Fundamental Limits of Stability Inference in High-Dimensional Complex Systems

Flaviano Morone; Kentaro Hoshisashi; Michela Costa; Paolo Barucca; Tim Rogers

arxiv: 2606.23644 · v1 · pith:YY6IAXDJnew · submitted 2026-06-22 · ❄️ cond-mat.dis-nn

Fundamental Limits of Stability Inference in High-Dimensional Complex Systems

Michela Costa , Kentaro Hoshisashi , Flaviano Morone , Tim Rogers , Paolo Barucca This is my paper

Pith reviewed 2026-06-26 05:44 UTC · model grok-4.3

classification ❄️ cond-mat.dis-nn

keywords stability inferenceOrnstein-Uhlenbeck processcriticalityhigh-dimensional systemslog-likelihood curvaturecomplex systemsdynamical modesparameter estimation

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The pith

The relative uncertainty on the estimated distance to instability diverges as that distance vanishes.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes fundamental limits on inferring how close a high-dimensional complex system is to a stability threshold from finite noisy recordings. Modeling dynamics via the multivariate Ornstein-Uhlenbeck process, it shows that precision is set by three factors: effective measurement budget from the number of samples relative to dimension and sampling interval, signal-to-noise ratio of deterministic interactions versus stochastic forcing, and the distance to criticality itself. As the slowest mode softens near the threshold, the log-likelihood curvature flattens along the stability direction, making relative uncertainty diverge. Temporal correlations reduce effective independent observations far below the nominal sample count, so inference fails when that effective count drops below system dimension even if raw data volume looks ample, and an optimal sampling interval appears that grows without bound at criticality.

Core claim

Using the multivariate Ornstein-Uhlenbeck process as the canonical linear model of relaxation near a stable fixed point, the attainable precision for estimating the distance to instability is governed by an effective measurement budget, the signal-to-noise ratio, and the distance to criticality, which simultaneously sets correlation times and degrades 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.

What carries the argument

The curvature of the log-likelihood along the stability direction in the multivariate Ornstein-Uhlenbeck process, which flattens as the slowest eigenvalue approaches zero.

Load-bearing premise

The system dynamics are accurately described by the multivariate Ornstein-Uhlenbeck process as the canonical linear model of relaxation near a stable fixed point, with stability set by the eigenvalues of the drift matrix.

What would settle it

A numerical simulation or analytic calculation in which the relative uncertainty on the stability distance stays finite as the distance to criticality is driven to zero while holding measurement budget and signal-to-noise ratio fixed.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Derives that relative uncertainty on distance to criticality diverges in high-dim OU processes, with an optimal sampling interval that also diverges near threshold.

read the letter

The one thing to know is that this paper derives diverging relative uncertainty on the estimated distance to criticality for multivariate OU processes, because the log-likelihood curvature flattens along the stability direction as the slowest eigenvalue approaches zero. They also show that an optimal sampling interval diverges as the system nears the threshold, and that effective independent samples can fall below dimension due to correlations even when raw data volume looks adequate.

The work does a good job making these limits quantitative in terms of effective sample budget, SNR, and distance to criticality. The dependence follows directly from the eigenvalue softening of the drift matrix and the associated Fisher information, which gives a clean account of why precision degrades. The reminder that temporal correlations reduce independent observations is useful for anyone working with high-dimensional time series.

Soft spots are limited. Everything rests on the OU process being an accurate model near a stable fixed point, which is the standard linear approximation but may not capture nonlinear or non-Gaussian effects in real data. The derivations are not fully reproduced in the abstract, so minor calculation slips in the high-dimensional regime cannot be ruled out, though the logic is consistent with standard OU likelihood results. No circularity or load-bearing fitting issues appear.

This is for researchers doing stability inference from noisy recordings in ecology, neuroscience, or finance. A reader who needs bounds on how precisely distance to instability can be recovered from finite data will find practical guidance. It deserves a serious referee because the result is a specific, falsifiable derivation with experimental design implications.

Recommendation: send to peer review.

Referee Report

0 major / 1 minor

Summary. The manuscript derives fundamental limits on inferring the distance to criticality (stability threshold) for high-dimensional systems from finite noisy observations, using the multivariate Ornstein-Uhlenbeck process as the model. Precision is shown to depend on an effective measurement budget (samples relative to dimension and sampling interval), signal-to-noise ratio, and the distance to criticality itself; as the slowest eigenvalue softens, log-likelihood curvature flattens along the stability direction, causing relative uncertainty to diverge, effective independent samples to drop below dimension, and an optimal sampling interval to exist that diverges near threshold.

Significance. If the derivations hold, the result supplies a parameter-free statistical bound on stability inference that is directly relevant to ecology, neuroscience, and finance. It explains why large raw data volumes can still yield unreliable stability estimates near criticality and identifies concrete experimental-design consequences (optimal sampling). The grounding in standard OU statistics without additional fitted parameters or ad-hoc entities is a clear strength.

minor comments (1)

The abstract states the central claims coherently, but the absence of explicit section or equation references in the provided materials prevents direct verification of the Fisher-information steps that underpin the divergence result.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading and for providing a concise and accurate summary of the manuscript's main results. We are pleased that the referee recognizes the potential relevance to ecology, neuroscience, and finance, as well as the grounding in standard OU statistics. Below we respond to the major comments.

Circularity Check

0 steps flagged

Derivation self-contained within OU process framework

full rationale

The paper derives its central results on inference precision limits directly from the eigenvalue structure and Fisher information of the multivariate Ornstein-Uhlenbeck process, without any reduction of predictions to fitted inputs by construction or load-bearing self-citations. The flattening of log-likelihood curvature near criticality and the divergence of relative uncertainty follow as mathematical consequences of the linear SDE model against external benchmarks of sample size, SNR, and effective observations. All stated consequences (optimal sampling interval, breakdown when effective count falls below dimension) are consistent derivations from the same model assumptions, with no internal equivalence to inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The derivation rests on the standard multivariate OU process as the model for linear relaxation; no free parameters or invented entities are introduced in the abstract.

axioms (1)

domain assumption Dynamics near a stable fixed point are described by the multivariate Ornstein-Uhlenbeck process.
Explicitly stated as the canonical linear model in the abstract.

pith-pipeline@v0.9.1-grok · 5801 in / 1099 out tokens · 30541 ms · 2026-06-26T05:44:11.919474+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

14 extracted references · 3 canonical work pages

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May, R.M.: Will a large complex system be stable? Nature238, 413–414 (1972)

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Nature483, 205–208 (2012)

Allesina, S., Tang, S.: Stability criteria for complex ecosystems. Nature483, 205–208 (2012)

2012

[3] [3]

Physical Review E100, 032307 (2019) 21

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Princeton University Press, ??? (2012)

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Mora, T., Bialek, W.: Are biological systems poised at criticality? Journal of Statistical Physics144, 268–302 (2011)

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Nature461, 53–59 (2009)

Scheffer, M.,et al.: Early-warning signals for critical transitions. Nature461, 53–59 (2009)

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Journal of Statistical Mechanics2011, 10012 (2011)

Mastromatteo, I., Marsili, M.: On the criticality of inferred models. Journal of Statistical Mechanics2011, 10012 (2011)

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Marchenko, V.A., Pastur, L.A.: Distribution of eigenvalues for some sets of random matrices. Matematicheskii Sbornik114(4), 507–536 (1967)

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Ferreira, L.S., Metz, F.L., Barucca, P.: Random matrix ensemble for the covari- ance matrix of ornstein-uhlenbeck processes with heterogeneous temperatures. Physical Review E111, 014151 (2025) https://doi.org/10.1103/PhysRevE.111. 014151

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work page doi:10.1103/physreve.98.012136 2017

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2018