The Rise and Fall of Ideas' Popularity

Alessio Muscillo; Claudio Pacati; Paolo Pin; Piero Mazzarisi

REVIEW 3 major objections 4 minor 35 references

The Rise and Fall of Ideas' Popularity

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read A minimal feedback rule that makes the rate of losing interest rise with overexposure and fall with remaining potential turns the SIRS opinion model into a source of stable popularity cycles, and these cycles match detrended Google Trends…

desk verdict Tractable new SIRS variant with a honest theory section, but the Google Trends validation is too asymmetric to carry the paper's empirical claim. read the letter →

arxiv 2411.18541 v2 pith:CYZVG6C3 submitted 2024-11-27 econ.GN q-fin.EC

Piero Mazzarisi , Alessio Muscillo , Claudio Pacati , Paolo Pin This is my paper · ORCID

classification econ.GNq-fin.EC MSC 34C2334C2537G15

keywords IdeadiffusionSIRSmodelOpiniondynamicsEndogenousfluctuationsLimitcyclesHopfbifurcationGoogleTrendsDynamictimewarping

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 claims that the rise and fall of an idea's popularity need not be imposed from outside: a minimal internal feedback can make popularity oscillate on its own. The authors take the epidemiological SIRS model, read the three compartments as people open to an idea, actively spreading it, and temporarily tired of it, and let the rate at which spreaders lose interest depend on the current state: it rises with the number of spreaders (interest saturation) and falls with the number of people still open to the idea (influencing enthusiasm). With that feedback, the model's steady state becomes unstable and the system settles into stable limit cycles, so popularity naturally revives and collapses without any external shock. They then compare the model's oscillations with detrended and deseasonalized Google Trends residuals for 1,000 queries and report that the model fits better than a random walk in 67% of cases. If this is right, it gives a one-equation mechanism for fads, virality, and opinion cycles that many current models relegate to exogenous noise.

What carries the argument

The load-bearing object is the endogenous recovery rate $\Gamma(t)$, whose log evolves through $\frac{d\Gamma}{dt} = \Gamma(\alpha I - \delta S)$ alongside the SIRS equations for $S$ and $I$. This single state-dependent rate encodes the two psychological mechanisms: the positive $\alpha I$ term says widespread promotion accelerates loss of interest (interest saturation), and the negative $\delta S$ term says a large pool of potential adopters makes current promoters hold on longer (influencing enthusiasm). The rate is what destabilizes the otherwise stable SIRS fixed point; the Hopf bifurcation proof for $\alpha = \delta$ shows that the resulting oscillations are genuine stable limit cycles, not transient decay or noise.

What would settle it

Run the same per-query $\beta$-fitting and DTW comparison on control series that have the same length, variance, smoothness, and autocorrelation as the real residuals but are generated by a stationary process with no feedback (for example, shuffled residuals or matched ARMA simulations). If the fitted model beats random walks on those controls at a rate close to 67%, the reported result is an artifact of fitting and series texture rather than evidence for the feedback mechanism.

Watch

Extended reading notes

Core claim

The central claim is that the feedback rule $\frac{d\Gamma}{dt} = \Gamma(\alpha I - \delta S)$, appended to the standard SIRS equations, is enough to turn the system into a generator of sustained popularity cycles. Here $\Gamma$ is the recovery rate, $I$ the fraction actively promoting the idea, $S$ the fraction still open to it, and $\alpha, \delta > 0$. The paper proves, as Proposition 1, that the unique interior fixed point is locally unstable exactly when $\alpha > \beta + \xi$ and $\delta \geq \frac{\alpha^2\xi}{(\alpha-\beta)(\alpha-\beta-\xi)}$, and, as Proposition 2, that in the restricted case $\alpha = \delta$ a Hopf bifurcation occurs at $\alpha = \beta + \xi + \sqrt{\xi(\beta+\xi)}$ provided $\beta > \frac{11}{25}\xi$, so stable limit cycles emerge from the fixed point. The empirical section then claims that these periodic solutions, with the transmission rate $\beta$ chosen per query to minimize Dynamic Time Warping distance, match the residuals of 1,000 Google Trends series after trend and seasonality removal better than the average of 500 random walks in 67% of cases. The intended upshot is that observed popularity volatility can be an intrinsic property of the diffusion process rather than a sequence of exogenous shocks.

Load-bearing premise

The empirical evidence depends on the assumption that detrended and deseasonalized Google Trends residuals stand in for the model's infected population, and that fitting the transmission rate $\beta$ per query to minimize in-sample Dynamic Time Warping distance, then comparing the fitted model against unfitted random walks, is a valid way to show the residuals' oscillations are endogenous.

Editorial extensions

If this is right

Popularity cycles can be explained without external shocks: the same SIRS-plus-feedback system produces recurring booms and busts from its own dynamics.
There is a sharp parameter threshold: once $\alpha$ and $\delta$ are large enough to satisfy the instability condition, small perturbations grow into limit cycles rather than decaying, so the model predicts a qualitative switch from damping to sustained oscillation.
The Hopf bifurcation locates that switch explicitly in the symmetric case, so for $\alpha = \delta$ the period and amplitude of the cycle are determined by the model's parameters and can be computed from the local crossing.
The empirical comparison implies that a substantial share of Google search dynamics after detrending is structured oscillation rather than pure noise, since the model beats a random walk in 67% of 1,000 queries.

Reading between the lines

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

The same feedback could plausibly be inserted into simpler or network-based compartmental models, where it would likely produce oscillations as well; the paper does not test this.
A stricter out-of-sample test would split each time series, fit $\beta$ on the first half, and evaluate the DTW distance on the second half; the paper's in-sample fitting may overstate the 67% figure.
The model predicts that ideas with strong saturation and strong enthusiasm parameters should show more regular, longer-lasting cycles, which could be checked by sorting Google Trends queries by estimated cycle regularity.

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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

Rough take: the mathematical core is a genuine, small extension that deserves serious attention; the empirical validation is too weak to support the paper's central claim about real-world popularity dynamics.

What's actually new: the recovery rate feedback in Eq. (4) — dΓ/dt = Γ(αI − δS) — is a clean way to get endogenous oscillations out of a three-variable SIRS system. The stability condition (Prop 1) and the Hopf bifurcation result (Prop 2, with the β > (11/25)ξ condition) are new and, as far as I can tell, correct. The proof in Appendix A is standard but careful. The paper also does a solid job placing the model in the diffusion and opinion dynamics literature, and it does not oversell the theoretical contribution: it is a tractable extension, not a revolution.

The soft spots are on the empirical side, and they are load-bearing. The DTW comparison in Section 4 fits β per query in-sample to minimize the distance, then compares against the average of 500 random walks that have no fitted parameters. That asymmetry alone could explain the 67% result: any smooth curve with a free parameter will generally beat an unfitted random walk under DTW. The model is also always tuned to be just past the Hopf threshold (α = β + ξ + √(ξ(β+ξ)) + 0.01), so it is effectively a family of smooth periodic curves, not a sharp test of the specific mechanism. There are no error bars, no out-of-sample splits, and no comparison against an equally flexible null (say, an AR(2) or a fitted sinusoid). So the claim that the model 'captures' Google Trends residuals is not supported by the evidence as presented.

One more point: the theoretical claim of 'stable limit cycles' is stated without proving supercriticality. Proposition 2 proves a Hopf bifurcation, not that the cycles are attracting. The numerics suggest it, but a referee should ask for the first Lyapunov coefficient or a softer statement of what is proved. Also, the code is referenced as 'this link' without a URL, which is a reproducibility problem.

Bottom line: send it to peer review. The model is worth having and the theory section is solid enough to justify referee time, but the empirical validation needs substantial rework before the data claim can stand.

Referee Report

3 major / 4 minor

Summary. This paper proposes a three-dimensional ODE extension of the SIRS model in which the recovery rate Gamma is not constant but evolves according to dGamma/dt = Gamma(alpha I - delta S), encoding interest saturation and influencing enthusiasm. The authors characterize the local stability of the unique interior fixed point (Proposition 1), prove a Hopf bifurcation for the restricted case alpha = delta (Proposition 2), and report numerical evidence of limit cycles. They then apply the model to 1000 Google Trends queries: after detrending and deseasonalizing, they compare the residuals to the model's I(t) (with the transmission rate beta fitted per query) and to unfitted random walks using Dynamic Time Warping, reporting that the model matches better in 67% of cases. The central claims are that the feedback mechanism generates endogenous periodic cycles and that these cycles are visible in real-world search-volume residuals.

Significance. If fully established, the paper would provide a simple and tractable mechanism for endogenous oscillations in idea diffusion, with an explicit empirical implication and public code on GitHub. The analytical derivation of local instability and the Hopf bifurcation is a solid start, and the manuscript is careful to separate trend and seasonality before testing for oscillatory residuals. However, two load-bearing gaps currently limit the contribution: the proof does not establish that the bifurcating cycles are stable (attracting), and the empirical validation compares an in-sample fitted model against an unfitted baseline, so the 67% result does not yet discriminate the proposed feedback mechanism from generic smoothness or flexible curve fitting. These issues are fixable, which is why I recommend major revision rather than rejection.

major comments (3)

[Section 3.2, Proposition 2] Proposition 2 proves that the fixed point loses stability via a Hopf bifurcation at alpha = beta + xi + sqrt(xi(beta + xi)), but it does not prove that the resulting limit cycles are stable. The text in Section 3 states that the feedback mechanism leads to 'stable limit cycles', and Section 3.1 says extensive numerics support 'stable limit cycles appear', but a Hopf bifurcation can be subcritical, producing unstable cycles, or supercritical, producing attracting cycles. Please compute the first Lyapunov coefficient (or provide a center-manifold reduction) to determine the criticality, or explicitly label the stability claim as numerical only. This is load-bearing because the paper's theoretical contribution is the existence of persistent, observable cycles, not merely the loss of stability of a fixed point.
[Section 4 and Appendix C.3] The empirical comparison is asymmetric: for each query, beta is chosen by minimizing the in-sample DTW distance between the residual series and the model, while the random-walk baseline is generated unfitted with no parameter adjustment. The model therefore benefits from per-series optimization, and DTW already permits arbitrary time-axis warping. As a result, the reported 67% closer-match rate does not establish that residuals have the specific SIRS feedback structure; it may only reflect that a smooth curve with one free parameter, when optimally warped, is closer to a smooth residual than the average of unfitted random walks. Please add an out-of-sample evaluation (e.g., split the series, fit beta on a training segment, measure DTW on a holdout segment) and compare against at least one equally flexible null model that is fitted in the same way, such as a sinusoid with fitted amplitude, frequency, and phase, or an AR(2) process fitted per query. Also report standard errors or confidence intervals for the 67% statistic.
[Section 4, Figure 3] The definition of the success rate is ambiguous. The text says residuals 'systematically show a closer match' to the model in 67% of cases, but the preceding explanation defines a blue circle as the case where the model DTW distance is smaller than the first quartile of the random-walk distance distribution. These are two different counting rules: the fraction below the diagonal (model distance < random-walk average distance) and the fraction below the first quartile. Please specify exactly which criterion gives 67%, and report both counts if appropriate. This matters for interpreting the strength of the empirical claim.

minor comments (4)

[Section 3.1, after Figure 1] There is a typo: 'computaions' should be 'computations'.
[Section 4] The abbreviation 'DWT' is used inconsistently; the text refers to 'best value for beta ... in terms of DWT' and Figure 3's caption uses 'DWT distances', while the standard abbreviation defined in the paper is DTW (Dynamic Time Warping).
[Appendix C.3] The parameter delta is not specified in the empirical setup; it would be helpful to state explicitly whether the restricted model alpha = delta is used (as in Proposition 2), since the Hopf bifurcation result applies only in that case.
[Section 3.2, Eq. (20)] The derivation of the sufficient condition for Delta > 0 is algebraically dense; adding a brief explanatory sentence about the sign of the numerator and denominator would improve readability.

Circularity Check

1 steps flagged · score 6.0 of 10

Empirical validation is an in-sample fit: β is chosen per query to minimize the very DTW distance later reported as evidence, so the 67% result compares a fitted oscillator against an unfitted random-walk baseline.

fitted input called prediction [Appendix C.3 (Distance Computation for Residuals) and Section 4 (Empirical application)]
"For each word, we computed the DTW distance between its residuals and the model corresponding to each β. We then selected the model with the β that minimized this distance, ensuring the closest match between the residuals and our model of endogenous oscillations. ... Our analysis shows that the endogenous cyclical effects capture the underlying patterns of Google searches statistically better than what would be expected from random fluctuations in the 67% of cases."

The model's transmission rate β is selected per word by minimizing exactly the DTW distance that is then used as the evidence of fit. The reported 'model' DTW is therefore the minimum of a one-parameter family of distances by construction, not an independent prediction or out-of-sample evaluation. The random-walk baseline receives no such fitting: its 500 simulations are generated independently with no parameters optimized against the residuals. Comparing a per-series minimized distance to an unfitted baseline average makes the 67% statistic reflect fitting flexibility plus the smoothness of a periodic trajectory under DTW time-warping, rather than providing independent confirmation that the residuals have the specific SIRS feedback structure.

full rationale

The theoretical part of the paper is self-contained and not circular: the SIRS-with-feedback system (5)-(7) is stated directly, the fixed point is computed, and Proposition 1 and Proposition 2 are proved from the Jacobian and Routh-Hurwitz/Cardano analysis within the paper. No load-bearing self-citation chain or imported uniqueness theorem is involved, and the limit-cycle result is a genuine mathematical derivation. The circularity concern is confined to the empirical validation in Section 4 and Appendix C.3. There, the model is fitted to each Google Trends residual series by choosing β to minimize the DTW distance, and that minimized in-sample distance is then compared against the average DTW distance of 500 unfitted random walks. This is the pattern of a fitted input being presented as a prediction: the success metric is the fitting objective, the comparison baseline is not given the same fitting opportunity, and DTW's time-axis warping further inflates the flexibility of a smooth oscillatory trajectory. The 67% result therefore does not independently confirm endogenous SIRS cycles; it is consistent with the weaker statement that a fitted smooth periodic curve can beat an unfitted random-walk average under a warping-invariant distance. For this reason the empirical claim is partially circular, while the theoretical derivation remains independent.

Assumptions & free parameters 5 free parameters · 3 assumptions · 0 invented entities

The model introduces a new differential equation for the recovery rate. All parameters are either free or set by hand; no new unobservable entities are postulated. The main load-bearing assumptions are the well-mixed deterministic population and the linear log-feedback form.

free parameters (5)

β (transmission rate) = per-query values in [0.01,0.3] (not reported)
Chosen per query to minimize the DTW distance between the model's I(t) and the residuals; this is a fitted parameter in the empirical validation.
ξ (rate of losing immunity) = 0.1
Set by hand in the empirical application (Appendix C.3); no justification is given.
α (saturation feedback strength) = β+ξ+sqrt(ξ(β+ξ))+0.01
Determined by a formula rather than fitted, but the +0.01 is an arbitrary choice to place the system just past the Hopf point.
δ (enthusiasm feedback strength) = unclear (assumed equal to α)
Appendix C does not state δ; presumably δ=α (restricted model), but this is a hand choice not made explicit.
Rolling window length for trend = 52 weeks
This choice affects the residual series and therefore the DTW comparison; it is set by hand.

assumptions (3)

domain assumption The population is well-mixed and large enough that deterministic ODEs describe the dynamics.
Used throughout Section 3; neglects network structure, heterogeneity, and stochastic fluctuations.
ad hoc to paper The log recovery rate changes linearly with I and S (Eq. 4).
This functional form is assumed, not derived from individual behavior; it is the key driver of the oscillations.
standard math Routh-Hurwitz stability criterion and Cardano's formula for cubic equations are valid tools for the eigenvalue analysis.
Used in Appendices A.1 and A.2 to establish local stability and the Hopf bifurcation.

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Cite this review

Pith. "Pith review of The Rise and Fall of Ideas' Popularity." pith.science (2026). https://pith.science/paper/CYZVG6C3

@misc{pith2026241118541,
  author       = {Pith},
  title        = {Pith review of: The Rise and Fall of Ideas' Popularity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CYZVG6C3}},
  note         = {Machine review of arXiv:2411.18541}
}

read the original abstract

In the dynamic landscape of contemporary society, the popularity of ideas, opinions, and interests fluctuates rapidly. Traditional dynamical models in social sciences often fail to capture this inherent volatility, attributing changes to exogenous shocks rather than intrinsic features of the system. This paper introduces a novel, tractable model that simulates the natural rise and fall of ideas' popularity, offering a more accurate representation of real-world dynamics. Building upon the SIRS (Susceptible, Infectious, Recovered, Susceptible) epidemiological model, we incorporate a feedback mechanism that allows the recovery rate to vary dynamically based on the current state of the system. This modification reflects the cyclical nature of idea adoption and abandonment, driven by social saturation and renewed interest. Our model successfully captures the rapid and recurrent shifts in popularity, providing valuable insights into the mechanisms behind these fluctuations. This approach offers a robust framework for studying the diffusion dynamics of popular ideas, with potential applications across various fields such as marketing, technology adoption, and political movements.

Figures

Figures reproduced from arXiv: 2411.18541 by the authors.

**Figure 1.** Left: layout in the (α, δ) parameter space, for β = 0.5 and ξ = 0.4, of the region of instability (red), and of stability due to the first condition (yellow), to the second (blue) or to both conditions (green) of Eq. (14). The five dots represent the parameters used to produce the graph in the right panel (black dot), and Figures 5 (yellow dot), 6 (blue dot), 7 (green dot), 8 (red dot) in Appendix B. The black star … view at source ↗

**Figure 2.** Seasonal decomposition for the word ‘economy’ searched in Google Trends. After removing the trend and annual seasonality, we are left with the residuals (bottom panel). See Appendix C.1 for additional information. value in the investigated period, which is set to 100. We show the research query for ‘economy’ in a 5-year period from September 15th, 2019 to September 15th, 2024 in the top panel of [PITH_FULL_IMAGE:fi… view at source ↗

**Figure 3.** Scatter plot showing the relationship between the Dynamic Time Warping (DTW) distance of residuals to a random walk (x-axis) and the DTW distance to the model (y-axis). (See Appendix C.3 for additional information.) Each circle represents a word from the dataset. Marginal histograms indicate the distribution of DTW distances for both reference processes. The dotted diagonal line represents the case when the average … view at source ↗

Figures from the paper (8 more)

**Figure 4.** Figure 4: Layout in the (α, δ) parameter space, for β = 0.5 and ξ = 0.4, of the region of instability (red), and of stability due to the first condition (yellow), to the second (blue) or to both conditions (green). The five dots represent the parameters used to produce Figures 5…

**Figure 5.** Figure 5: SIRS model with feedback, with fixed point satisfying the first stability condition: α = 0.65, β = 0.5, δ = 0.6, ξ = 0.4, S(0) = 0.9, I(0) = Γ(0) = 0.1. 25 [PITH_FULL_IMAGE:figures/full_fig_p025_5.png]

**Figure 6.** Figure 6: SIRS model with feedback, with fixed point satisfying the second stability condition: α = 0.95, β = 0.5, δ = 0.6, ξ = 0.4, S(0) = 0.9, I(0) = Γ(0) = 0.1. 26 [PITH_FULL_IMAGE:figures/full_fig_p026_6.png]

**Figure 7.** Figure 7: SIRS model with feedback, with fixed point satisfying both stability conditions: α = 0.45, β = 0.5, δ = 3, ξ = 0.4, S(0) = 0.9, I(0) = Γ(0) = 0.1. 27 [PITH_FULL_IMAGE:figures/full_fig_p027_7.png]

**Figure 8.** Figure 8: SIRS model with feedback, with fixed point not satisfying the stability conditions: α = 1.5, β = 0.5, δ = 2.5, ξ = 0.4, S(0) = 0.9, I(0) = Γ(0) = 0.1. 28 [PITH_FULL_IMAGE:figures/full_fig_p028_8.png]

**Figure 9.** Figure 9: SIRS model with feedback, with fixed point not satisfying the stability conditions: α = 1.75, β = 0.5, δ = 1.75, ξ = 0.4, S(0) = 0.9, I(0) = Γ(0) = 0.1. 29 [PITH_FULL_IMAGE:figures/full_fig_p029_9.png]

**Figure 10.** Figure 10: Evolution of the solution of the restricted version of the SIRS model with feedback when crossing a Hopf bifurcation point in the parameter space; β = 0.5, ξ = 0.4, S(0) = 0.9, I(0) = Γ(0) = 0.1, α = δ from 1 (upper left – stable), to 1.5 (center – Hopf bifurcation), …

**Figure 11.** Figure 11: Evolution of the solution (projection on the (S, I) plane) of the restricted version of the SIRS model with feedback when crossing a Hopf bifurcation point in the parameter space; β = 0.5, ξ = 0.4, S(0) = 0.9, I(0) = Γ(0) = 0.1, α = δ from 1 (upper left – stable), to …

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

, Como, G

Acemo g lu, D. , Como, G. , Fagnani, F. and Ozdaglar, A. (2013). Opinion fluctuations and disagreement in social networks. Mathematics of Operations Research, 38 (1), 1--27

work page 2013

[2] [2]

Anderson, R. M. and May, R. M. (1991). Infectious diseases of humans: dynamics and control. Oxford university press

work page 1991

[3] [3]

and Godart, F

Aspers, P. and Godart, F. (2013). Sociology of fashion: Order and change. Annual Review of Sociology, 39, 171--192

work page 2013

[4] [4]

, Abdel-Kader, H

Badr, N. , Abdel-Kader, H. and Ali, A. H. (2021). Diffusion models for social analysis, influence and learning. IJCI. International Journal of Computers and Information, 8 (2), 162--169

work page 2021

[5] [5]

and Verdier, T

Bisin, A. and Verdier, T. (2001). The economics of cultural transmission and the dynamics of preferences. Journal of Economic Theory, 97 (2), 298--319

work page 2001

[6] [6]

Brock, W. A. and Hommes, C. H. (1998). Heterogeneous beliefs and routes to chaos in a simple asset pricing model. Journal of Economic Dynamics and Control, 22 (8-9), 1235--1274

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, Witmer, T

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