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Detecting and forecasting tipping points from sample variance alone

T0 review · 3 major / 4 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read A new method, TIPMOC, detects approaching bifurcations and forecasts their location using only the sample variance, with high detection rates and low false positives.

desk verdict A clean, honest small-step methods paper: TIPMOC turns variance-based EWS into a sequential power-law model comparison with strong detection performance and modest forecasting; the statistical formalism is approximate and the missing confidence interval for u_c needs attention, but it deserves a real referee. read the letter →

arxiv 2602.10817 v2 pith:EJJATPQ6 submitted 2026-02-11 physics.soc-ph

Naoki Masuda This is my paper · ORCID
classification physics.soc-ph
keywords tippingpointsearlywarningsignalsbifurcationdetectionsamplevariancepower-lawdivergenceAICcmodelselectioncriticalslowingdownstochasticdynamicalsystems
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper introduces TIPMOC, a parametric framework that treats the sample variance as a control parameter changes and, at each step, decides whether the variance is diverging as a power law—an unmistakable sign of an approaching codimension-one bifurcation—or merely rising linearly. When the power-law fit wins by a wide margin for several consecutive steps, TIPMOC sounds an alarm and extrapolates the variance's divergence point as the predicted bifurcation location. The claim is that this simple, transparent procedure works across saddle-node, transcritical, and Hopf bifurcations, handles unevenly spaced control-parameter values and colored noise, and almost never raises false alarms on systems that merely grow noisier without bifurcating. A sympathetic reader would care because it offers a statistically controlled upgrade to the classic variance-based early warning signal, producing a forecast of when and where a tipping point will occur rather than only a vague 'something is changing.'

What carries the argument

The load-bearing object is the power-law divergence law for stationary variance near a codimension-one bifurcation, V(u)∝(uc−u)^{−γ} with γ=1/2 for saddle-node and γ=1 for transcritical, pitchfork, and Hopf. The method fits this curve to the observed pairs (u,Vhat) by optimizing over the divergence point ûc and baseline b, then uses the corrected Akaike Information Criterion (AICc) to compare the power-law model against a linear model. The alarm rule—three consecutive ΔAICc crossings of −10—is the decision mechanism that converts curve-fitting into a sequential early-warning detector.

What would settle it

Take a system with no bifurcation but a variance that increases linearly with the control parameter, generate 100 independent realizations, and run TIPMOC: the claim of low false positives predicts zero alarms. Conversely, for a system known to undergo a saddle-node bifurcation, run TIPMOC with L=50 samples instead of L=100; the claim that the method is data-efficient at this scale predicts detection in a large majority of runs, whereas a sharp drop in detection rate would falsify that part of the claim.

Watch

Extended reading notes

Core claim

The central discovery is that by sequentially fitting a power-law function V(u)=a(ûc−u)^−γ+b to the observed sample variance and comparing its AICc score against a linear fit, one can identify with high probability that a codimension-one bifurcation is approaching, and estimate the bifurcation point ûc, before the system actually tips. The method exploits the known mathematical fact that true stationary variance diverges as (uc−u)^−1/2 for saddle-node and as (uc−u)^−1 for transcritical, pitchfork, and Hopf bifurcations. TIPMOC declares an alarm only when the power-law model beats the linear model by an AICc margin of at least 10 for three consecutive control-parameter values, a guard against

Load-bearing premise

The entire statistical comparison rests on the assumption that the computed sample variance at each control-parameter value is a faithful estimate of the true stationary variance—meaning the system is at stationarity, the L=100 samples are approximately independent (Tskip large enough), and no unrelated non-stationarity intrudes; if these break, the variance estimates become biased and the AICc comparison loses its meaning.

Editorial extensions

If this is right

  • TIPMOC can be used as an add-on to any workflow that already computes sample variance, turning a trend that is usually summarized by Kendall's τ into a formal test with a predicted bifurcation location.
  • The method supplies a point estimate ûc of the tipping threshold, which classic scalar EWSs do not provide.
  • Because it only needs pairs (u, Vhat), it applies to other power-law-divergent EWSs such as standard deviation, node-averaged variance, and leading eigenvalue of the covariance matrix.
  • The three-consecutive-crossing rule with a conservative AICc threshold yields near-zero false positives in non-bifurcating systems where variance rises linearly, addressing a documented weakness of Kendall's τ.
  • TIPMOC remains effective when control-parameter values are unevenly spaced and when dynamical noise is colored, though the accuracy of ûc degrades under colored noise.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A natural extension is to use the fitted exponent γ to classify bifurcation type; the paper notes this is currently unreliable, but a smoothed or windowed version of TIPMOC might make γ estimates stable enough.
  • TIPMOC assumes the sample variance is computed at stationarity for each u; in real time-series where u drifts continuously, one could combine rolling windows with TIPMOC, but the interplay of non-stationarity and the AICc comparison remains untested here.
  • Because the method compares only two parametric curves, it could be extended to a Bayesian model selection (e.g., Bayes factors) that outputs posterior probabilities and intervals for ûc, giving the uncertainty quantification the paper currently lacks.
  • The low false-positive rate in the OU process suggests TIPMOC might serve as a screening tool in high-dimensional monitoring, flagging only those observables whose variance truly diverges; whether this holds with short time series (L < 100) is an open empirical question.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper introduces TIPMOC, a sequential framework that fits a divergent power-law model to the sample variance as a control parameter changes and compares it, via AICc, with a linear trend. When the power-law fit is preferred by a conservative threshold for three consecutive steps, TIPMOC declares an impending bifurcation and outputs an estimated bifurcation point. The method is tested on five stochastic dynamical systems (double-well, over-harvesting with saddle-node, linear-grazing transcritical, Rosenzweig–MacArthur Hopf, and mutualistic network), on unevenly spaced control parameters, and on colored noise, with detection rates between 93% and 100% and no false positives in two non-bifurcating null models.

Significance. If the results hold, TIPMOC is a valuable, transparent addition to the EWS toolbox: it uses only a classical scalar summary (sample variance), avoids explicit dynamical-system fitting, and provides both a detection decision and a point forecast of the bifurcation location. The power-law divergence is derived from standard bifurcation normal forms, the code and data are publicly available, and the simulation battery spans several bifurcation types. The main strengths are the simplicity of the approach, the reproducible implementation, and the explicit false-positive tests against non-bifurcating models with rising variance.

major comments (3)
  1. [Methods, 'Fitting of a power-law function' and 'AICc'] The AICc comparison is not a likelihood-based model comparison. The power-law parameters are obtained by maximizing the Pearson correlation in log-space and then linear regression, not by maximum likelihood or least squares in the original space, whereas the linear model is fit by ordinary least squares. Eq. (14) is therefore applied to a heuristic fit, as the manuscript acknowledges. This makes the 'statistical' interpretation of the ΔAICc threshold unclear. The two false-positive nulls are useful but do not calibrate the threshold generally. Please either implement a proper nonlinear least-squares/MLE fit and show that the heuristic approximation does not materially change ΔAICc, or explicitly present TIPMOC as an empirically validated heuristic score rather than an AICc-based test.
  2. [Methods, 'Simulation methods and computation of early warning signals'] The assumption that Tskip=1 yields 'approximately uncorrelated' samples is not verified and is questionable near a bifurcation, where the relaxation time diverges. For example, in the double-well system at u≈2.7, |Re(λ)|≈1.2, implying a lag-1 autocorrelation of roughly e^{-1}≈0.37 for Tskip=1, which is not negligible. Correlated samples inflate the variability of the variance estimates and violate the i.i.d.-residual assumption underlying Eq. (14). Please report autocorrelation diagnostics for each model, or select Tskip adaptively, and show that the main detection and false-positive results are robust to Tskip.
  3. [Introduction and Results (Eq. (1), Fig. 3, Table 1)] The introduction motivates the need for estimating the bifurcation value 'with quantified uncertainty (e.g., confidence intervals)', but TIPMOC outputs only a point estimate û_c. Table 1 reports across-run means and standard deviations, which do not provide uncertainty for a single observed sequence. Because the forecasting claim depends on the reliability of û_c, please either provide a within-sequence uncertainty measure (e.g., bootstrap or profile-likelihood interval) or state explicitly that no such uncertainty is available and temper the forecasting language accordingly.
minor comments (4)
  1. [Fig. 2] The text says 'The power-law and linear fits ... are shown as the blue and red dashed lines, respectively, in Fig. 2(b)', but those fits appear in Fig. 2(a); Fig. 2(b) shows only ΔAICc. Please correct the cross-reference.
  2. [Methods, 'Sample variance for the normal form of a Hopf bifurcation'] The text states that the variances are 'both equal to −σ/(2u)(>0)', but Eq. (6) gives −σ²/(2u). The missing square on σ appears to be a typo.
  3. [Methods, 'Simulation methods' / Table 1] The 'random u' implementation draws sorted uniform values, not exponentially spaced values; the text says the intervals 'approximately obey an exponential distribution.' This is a minor wording issue, but the simulation protocol should be described accurately.
  4. [Discussion] For the colored-noise double-well row, the mean û_c (3.687) exceeds the deterministic uc and the standard deviation is very large (3.859). The text notes this, but a median and interquartile range would better convey the distribution and would be less sensitive to outliers.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the power-law divergence is derived from standard bifurcation normal forms, and u_c is an extrapolated model parameter, not a restatement of the inputs.

full rationale

The central derivation is self-contained. Equation (1) is motivated by the standard result V(u) ∝ 1/|Re(λ)| for codimension-one bifurcations, and the saddle-node, transcritical, and pitchfork exponents are computed directly from normal forms; the Hopf case is solved from the Lyapunov equation in Methods. TIPMOC then fits this four-parameter curve to observed (u, Vhat) pairs and compares it to a linear fit via AICc. The reported u_c is the fitted divergence point of the power-law curve—an extrapolated parameter that is not present in the input data—and detection is a model-selection decision validated against non-bifurcating controls (OU and K=2, 0% false positives in 100 runs each). Self-citations [36,57,58,73,74] appear only for network-specific extensions or for the particular LFR network instance; they are not used to justify the core power-law model, to exclude alternative models, or to import a uniqueness result. The acknowledged approximations (AICc computed from a heuristic fit; no confidence interval for u_c) are limitations, not circularities. No load-bearing step reduces to its own input by construction.

Assumptions & free parameters 8 free parameters · 4 assumptions · 0 invented entities

The method's free parameters are largely the fitted constants of the power-law curve plus a few hand-chosen thresholds. The statistical validity of the AICc comparison rests on the approximation that a heuristic fit behaves like an MLE. No new physical entities are postulated.

free parameters (8)
  • a (power-law amplitude)
    Fitted to variance data in Eq. (1).
  • u_c (divergence point)
    Fitted to variance data; serves as the forecast of bifurcation location.
  • gamma (power-law exponent)
    Fitted rather than fixed to theoretical 0.5 or 1; central to identifying the scaling and distinguishing bifurcation types.
  • b (offset)
    Fitted to variance data.
  • AICc threshold (=10) = 10
    Chosen by hand following Burnham & Anderson; not derived from data.
  • consecutive crossings (=3) = 3
    Chosen to guard against isolated outliers.
  • initial fitting window (ℓ0=8) = 8
    Chosen to ensure enough points for fitting.
  • grid search bounds for u_c and b = u_c max = u_l + 10(u_l-u_1); b range specified
    Chosen to bound the optimization; affects the fit and thus detection and forecasts.
assumptions (4)
  • domain assumption Stationary variance of a stochastic differential equation near a codimension-one bifurcation scales inversely with |Re(lambda)|.
    Used to justify power-law form Eq. (1) near the bifurcation; standard linearization result, but assumes the noise is small and the normal form applies.
  • domain assumption The sample variance Vhat(u) computed from L=100 samples approximately equals the true stationary variance of the observable at that u.
    Assumes stationarity at each u, approximate independence of samples after Tskip, and L large enough for a reliable estimate.
  • ad hoc to paper AICc computed from residual sum of squares of the heuristic fit approximates the AICc of the true maximum-likelihood model.
    The paper uses grid search + linear regression, not maximum likelihood, to fit the power-law; it then uses RSS as if it were from an MLE to compute AICc (Methods, 'Fitting of a power-law function').
  • domain assumption Residuals of both the linear and power-law models are i.i.d. normal.
    Needed for the AICc likelihood expression; not tested.

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Cite this review

Pith. "Pith review of Detecting and forecasting tipping points from sample variance alone." pith.science (2026). https://pith.science/paper/EJJATPQ6

@misc{pith2026260210817,
  author       = {Pith},
  title        = {Pith review of: Detecting and forecasting tipping points from sample variance alone},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EJJATPQ6}},
  note         = {Machine review of arXiv:2602.10817}
}
read the original abstract

Anticipating tipping points in complex systems is a fundamental challenge across domains. Traditional early warning signals (EWSs) based on critical slowing down, such as increasing sample variance, are widely used, but their ability to reliably indicate imminent bifurcations and forecast their timing remains limited. Here, we introduce TIPMOC (TIpping via Power-law fits and MOdel Comparison), a parametric framework designed to statistically detect the approach of a bifurcation and estimate its future location using only the sample variance. TIPMOC exploits the mathematical property that variance diverges with a characteristic power-law form near codimension-one bifurcations. By sequentially monitoring system variance as a control parameter changes, TIPMOC statistically adjudicates between linear and power-law divergence at each step. When evidence favors power-law divergence, TIPMOC forecasts the impending tipping point and estimates its position; otherwise, it avoids false positives. Through numerical simulations, we demonstrate TIPMOC's robustness and accuracy in both detection and timing prediction across different types of dynamics and bifurcation, whereas the accuracy of timing prediction is limited. TIPMOC shows low false positive rates and performs well even with uneven sampling and colored noise. This method thus enhances the interpretability and practical utility of classical EWSs, serving as both a transparent add-on and a stand-alone statistical tool for forecasting regime shifts in diverse complex systems.

Figures

Figures reproduced from arXiv: 2602.10817 by the authors.

Figure 1
Figure 1. Schematic of a power-law fit used by TIPMOC. The value of the control parameter at which the fitted power law diverges, ˆuc, is the estimated bifurcation point. approaching19, 21, addressing the first two limitations. Binary classifiers trained with machine learning can realize this goal as well26. Second, some model-based21, 30, 31 and machine learning24, 27, 28 EWS methods estimate the value of u at which the bifu… view at source ↗
Figure 2
Figure 2. Detection of the saddle-node bifurcation point for the stochastic double-well system. (a) Vˆ as a function of u, shown as circles, for one simulation. An impending bifurcation is detected at udet = 2.702, and the bifurcation point is estimated as uˆc = 3.185. The solid line represents the power-law fit to all the observed (u, Vˆ ) pairs. The blue dashed line represents the power-law fit at detection (i.e., fit using… view at source ↗
Figure 3
Figure 3. Results of early detection of the bifurcation point for 100 runs of the stochastic double-well dynamics. (a) Relationship between the control parameter at detection, i.e., udet, and the bifurcation point predicted at u = udet, i.e., ˆuc. (b) Relationship between Kendall’s τ and ˆuc. Each circle corresponds to one run. The horizontal dashed lines show the deterministic saddle-node bifurcation point, uc ≈ 3.079, in th… view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Vˆ for three runs of different dynamical systems and detection of impending bifurcations. (a) Over-harvesting model with K = 10, showing a saddle-node bifurcation. (b) Over-harvesting model with linear grazing, showing a transcritical bifurcation. (c) Rosenzweig-MacArt…

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Works this paper leans on

58 extracted references · 1 linked inside Pith

  1. [2]

    R., Dakos, V

    Scheffer, M., Carpenter, S. R., Dakos, V . & van Nes, E. H. Generic indicators of ecological resilience: inferring the chance of a critical transition.Annu. Rev. Ecol. Evol. Syst.46, 145–167 (2015)

    2015

  2. [3]

    Liu, Y ., Kumar, M., Katul, G. G. & Porporato, A. Reduced resilience as an early warning signal of forest mortality.Nat. Clim. Chang.9, 880–885 (2019)

    2019

  3. [4]

    Wunderling, N.et al.Recurrent droughts increase risk of cascading tipping events by outpacing adaptive capacities in the Amazon rainforest.Proc. Natl. Acad. Sci. USA119, e2120777119 (2022)

    2022

  4. [5]

    Observation-based early-warning signals for a collapse of the Atlantic Meridional Overturning Circulation.Nat

    Boers, N. Observation-based early-warning signals for a collapse of the Atlantic Meridional Overturning Circulation.Nat. Clim. Chang.11, 680–688 (2021)

    2021

  5. [6]

    M.et al.Remotely sensing potential climate change tipping points across scales.Nat

    Lenton, T. M.et al.Remotely sensing potential climate change tipping points across scales.Nat. Commun.15, 343 (2024). 7.Southall, E., Brett, T. S., Tildesley, M. J. & Dyson, L. Early warning signals of infectious disease transitions: a review.J. R. Soc. Interface18, 20210555 (2021)

    2024

  6. [8]

    Public Heal.3, e0002253 (2023)

    Delecroix, C.et al.The potential of resilience indicators to anticipate infectious disease outbreaks, a systematic review and guide.PLoS Glob. Public Heal.3, e0002253 (2023)

    2023

  7. [9]

    A.et al.Critical slowing down as early warning for the onset and termination of depression.Proc

    van de Leemput, I. A.et al.Critical slowing down as early warning for the onset and termination of depression.Proc. Natl. Acad. Sci. USA111, 87–92 (2014)

    2014

  8. [10]

    & Bacilieri, A

    Dablander, F., Pichler, A., Cika, A. & Bacilieri, A. Anticipating critical transitions in psychological systems using early warning signals: Theoretical and practical considerations.Psychol. Methods28, 765–790 (2023). 12/15

    2023

Show all 58 references
  1. [11]

    Liu, X.et al.Detection for disease tipping points by landscape dynamic network biomarkers.Nat. Sci. Rev.6, 775–785 (2019)

    2019

  2. [12]

    & Chen, L

    Aihara, K., Liu, R., Koizumi, K., Liu, X. & Chen, L. Dynamical network biomarkers: Theory and applications.Gene808, 145997 (2022)

    2022

  3. [13]

    & Hastings, A

    Boettiger, C., Ross, N. & Hastings, A. Early warning signals: the charted and uncharted territories.Theor . Ecol.6, 255–264 (2013)

    2013

  4. [14]

    R., van Nes, E

    Dakos, V ., Carpenter, S. R., van Nes, E. H. & Scheffer, M. Resilience indicators: prospects and limitations for early warnings of regime shifts.Phil. Trans. R. Soc. B370, 20130263 (2015)

    2015

  5. [15]

    16.Scheffer, M.et al.Anticipating critical transitions.Science338, 344–348 (2012)

    Dakos, V .et al.Methods for detecting early warnings of critical transitions in time series illustrated using simulated ecological data.PLoS ONE7, e41010 (2012). 16.Scheffer, M.et al.Anticipating critical transitions.Science338, 344–348 (2012)

    2012

  6. [17]

    V ., Kachhara, S

    George, S. V ., Kachhara, S. & Ambika, G. Early warning signals for critical transitions in complex systems.Phys. Scripta 98, 072002 (2023)

    2023

  7. [18]

    & Laepple, T

    Rietkerk, M., Skiba, V ., Weinans, E., Hébert, R. & Laepple, T. Ambiguity of early warning signals for climate tipping points.Nat. Clim. Chang.15, 479–488 (2025)

    2025

  8. [19]

    & Hastings, A

    Boettiger, C. & Hastings, A. Quantifying limits to detection of early warning for critical transitions.J. R. Soc. Interface9, 2527–2539 (2012)

    2012

  9. [20]

    Perretti, C. T. & Munch, S. B. Regime shift indicators fail under noise levels commonly observed in ecological systems. Ecol. Appl.22, 1772–1779 (2012). 21.Heßler, M. & Kamps, O. Bayesian on-line anticipation of critical transitions.New J. Phys.24, 063021 (2022)

    2012

  10. [22]

    & Kamps, O

    Heßler, M. & Kamps, O. Quantifying resilience and the risk of regime shifts under strong correlated noise.PNAS Nexus2, pgac296 (2022)

    2022

  11. [23]

    & Kamps, O

    Heßler, M. & Kamps, O. Quantifying local stability and noise levels from time series in the US Western Interconnection blackout on 10th August 1996.Nat. Commun.16, 6246 (2025)

    1996

  12. [24]

    & Lai, Y .-C

    Kong, L.-W., Fan, H.-W., Grebogi, C. & Lai, Y .-C. Machine learning prediction of critical transition and system collapse. Phys. Rev. Res.3, 013090 (2021)

    2021

  13. [25]

    & Ott, E

    Patel, D. & Ott, E. Using machine learning to anticipate tipping points and extrapolate to post-tipping dynamics of non-stationary dynamical systems.Chaos33, 023143 (2023)

    2023

  14. [26]

    & Boers, N

    Huang, Y ., Bathiany, S., Ashwin, P. & Boers, N. Deep learning for predicting rate-induced tipping.Nat. Mach. Intell.6, 1556–1565 (2024)

    2024

  15. [27]

    Panahi, S.et al.Machine learning prediction of tipping in complex dynamical systems.Phys. Rev. Res.6, 043194 (2024)

    2024

  16. [28]

    Liu, Z.et al.Early predictor for the onset of critical transitions in networked dynamical systems.Phys. Rev. X14, 031009 (2024)

    2024

  17. [29]

    & Epureanu, B

    Chen, S., Ghadami, A. & Epureanu, B. I. Practical guide to using Kendall’s τ in the context of forecasting critical transitions.R. Soc. Open Sci.9, 211346 (2022)

    2022

  18. [30]

    & Gao, J

    Zhang, H., Wang, Q., Zhang, W., Havlin, S. & Gao, J. Estimating comparable distances to tipping points across mutualistic systems by scaled recovery rates.Nat. Ecol. Evol.6, 1524–1536 (2022). 31.Grziwotz, F.et al.Anticipating the occurrence and type of critical transitions.Sci...

    2022

  19. [32]

    Carpenter, S. R. & Brock, W. A. Rising variance: a leading indicator of ecological transition.Ecol. Lett.9, 311–318 (2006)

    2006

  20. [33]

    M., Bauch, C

    Bury, T. M., Bauch, C. T. & Anand, M. Detecting and distinguishing tipping points using spectral early warning signals.J. R. Soc. Interface17, 20200482 (2020). 34.Kuznetsov, Y . A.Elements of Applied Bifurcation Theory(Springer-Verlag, New York, NY , 2000), second edn. 35.Stro...

    2020

  21. [36]

    & MacLaren, N

    Masuda, N., Aihara, K. & MacLaren, N. G. Anticipating regime shifts by mixing early warning signals from different nodes.Nat. Commun.15, 1086 (2024). 13/15

    2024

  22. [37]

    Burnham, K. P. & Anderson, D. R.Model Selection and Multimodel Inference(Springer-Verlag, New York, NY , 2002), second edn

    2002

  23. [38]

    Symonds, M. R. E. & Moussalli, A. A brief guide to model selection, multimodel inference and model averaging in behavioural ecology using Akaike’s information criterion.Behav. Ecol. Sociobiol.65, 13–21 (2011)

    2011

  24. [39]

    Time-series-analysis-based detection of critical transitions in real-world non-autonomous systems.Chaos34, 072102 (2024)

    Lehnertz, K. Time-series-analysis-based detection of critical transitions in real-world non-autonomous systems.Chaos34, 072102 (2024)

    2024

  25. [40]

    & Kurths, J

    Witt, A., Neiman, A. & Kurths, J. Characterizing the dynamics of stochastic bistable systems by measures of complexity. Phys. Rev. E55, 5050–5059 (1997). 41.Gammaitoni, L., Hänggi, P., Jung, P. & Marchesoni, F. Stochastic resonance.Rev. Mod. Phys.70, 223–287 (1998). 42.Horsthe...

    1997

  26. [43]

    & Schimansky-Geier, L

    Lindner, B., Garcıa-Ojalvo, J., Neiman, A. & Schimansky-Geier, L. Effects of noise in excitable systems.Phys. Rep.392, 321–424 (2004). 44.Noy-Meir, I. Stability of grazing systems: an application of predator-prey graphs.J. Ecol.63, 459–481 (1975). 45.May, R. M. Thresholds and ...

    2004

  27. [46]

    & Jayaprakash, C

    Guttal, V . & Jayaprakash, C. Changing skewness: an early warning signal of regime shifts in ecosystems.Ecol. Lett.11, 450–460 (2008)

    2008

  28. [47]

    & Tabbaa, O

    Guttal, V ., Jayaprakash, C. & Tabbaa, O. P. Robustness of early warning signals of regime shifts in time-delayed ecological models.Theor . Ecol.6, 271–283 (2013)

    2013

  29. [48]

    Kéfi, S., Dakos, V ., Scheffer, M., Van Nes, E. H. & Rietkerk, M. Early warning signals also precede non-catastrophic transitions.Oikos122, 641–648 (2013)

    2013

  30. [49]

    S., Sharma, Y

    Dutta, P. S., Sharma, Y . & Abbott, K. C. Robustness of early warning signals for catastrophic and non-catastrophic transitions.Oikos127, 1251–1263 (2018)

    2018

  31. [50]

    M.et al.Deep learning for early warning signals of tipping points.Proc

    Bury, T. M.et al.Deep learning for early warning signals of tipping points.Proc. Natl. Acad. Sci. USA118, e2106140118 (2021)

    2021

  32. [51]

    Rosenzweig, M. L. & MacArthur, R. H. Graphical representation and stability conditions of predator-prey interactions. Am. Nat.97, 209–223 (1963)

    1963

  33. [52]

    Rosenzweig, M. L. Paradox of enrichment: destabilization of exploitation ecosystems in ecological time.Science171, 385–387 (1971). 53.Gao, J., Barzel, B. & Barabási, A. L. Universal resilience patterns in complex networks.Nature530, 307–312 (2016)

    1971

  34. [54]

    & Chen, W

    Zhuge, C., Li, J. & Chen, W. Deep learning for predicting the occurrence of tipping points.Royal Soc. Open Sci.12, 242240 (2025)

    2025

  35. [55]

    A.et al.Early warning signals have limited applicability to empirical lake data.Nat

    O’Brien, D. A.et al.Early warning signals have limited applicability to empirical lake data.Nat. Commun.14, 7942 (2023). 56.Qin, S. & Tang, C. Early-warning signals of critical transition: Effect of extrinsic noise.Phys. Rev. E97, 032406 (2018)

    2023

  36. [57]

    G., Kundu, P

    MacLaren, N. G., Kundu, P. & Masuda, N. Early warnings for multi-stage transitions in dynamics on networks.J. R. Soc. Interface20, 20220743 (2023)

    2023

  37. [58]

    Yu, S., MacLaren, N. G. & Masuda, N. Using covariance of node states to design early warning signals for network dynamics.Philos. Transactions Royal Soc. Ain press (2026). arXiv:2405.15982

    2026 arXiv

  38. [59]

    C., Strang, A

    Patterson, A. C., Strang, A. G. & Abbott, K. C. When and where we can expect to see early warning signals in multispecies systems approaching tipping points: Insights from theory.Am. Nat.198, E12–E26 (2021)

    2021

  39. [60]

    Brock, W. A. & Carpenter, S. R. Variance as a leading indicator of regime shift in ecosystem services.Ecol. Soc.11, 9 (2006)

    2006

  40. [61]

    Identifying best-indicator species for abrupt transitions in multispecies communities.Ecol

    Dakos, V . Identifying best-indicator species for abrupt transitions in multispecies communities.Ecol. Ind.94, 494–502 (2018)

    2018

  41. [62]

    B., Drake, J

    Chen, S., O’Dea, E. B., Drake, J. M. & Epureanu, B. I. Eigenvalues of the covariance matrix as early warning signals for critical transitions in ecological systems.Sci. Rep.9, 2572 (2019). 63.Bury, T. M.et al.Predicting discrete-time bifurcations with deep learning.Nat. Commun...

    2019

  42. [64]

    Deb, S.et al.Optimal sampling of spatial patterns improves deep learning-based early warning signals of critical transitions. R. Soc. Open Sci.11, 231767 (2024)

    2024

  43. [65]

    A.et al.Slow down and be critical before using early warning signals in psychopathology.Nat

    Helmich, M. A.et al.Slow down and be critical before using early warning signals in psychopathology.Nat. Rev. Psychol. 3, 767–780 (2024)

    2024

  44. [66]

    & Guttal, V

    Chen, N., Jayaprakash, C., Yu, K. & Guttal, V . Rising variability, not slowing down, as a leading indicator of a stochastically driven abrupt transition in a dryland ecosystem.Am. Nat.191, E1–E14 (2018). 67.Carpenter, S. R.et al.Early warnings of regime shifts: a whole-ecosys...

    2018

  45. [68]

    C., Psychosystems, ESM Group & EWS Group

    Wichers, M., Groot, P. C., Psychosystems, ESM Group & EWS Group. Critical slowing down as a personalized early warning signal for depression.Psychother. Psychosom.85, 114–116 (2016)

    2016

  46. [69]

    & Qureshi, M

    Gaji´c, Z. & Qureshi, M. T. J.Lyapunov Matrix Equation in System Stability and Control(Academic Press, San Diego, CA, 1995). 70.Gardiner, C.Stochastic Methods(Springer-Verlag, Berlin, 2009), fourth edn

    1995

  47. [71]

    H., Donangelo, R., Fort, H

    Dakos, V ., van Nes, E. H., Donangelo, R., Fort, H. & Scheffer, M. Spatial correlation as leading indicator of catastrophic shifts.Theor . Ecol.3, 163–174 (2010)

    2010

  48. [72]

    & Radicchi, F

    Lancichinetti, A., Fortunato, S. & Radicchi, F. Benchmark graphs for testing community detection algorithms.Phys. Rev. E78, 046110 (2008)

    2008

  49. [73]

    G., Barzel, B

    MacLaren, N. G., Barzel, B. & Masuda, N. Observing network dynamics through sentinel nodes.Nat. Commun.16, 10211 (2025)

    2025

  50. [74]

    G., Aihara, K

    MacLaren, N. G., Aihara, K. & Masuda, N. Applicability of spatial early warning signals to complex network dynamics.J. R. Soc. Interface22, 20240696 (2025). 15/15

    2025

Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.