diverges as that distance vanishes. Critically, temporal correlations near instability reduce the effective number of independent observations far below the nominal sample count, and inference breaks down when this effective count falls below the system dimension, even when the raw data volume appears sufficient. A direct consequence is the existence of an optimal sampling interval that diverges as the system approaches criticality, with practical implications for experimental design.