Simon's model does not produce Zipf's law: The fundamental rich-get-richer mechanism for any power-law size ranking

Julia Witte Zimmerman; Laurent H\'ebert-Dufresne; Pablo Rosillo-Rodes; Peter Sheridan Dodds

arxiv: 2604.13184 · v1 · submitted 2026-04-14 · ⚛️ physics.soc-ph

Simon's model does not produce Zipf's law: The fundamental rich-get-richer mechanism for any power-law size ranking

Pablo Rosillo-Rodes , Julia Witte Zimmerman , Laurent H\'ebert-Dufresne , Peter Sheridan Dodds This is my paper

Pith reviewed 2026-05-10 13:46 UTC · model grok-4.3

classification ⚛️ physics.soc-ph

keywords Zipf's lawrich-get-richerinnovation ratepower-law size rankingSimon modeltype emergencepreferential attachmentword frequency

0 comments

The pith

Simon's classic rich-get-richer model collapses to a winner-takes-all outcome instead of producing Zipf's law, and a specific decaying innovation rate is required to generate any power-law size ranking.

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

Simon's 1955 model for system growth combines preferential attachment with a constant rate of new type introduction, yet this combination fails to sustain the observed inverse relationship between size and rank. In the limit of vanishing innovation the model produces a single dominant type rather than the balanced distribution where sizes fall as the inverse of rank. The authors derive the precise time-dependent rate at which new types must appear to enforce power-law rankings for any exponent, showing that the rate must fall as one over the logarithm of the current number of types to obtain Zipf's law specifically. This same rate governs the appearance of new types in every system that displays power-law size rankings, regardless of the details of how sizes grow.

Core claim

For pure rich-get-richer dynamics the innovation probability ρ_t must be chosen to decay in a controlled way with time so that the resulting size distribution follows S ∝ r^{-α} for arbitrary α ≥ 0; in particular, ρ_t ∼ 1/ln N produces the Zipf case α = 1. This time-dependent innovation rate, rather than any constant rate, is the mechanism that controls type emergence across all power-law systems.

What carries the argument

The time-dependent innovation rate ρ_t, defined as the probability of introducing a new type at each growth step, derived so that the cumulative size distribution remains a pure power law at every time.

If this is right

Zero innovation forces every rich-get-richer system into a single dominant type.
To maintain Zipf's law the probability of adding a new type must fall exactly as one over the logarithm of the number of types already present.
The same innovation schedule reproduces word-frequency rankings in novels far more accurately than a constant-rate model.
Any mechanism that produces power-law size rankings must implicitly obey this time-dependent innovation schedule.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Empirical counts of new entrants in markets or ecosystems could be tested directly against the predicted 1/ln N decay to check whether the schedule is mechanism-independent.
The derived rate supplies a minimal dynamical model that can be inserted into simulations of any growing system to enforce realistic diversity growth.
If the schedule holds, then slowing the appearance of new types over time becomes a necessary condition for preserving observed power-law rankings rather than an optional parameter.

Load-bearing premise

That the innovation rate required to stabilize power-law rankings inside a rich-get-richer process is in fact the universal driver of new-type appearance in every system that obeys a power-law size ranking.

What would settle it

Direct measurement of the rate at which new distinct words appear (or new firms, cities, or species) in a growing data set; if that rate does not decay proportionally to 1 over the log of the number of existing types while the size-rank plot remains a power law, the claimed universality fails.

Figures

Figures reproduced from arXiv: 2604.13184 by Julia Witte Zimmerman, Laurent H\'ebert-Dufresne, Pablo Rosillo-Rodes, Peter Sheridan Dodds.

**Figure 1.** Figure 1: Simulations of Simon’s model fail to reproduce the size ranking Sr,t,α in the zero innovation limit. As ρ → 0 (A– C), the size ranking fails to approach a power law with α = 1 − ρ → 1. Instead, the first-mover advantage dominates and overestimates S0,t,α in a factor of 1/ρ. This effect intensifies as the innovation probability decreases, arriving at α → ∞ when ρ = 0 (D). smoothly decreases as the system gr… view at source ↗

**Figure 2.** Figure 2: The generalized innovation rate ρt,α in Eq. (17) correctly reproduces the size–ranking Sr,t,α for all values of α, including α ≥ 1. Unlike Simon’s model, which misrepresents the α = 1 − ρ = 1 case by introducing an outsized first-mover advantage and shifting the size ranking toward α → ∞, the generalized model we introduce in this work remains consistent across all regimes. We run 100 realizations of both … view at source ↗

**Figure 3.** Figure 3 [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

read the original abstract

Many complex systems are composed of disparate, interacting types of varying sizes: Species abundances in ecosystems, firm sizes in markets, city populations in countries, word counts in language, etc. A longstanding mystery of complex systems is Zipf's law, which is the empirical observation that component size decreases as the inverse of component rank -- $S \propto r^{-1}$ -- and its generalization $S \propto r^{-\alpha}$ for $\alpha \ge 0$. Herbert Simon's 1955 theoretical rich-get-richer mechanism for system growth has prevailed as capturing the essential process. But Simon's analysis is in fact flawed: In the limit of zero innovation, the model leads to a winner-takes-all system with $\alpha \rightarrow \infty$, rather than $\alpha \rightarrow 1$. Here, for pure rich-get-richer systems, we derive the time-dependent innovation rate $\rho_t$ that correctly produces power-law size rankings across all $\alpha \ge 0$. To produce Zipf's law, we uncover that $\rho_t$ must decay as the inverse of the log of the number of types, $1/\ln N$. We then show that our time-dependent innovation rate governs type emergence in any system obeying a power-law size-ranking, independent of the underlying mechanism. We demonstrate agreement between our model's output and word rankings in a collection of famous novels, while Simon's model fails. Going forward, our dynamic innovation rate mechanism provides the fundamental, Drosophila-like model for all rich-get-richer systems.

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

The paper fixes a real flaw in Simon's model by deriving a 1/ln N innovation rate for Zipf's law, but the claim that this rate is required for any power-law system rests on thin evidence.

read the letter

The core correction is straightforward and worth knowing: in the zero-innovation limit Simon's preferential attachment process collapses to one giant component with alpha going to infinity, not to the expected Zipf exponent. The authors solve the rate equations to get an explicit time-dependent innovation probability rho_t that produces any desired alpha, and for alpha=1 it decays as 1 over log of the number of types. They then test the output against word counts in several novels and show better agreement than the constant-rate version of Simon's model. That part is concrete and directly useful for anyone who has been plugging the 1955 equations into their own work on cities, firms, or species abundances. The derivation itself looks clean for the rich-get-richer case they consider. The softer spot is the assertion that the same rho_t governs type emergence in every system that happens to show a power-law rank-size relation, regardless of the growth mechanism. The abstract and the stress-test note both indicate the derivation stays inside the preferential-attachment master equation; there is no separate argument that starts only from S(r) proportional to r to the minus alpha plus linear total growth and derives the same rho_t without reference to attachment probabilities. That step is the one that feels like it needs more pages. Readers who already work with these models will find the explicit functional form handy and the data check informative. The paper is coherent on its own terms and engages the literature directly, so it is worth sending to referees even if the generality claim gets pushed back in review.

Referee Report

2 major / 2 minor

Summary. The manuscript argues that Simon's 1955 rich-get-richer model is flawed because the zero-innovation limit produces a winner-takes-all outcome with exponent α → ∞ rather than Zipf's law (α = 1). For pure preferential-attachment systems the authors derive a time-dependent innovation rate ρ_t that must decay as 1/ln N to generate power-law rank-size relations for any α ≥ 0. They then claim that this same ρ_t is the fundamental, mechanism-independent driver of type emergence in any system whose sizes obey a power-law ranking, and they report that the resulting model matches word-frequency rankings in a collection of novels while the original Simon model does not.

Significance. If the claimed independence of ρ_t from the underlying growth kernel can be rigorously established, the work would supply a compact, falsifiable explanation for the ubiquity of power-law size rankings across complex systems. The explicit derivation inside the Simon framework and the direct comparison with literary word counts are concrete strengths that allow quantitative testing. At present, however, the broader significance remains limited because the generality argument is not yet mechanism-agnostic.

major comments (2)

[§3] §3 (derivation of ρ_t): The innovation rate ρ_t ≈ 1/ln N is obtained by solving the rate equations under the specific preferential-attachment kernel of the Simon model. The subsequent assertion that this ρ_t governs type emergence in arbitrary power-law systems therefore requires an independent derivation that begins only from the observed rank-size relation S(r) ∝ r^{-α}, the linear growth of total size T(t), and the definition ρ_t = dN/dt, without reference to attachment probabilities. No such argument is supplied.
[§2] Zero-innovation limit (abstract and §2): The claim that α → ∞ when ρ_t → 0 contradicts the conventional reading of Simon's original analysis. An explicit solution of the master equation or rate equations in this limit must be shown to establish that the original model is indeed flawed rather than merely misstated.

minor comments (2)

[Abstract] The abstract refers to the model as 'Drosophila-like' without defining the intended analogy; a brief clarification of the model's intended role as a minimal, testable template would improve readability.
[Empirical comparison] Quantitative details of the word-ranking comparison (number of novels, goodness-of-fit metric, error analysis) are mentioned only qualitatively; adding these would strengthen the empirical section without altering the central argument.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thoughtful and constructive comments, which help clarify the scope and presentation of our results. We address each major comment in turn below.

read point-by-point responses

Referee: §3 (derivation of ρ_t): The innovation rate ρ_t ≈ 1/ln N is obtained by solving the rate equations under the specific preferential-attachment kernel of the Simon model. The subsequent assertion that this ρ_t governs type emergence in arbitrary power-law systems therefore requires an independent derivation that begins only from the observed rank-size relation S(r) ∝ r^{-α}, the linear growth of total size T(t), and the definition ρ_t = dN/dt, without reference to attachment probabilities. No such argument is supplied.

Authors: The derivation of ρ_t begins from the general assumptions of a power-law rank-size relation S(r) ∝ r^{-α}, linear growth of total size T(t), and the definition ρ_t = dN/dt. These conditions alone determine the required growth of the number of types N(t) needed to preserve the rank ordering as the system evolves. Solving for ρ_t under these constraints yields ρ_t ∼ 1/ln N for any α ≥ 0. The Simon model is employed only as a concrete dynamical realization to verify that this ρ_t indeed produces the target distribution; the functional form itself follows directly from the rank-size and growth assumptions and is therefore independent of the attachment kernel. We have added a short paragraph in the revised §3 that isolates this kernel-independent step. revision: yes
Referee: Zero-innovation limit (abstract and §2): The claim that α → ∞ when ρ_t → 0 contradicts the conventional reading of Simon's original analysis. An explicit solution of the master equation or rate equations in this limit must be shown to establish that the original model is indeed flawed rather than merely misstated.

Authors: Section 2 already contains the explicit limiting analysis. Setting ρ_t = 0 in the rate equations and solving the resulting master equation shows that the size distribution collapses to a single dominant type, with the exponent α diverging to infinity. This limit is obtained by direct substitution into the closed-form solution for the occupation probabilities and confirms a winner-takes-all outcome. The conventional reading of Simon (1955) implicitly retains a small positive constant innovation rate; our zero-innovation calculation demonstrates that the model does not recover Zipf’s law (α = 1) under the strict ρ_t → 0 condition stated in the abstract. No revision is required. revision: no

Circularity Check

1 steps flagged

ρ_t derived inside Simon preferential attachment to enforce power-law, then asserted as mechanism-independent

specific steps

fitted input called prediction [Abstract (and central derivation section)]
"Here, for pure rich-get-richer systems, we derive the time-dependent innovation rate ρ_t that correctly produces power-law size rankings across all α≥0. To produce Zipf's law, we uncover that ρ_t must decay as the inverse of the log of the number of types, 1/ln N. We then show that our time-dependent innovation rate governs type emergence in any system obeying a power-law size-ranking, independent of the underlying mechanism."

ρ_t is obtained by solving the master/rate equations under the specific preferential-attachment kernel (new units attach proportional to current size with probability 1−ρ_t). The same functional form is then declared to be required by the power-law rank-size relation alone, without an independent argument that uses only the observed S(r), total-size growth, and ρ_t=dN/dt. The 'any system' claim therefore restates the condition that was imposed inside the Simon framework.

full rationale

The paper correctly identifies that Simon's original fixed-ρ analysis yields α→∞ rather than Zipf's law. It then solves the rate equations under the rich-get-richer kernel with time-dependent ρ_t to recover any α, including the specific form ρ_t ∼ 1/ln N for α=1. The subsequent claim that this same ρ_t 'governs type emergence in any system obeying a power-law size-ranking, independent of the underlying mechanism' is presented without a separate derivation that starts only from S(r)∝r^{-α}, linear total-size growth, and ρ_t=dN/dt. Because the functional form of ρ_t is obtained exclusively inside the attachment model and then exported as universal, the generality step reduces to a re-statement of the condition imposed to produce the target ranking.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on a mathematical derivation within a pure rich-get-richer growth process plus the assertion that the resulting innovation schedule applies universally. No new physical entities are introduced. The time-dependent rate ρ_t is not a free parameter but is derived from the power-law requirement.

axioms (2)

domain assumption The system obeys a pure rich-get-richer growth rule in which existing types grow proportionally to their current size.
Invoked when stating the model that Simon's analysis is applied to and when deriving the required ρ_t.
domain assumption Power-law size rankings are the target outcome that the innovation rate must be chosen to produce.
Used to back-solve for ρ_t(N) and to claim the result is mechanism-independent.

pith-pipeline@v0.9.0 · 5601 in / 1500 out tokens · 56647 ms · 2026-05-10T13:46:53.124213+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

34 extracted references · 34 canonical work pages

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B. B. Mandelbrot. Final note on a class of skew distribution functions: analysis and critique of a model due to H. A. Simon.Information and Control, 4:198–216, 1961

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B. B. Mandelbrot. Post scriptum to ‘final note’. Information and Control, 4:300–304, 1961

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H. A. Simon. Reply to Dr. Mandelbrot’s post scriptum.Information and Control, 4:305–308, 1961

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P. S. Dodds, D. R. Dewhurst, F. F. Hazlehurst, C. M. Van Oort, L. Mitchell, A. J. Reagan, J. R. Williams, and C. M. Danforth. Simon’s fundamental rich-get-richer model entails a dominant first-mover advantage.Phys. Rev. E, 95:052301, May 2017

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Zanette and M

D. Zanette and M. Montemurro. Dynamics of Text Generation with Realistic Zipf’s Distribution. Journal of Quantitative Linguistics, 12(1):29–40, 2005

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Rosillo-Rodes, L

P. Rosillo-Rodes, L. H´ ebert-Dufresne, and P. S. Dodds. Complete asymptotic type-token relationship for growing complex systems with inverse power-law count rankings.Physical Review Research, 8(1):L012029, 2026

work page 2026
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T. M. Apostol. An Elementary View of Euler’s Summation Formula.The American Mathematical Monthly, 106(5):409–418, 1999

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[32]

Gerlach and F

M. Gerlach and F. Font-Clos. A Standardized Project Gutenberg Corpus for Statistical Analysis of Natural Language and Quantitative Linguistics. Entropy, 22(1):126, 2020

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http://az.lib.ru/, 2004

Lib.ru: Klassika. http://az.lib.ru/, 2004. Supported by the Federal Agency for Press and Mass Communications

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http://beq.ebooksgratuits.com

La Biblioth` eque ´ electronique du Qu´ ebec. http://beq.ebooksgratuits.com. Textes d’auteurs du domaine public. Fond´ ee en 1998. 9 Appendix A1 Limiting behavior of the generalized innovation rate Here, we derive the generalized innovation rate limits in Eq. 18. In the case of 0< α≪1, ρt,α ∼ 1−α 1− α Nt,α+1 ≃1−α.(A1) Whenα≫1, ρt,α ∼(N t,α + 1)1−α ≃N t,α....

work page 1998

[1] [1]

J. B. Estoup.Gammes st´ enographiques. Institut Stenographique de France, Paris, 1916

work page 1916

[2] [2]

G. K. Zipf. The psychology of language. In Encyclopedia of psychology, pages 332–341. Philosophical Library, 1946

work page 1946

[3] [3]

E. G. Altmann and M. Gerlach.Statistical Laws in Linguistics, pages 7–26. Springer International Publishing, Cham, 2016

work page 2016

[4] [4]

Arnon, S

I. Arnon, S. Kirby, J. A. Allen, C. Garrigue, E. L. Carroll, and E. C. Garland. Whale song shows language-like statistical structure.Science, 387(6734):649–653, 2025

work page 2025

[5] [5]

Camacho and R

J. Camacho and R. V. Sol´ e. Scaling in ecological size spectra.Europhysics Letters, 55(6):774, sep 2001

work page 2001

[6] [6]

M. A. Serrano and M. Bogu˜ n´ a. Topology of the world trade web.Phys. Rev. E, 68:015101, Jul 2003

work page 2003

[7] [7]

X. Gabaix. Power laws in economics: An introduction.Journal of Economic Perspectives, 30(1):185–206, 2016

work page 2016

[8] [8]

D. J. de Solla Price. Networks of scientific papers. Science, 149:510–515, 1965

work page 1965

[9] [9]

S. Redner. How popular is your paper? An empirical study of the citation distribution.The European Physical Journal B, 4:131–134, July 1998

work page 1998

[10] [10]

Mitzenmacher

M. Mitzenmacher. A brief history of generative models for power law and lognormal distributions. Internet mathematics, 1(2):226–251, 2004

work page 2004

[11] [11]

M. E. Newman. Power laws, Pareto distributions and Zipf’s law.Contemporary physics, 46(5):323–351, 2005

work page 2005

[12] [12]

Maillart, D

T. Maillart, D. Sornette, S. Spaeth, and G. von Krogh. Empirical tests of Zipf’s law mechanism in open source Linux distribution.Phys. Rev. Lett., 101(21):218701, 2008

work page 2008

[13] [13]

Roman and F

S. Roman and F. Bertolotti. A master equation for power laws.Royal Society open science, 9(12):220531, 2022

work page 2022

[14] [14]

Holehouse, S

J. Holehouse, S. Redner, V. C. Yang, P. L. Krapivsky, J. I. Arroyo, G. B. West, C. Kempes, and H. Youn. A generative model of function growth explains hidden self-similarities across biological and social systems.arXiv preprint arXiv:2509.14468, 2025

work page arXiv 2025

[15] [15]

Ferrer-i-Cancho and B

R. Ferrer-i-Cancho and B. Elvev˚ ag. Random texts do not exhibit the real Zipf’s law-like rank distribution. PLoS ONE, 5:e9411, 03 2010. 8

work page 2010

[16] [16]

A. D. Broido and A. Clauset. Scale-free networks are rare.Nature Communications, 10(1):1017, 2019

work page 2019

[17] [17]

Gerlach and E

M. Gerlach and E. G. Altmann. Testing Statistical Laws in Complex Systems.Phys. Rev. Lett., 122:168301, Apr 2019

work page 2019

[18] [18]

H. A. Simon. On a class of skew distribution functions.Biometrika, 42:425–440, 1955

work page 1955

[19] [19]

G. U. Yule. A mathematical theory of evolution, based on the conclusions of Dr J. C. Willis, F.R.S. Phil. Trans. B, 213:21–87, 1925

work page 1925

[20] [20]

R. K. Merton. The Matthew Effect in Science, II: Cumulative Advantage and the Symbolism of Intellectual Property.Isis, 79(4):606–623, 1988

work page 1988

[21] [21]

Albert, H

R. Albert, H. Jeong, and A.-L. Barab´ asi. Diameter of the world-wide web.Nature, 401(6749):130–131, 1999

work page 1999

[22] [22]

B. B. Mandelbrot. A note on a class of skew distribution function. Analysis and critique of a paper by H. A. Simon.Information and Control, 2:90–99, 1959

work page 1959

[23] [23]

H. A. Simon. Some further notes on a class of skew distribution functions.Information and Control, 3:80–88, 1960

work page 1960

[24] [24]

B. B. Mandelbrot. Final note on a class of skew distribution functions: analysis and critique of a model due to H. A. Simon.Information and Control, 4:198–216, 1961

work page 1961

[25] [25]

H. A. Simon. Reply to ‘final note’ by Benoˆ ıt Mandelbrot.Information and Control, 4:217–223, 1961

work page 1961

[26] [26]

B. B. Mandelbrot. Post scriptum to ‘final note’. Information and Control, 4:300–304, 1961

work page 1961

[27] [27]

H. A. Simon. Reply to Dr. Mandelbrot’s post scriptum.Information and Control, 4:305–308, 1961

work page 1961

[28] [28]

P. S. Dodds, D. R. Dewhurst, F. F. Hazlehurst, C. M. Van Oort, L. Mitchell, A. J. Reagan, J. R. Williams, and C. M. Danforth. Simon’s fundamental rich-get-richer model entails a dominant first-mover advantage.Phys. Rev. E, 95:052301, May 2017

work page 2017

[29] [29]

Zanette and M

D. Zanette and M. Montemurro. Dynamics of Text Generation with Realistic Zipf’s Distribution. Journal of Quantitative Linguistics, 12(1):29–40, 2005

work page 2005

[30] [30]

Rosillo-Rodes, L

P. Rosillo-Rodes, L. H´ ebert-Dufresne, and P. S. Dodds. Complete asymptotic type-token relationship for growing complex systems with inverse power-law count rankings.Physical Review Research, 8(1):L012029, 2026

work page 2026

[31] [31]

T. M. Apostol. An Elementary View of Euler’s Summation Formula.The American Mathematical Monthly, 106(5):409–418, 1999

work page 1999

[32] [32]

Gerlach and F

M. Gerlach and F. Font-Clos. A Standardized Project Gutenberg Corpus for Statistical Analysis of Natural Language and Quantitative Linguistics. Entropy, 22(1):126, 2020

work page 2020

[33] [33]

http://az.lib.ru/, 2004

Lib.ru: Klassika. http://az.lib.ru/, 2004. Supported by the Federal Agency for Press and Mass Communications

work page 2004

[34] [34]

http://beq.ebooksgratuits.com

La Biblioth` eque ´ electronique du Qu´ ebec. http://beq.ebooksgratuits.com. Textes d’auteurs du domaine public. Fond´ ee en 1998. 9 Appendix A1 Limiting behavior of the generalized innovation rate Here, we derive the generalized innovation rate limits in Eq. 18. In the case of 0< α≪1, ρt,α ∼ 1−α 1− α Nt,α+1 ≃1−α.(A1) Whenα≫1, ρt,α ∼(N t,α + 1)1−α ≃N t,α....

work page 1998