$1/f$ noise in extremal dynamics

Abha Singh; Avinash Chand Yadav; Rahul Chhimpa

REVIEW 5 major objections 3 minor 38 references

$1/f$ noise in extremal dynamics

T0 review · 5 major / 3 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read A two-species update rule turns the Bak-Sneppen evolution model into a source of $1/f^\alpha$ noise with $\alpha \approx 1$.

desk verdict A minimal min-max BS variant that reports α≈1 for global fitness noise is a genuinely new data point, but the visible text carries no data or scaling analysis, so the robustness and hyper-universality claims are unsupported as they stand. read the letter →

arxiv 2508.01327 v1 pith:IKV7SCNV submitted 2025-08-02 cond-mat.stat-mech

Rahul Chhimpa , Abha Singh , Avinash Chand Yadav This is my paper

classification cond-mat.stat-mech PACS 05.40.-a05.65.+b

keywords Bak-Sneppenmodelself-organizedcriticality1/fnoisepinkspectralexponentextremaldynamicsfitnessfluctuationslogarithmicautocorrelation

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

This paper argues that a minimal change to the Bak-Sneppen evolution model makes the global fitness signal behave as canonical pink noise, $1/f^\alpha$ with $\alpha \approx 1$. In the standard model one always updates the least-fit species and its two neighbors; the proposed rule instead updates only the two species with the minimum and maximum fitness. The authors report that this change turns the power spectrum of global fitness into $1/f$ noise and that the two-time autocorrelation decays logarithmically, the usual companion of pink noise. They also claim this result is robust and hyper-universal, and that local pairwise cross-power spectra carry the noise. If true, the paper supplies the first example of exactly pink noise in a Bak-Sneppen-type extremal dynamics model.

What carries the argument

The load-bearing object is the modified extremal update rule on a ring of $N$ species with uniformly random fitnesses: at each step, pick the current minimum-fitness and maximum-fitness species and draw new fitnesses for both from the uniform distribution. The paper then analyzes the time series of the global fitness through its power spectrum $S(f)$, and uses the two-time autocorrelation function as a consistency check for $1/f$ noise. The local cross-power spectra between individual species are the diagnostic that identifies where the noise originates. The named quantity is the spectral exponent $\alpha$, defined by $S(f) \sim 1/f^{\alpha}$; $\alpha \approx 1$ is pink noise, $\alpha = 0$ white noise, and $\alpha = 2$ Brownian noise.

What would settle it

Simulate the min-max rule for system sizes $N$ spanning more than a decade, and where possible in dimensions $d = 1,2,3,4$, then compute $\alpha$ from the power spectrum. If $\alpha$ systematically drifts away from 1 as $N$ grows or as $d$ changes, the central hyper-universal pink-noise claim collapses; a second check is that the autocorrelation must remain logarithmic rather than becoming exponential.

Watch

Extended reading notes

Core claim

The central claim is that extremal dynamics can produce the canonical $1/f$ noise without changing the spirit of the Bak-Sneppen model: at each time step, reassign new random fitness values to the species with the smallest and the largest current fitness, rather than to the least fit and its neighbors. The global fitness then has a power spectrum $S(f) \sim 1/f^{\alpha}$ with the spectral exponent $\alpha$ close to 1, and the two-time autocorrelation decays as a logarithm. The paper identifies non-trivial local cross-power spectra as the dominant contribution to this noise, and asserts that the $\alpha \approx 1$ behavior is robust and hyper-universal, in contrast to earlier BS variants where $\alpha$ depends on dimension or variant details. The discovery is an extension of the SOC framework: it shows the update rule itself, not just the extremal threshold, controls the color of the noise.

Load-bearing premise

The claim that the pink noise is universal depends on the assumption that the measured spectral exponent near 1 is not an artifact of the limited system sizes simulated, and that it stays near 1 for larger systems, other dimensions, and other parameter values.

Editorial extensions

If this is right

The global fitness time series in the modified model should show a power spectrum consistent with $1/f^\alpha$, $\alpha \approx 1$, over the simulated frequency range.
The two-time autocorrelation of global fitness should decay logarithmically with time, the companion signature of pink noise.
If the behavior is hyper-universal, the spectral exponent should remain $\alpha \approx 1$ across dimensions and parameter choices, unlike the original BS model where it varies up to the upper critical dimension $D_u = 4$.
The dominance of non-trivial local cross-power spectra implies the pink noise is a collective effect of pairs of species rather than a single-site fluctuation feature.

Reading between the lines

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

Editorial extension: because the update targets the extremes, the $1/f$ spectrum may be derivable from order statistics of independent uniform fitnesses, which would give an analytic handle the paper does not pursue.
Editorial extension: a random-neighbor (mean-field) version of the min-max rule would test hyper-universality in a regime where the standard BS model gives $\alpha \approx 2$; if $\alpha$ stays near 1 there, the claim is much stronger.
Editorial extension: the logarithmic autocorrelation suggests an underlying superposition of relaxation times; fitting the distribution of waiting times between extreme updates could expose the same mechanism.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague,

Here is my honest read of arXiv:2508.01327. The genuinely new thing is the update rule: at each time step, replace the species with the current minimum and maximum fitness. That is a natural, minimal variant of Bak–Sneppen that nobody seems to have tried, and the reported α≈1 for global fitness noise would be a useful data point—the authors correctly note that earlier BS variants give α≠1 (1.2 for standard BS, 2 for random-neighbor, etc.). The introduction is clear and well-grounded; the discussion of 1/f mechanisms and the solvable MTZ case is appropriate.

What I cannot evaluate is the result itself. The text I was given stops after the introduction: no methods, no figures, no system-size analysis, no error bars. The abstract asserts robustness and hyper-universality, but those are strong claims. In this model family, the spectral exponent is known to be sensitive to dynamics and dimension, and α is defined only over a finite frequency window. Without a scaling plot of α versus system size and a clear statement of the fitting window, α≈1 is just a number. The logarithmic autocorrelation decay is consistent with many spectral shapes, so it does not pin α=1 by itself. The stress-test note's concern is therefore legitimate, and I don't think it is pedantic.

That said, I see no circularity or obvious error in the reasoning from what is visible. The authors are not fitting α as a parameter; they report a measured exponent. Their literature claim—no prior α=1 in BS variants—is plausible, and citing their own 2023 PRE paper is appropriate context.

Bottom line: this is a simple, potentially nice observation that deserves a serious referee, but not on the strength of the abstract alone. If the full manuscript includes finite-size scaling for α, a clear definition of the frequency window, and a test of hyper-universality across dimensions or parameters, it could be a solid PRE-style paper. If those are missing, the claims should be softened. I would send it to review, with a request that referees demand the scaling analysis.

For me: I'd bring it to reading group to discuss spectral exponent extraction, but I wouldn't cite it in my own work until the full data are out. The paper is coherent on its own terms, so serious thinker: yes.

Referee Report

5 major / 3 minor

Summary. The paper proposes a variant of the Bak–Sneppen model in which, at each time step, only the species with minimum and maximum fitness are updated. It claims that the global fitness fluctuations then exhibit 1/f^alpha noise with alpha approximately 1 (pink noise), that the two-time autocorrelation function decays logarithmically, that the behavior is robust and hyper-universal, and that non-trivial local fitness cross-power spectra dominate. The provided text consists of the abstract and the introduction only; no numerical methods, simulation data, spectral fits, finite-size scaling tests, or derivations are included.

Significance. If the central claims were fully substantiated, this would be a useful contribution: the update rule is simple, falsifiable, and would fill a gap explicitly identified in the introduction, where previous BS variants yield alpha values of about 1.2 to 2 rather than 1. The paper also situates itself honestly against prior work [33] and [34], and the spectral exponent is an empirical output rather than a fitted parameter, so circularity is not a concern. However, as submitted, the evidence for every quantitative claim is absent, so the significance cannot yet be assessed.

major comments (5)

[I (Introduction), final paragraph] The model is specified only by the sentence 'We suggest evolving only two species (having minimum and maximum fitness) at each time step.' Essential details are left unspecified: whether the two updates are simultaneous or sequential, how ties at the minimum or maximum are broken, whether the chosen species receive fresh independent fitnesses, how the 'global fitness' observable is defined (mean, sum, or minimum), and what system size, dimension, and boundary conditions are used. Without these details the numerical claim cannot be reproduced or checked.
[Abstract and I (Introduction)] The claimed spectral exponent alpha approximately 1 is stated but never supported by data. No power spectrum is shown, and no information is given about the sampling rate, frequency window, detrending procedure, ensemble size, or fitting method. Because the introduction itself reports alpha approximately 1.2 for the standard BS model and alpha approximately 2 in the random-neighbor mean-field limit [33], the difference between those values and alpha approximately 1 is exactly the sort of effect that could be a finite-size or finite-frequency-window artifact. A finite-size scaling analysis across system size and dimension, with explicit spectral fits and error estimates, is needed before the central claim can be accepted.
[Abstract and I (Introduction)] The claim that the two-time autocorrelation function decays logarithmically is made without showing any computed autocorrelation data. Moreover, a logarithmically decaying autocorrelation over a finite observation window does not uniquely determine a power spectrum with alpha = 1; the relation between the measured C(t) and the fitted spectral exponent must be demonstrated explicitly for the finite time series used in the study.
[Abstract] The statement that the 1/f noise is 'robust and hyper-universal' is unsupported. No variation of system size, spatial dimension, parameter values, initial conditions, or disorder is reported. The introduction itself notes that the BS spectral exponent is dimension-dependent up to the upper critical dimension [34], so hyper-universality requires explicit demonstration rather than assertion.
[Abstract] The claim of 'dominance of non-trivial local fitness cross-power spectra' is not defined or derived anywhere in the provided text. The manuscript must define the local cross-power-spectrum observable, explain how it is computed, and show the spectra before the abstract's claim can be evaluated.

minor comments (3)

[General] The provided text ends mid-sentence with 'the variance grows logarithmically with the' and jumps directly into the references; the manuscript appears to be missing its model definition, methods, results, and discussion sections.
[I (Introduction)] The notation for the power law is typeset inconsistently as '1/f^alpha', '1 /f alpha', and '1/f alpha'; please use a single consistent mathematical notation and define alpha's range once.
[References] References [35]–[37] appear in the reference list but have no visible in-text citation in the available text; please check the completeness of the citation list.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the claimed 1/f result is an empirical output of the proposed dynamics, not a restatement of its inputs.

full rationale

The paper proposes a variant of the Bak-Sneppen model in which only the minimum- and maximum-fitness species are updated, and it reports that the global fitness power spectrum has alpha approximately 1, with logarithmic autocorrelation decay. There is no derivation chain in which an output quantity is defined in terms of the target quantity: the spectral exponent is presented as a measured outcome, not as a fitted parameter or as an analytical consequence that assumes the desired result. The only self-citation is [33] (Singh, Chhimpa, and Yadav, Phys. Rev. E 108, 044109), used in the Introduction to quote previously reported spectral exponents for the standard BS model and its variants; this context is not load-bearing for the new proposal, and no uniqueness theorem or ansatz is imported from that work to force the alpha-approximately-1 result. The statements 'robust and hyper-universal' are strong empirical claims, and the visible text lacks the simulation details needed to support them, but missing support is a rigor concern rather than circularity. No equation equates the prediction to an input, and no fitted input is renamed as a prediction. Accordingly, no significant circularity is present.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

No free parameters are identifiable from the abstract; the model rule and stationarity are the main assumptions. No new entities are introduced.

assumptions (3)

domain assumption Fitness values are assigned from a uniform distribution initially and after each update.
Standard in Bak-Sneppen type models; not stated in the abstract as a variable.
ad hoc to paper The dynamics update exactly two species, the minimum and maximum fitness, at each time step.
This is the proposed new rule and is the core model definition.
domain assumption The system reaches a stationary state and the global fitness time series is stationary for spectral analysis.
Stationarity is required to interpret the power spectrum; not proven in the abstract.

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Cite this review

Pith. "Pith review of $1/f$ noise in extremal dynamics." pith.science (2026). https://pith.science/paper/IKV7SCNV

@misc{pith2026250801327,
  author       = {Pith},
  title        = {Pith review of: $1/f$ noise in extremal dynamics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IKV7SCNV}},
  note         = {Machine review of arXiv:2508.01327}
}

abstract

The Bak-Sneppen (BS) evolution model remains a well-studied example of self-organized criticality (SOC). We propose a simple variant of the BS model, where the global fitness fluctuations show $1/f^{\alpha}$ noise with a spectral exponent nearly equal to 1 (pink noise). To further corroborate, we compute the two-time autocorrelation function that decays logarithmically. The $1/f$ noise in the global fitness is robust and hyper-universal. We identify the dominance of non-trivial local fitness cross-power spectra.

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Reference graph

Works this paper leans on

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [33]

Davidsen \ and\ author N

author author J. Davidsen \ and\ author N. L\"uthje ,\ title Comment on `` 1/f noise in the Bak-Sneppen model'' , \ 10.1103/PhysRevE.63.063101 journal journal Phys. Rev. E \ volume 63 ,\ pages 063101 ( year 2001 ) NoStop

work page doi:10.1103/physreve.63.063101 2001

[2] [34]

Singh , author R

author author A. Singh , author R. Chhimpa , \ and\ author A. C. \ Yadav ,\ title Fitness fluctuations in the Bak-Sneppen model , \ 10.1103/PhysRevE.108.044109 journal journal Phys. Rev. E \ volume 108 ,\ pages 044109 ( year 2023 ) NoStop

work page doi:10.1103/physreve.108.044109

[3] [1]

merlin.mbs apsrev4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked

FUNCTION id.bst "merlin.mbs apsrev4-1.bst 2010-07-25 4.21a (PWD, AO, DPC) hacked" ENTRY address archive archivePrefix author bookaddress booktitle chapter collaboration doi edition editor eid eprint howpublished institution isbn issn journal key language month note number organization pages primaryClass publisher school SLACcitation series title translati...

work page 2010

[4] [2]

Dutta \ and\ author P

author author P. Dutta \ and\ author P. M. \ Horn ,\ title Low-frequency fluctuations in solids: 1 f noise , \ 10.1103/RevModPhys.53.497 journal journal Rev. Mod. Phys. \ volume 53 ,\ pages 497 ( year 1981 ) NoStop

doi:10.1103/revmodphys.53.497 1981

[5] [3]

author author R. F. \ Voss \ and\ author J. Clarke ,\ title ‘1/f noise’ in music and speech , \ https://doi.org/10.1038/258317a0 journal journal Nature (London) \ volume 258 ,\ pages 317 ( year 1975 ) NoStop

work page doi:10.1038/258317a0 1975

[6] [4]

author author D. J. \ Levitin , author P. Chordia , \ and\ author V. Menon ,\ title Musical rhythm spectra from Bach to Joplin obey a 1/f power law , \ 10.1073/pnas.1113828109 journal journal Proc. Natl. Acad. Sci. USA \ volume 109 ,\ pages 3716 ( year 2012 ) NoStop

doi:10.1073/pnas.1113828109

[7] [5]

Yu , author R

author author Y. Yu , author R. Romero , \ and\ author T. S. \ Lee ,\ title Preference of Sensory Neural Coding for 1/f Signals , \ 10.1103/PhysRevLett.94.108103 journal journal Phys. Rev. Lett. \ volume 94 ,\ pages 108103 ( year 2005 ) NoStop

work page doi:10.1103/physrevlett.94.108103

[8] [6]

author author R. V. \ Sol\'e , author S. C. \ Manrubia , author M. Benton , \ and\ author P. Bak ,\ title Self-similarity of extinction statistics in the fossil record , \ https://doi.org/10.1038/41996 journal journal Nature (London) \ volume 388 ,\ pages 764 ( year 1997 ) NoStop

work page doi:10.1038/41996