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

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 $S \propto r^{-α}$ for $α\ge 0$. 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 $α\rightarrow \infty$, rather than $α\rightarrow 1$. 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 $α\ge 0$. 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. 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.

• The paper reveals that Simon’s traditional rich-get-richer mechanism fails to generate Zipf’s law in the zero-innovation limit.
• It introduces a time-dependent innovation rate that adjusts dynamically to produce stable power-law rank distributions.
• Empirical validation with word frequency data in novels confirms the new model’s robust fit to real-world scaling phenomena.

Simon’s Model and Its Incompatibility With Zipf’s Law

The paper “Simon’s model does not produce Zipf’s law: The fundamental rich-get-richer mechanism for any power-law size ranking” (2604.13184) delivers a rigorous critique of the canonical rich-get-richer (RGR) model, rigorously demonstrating its critical failure to generate Zipf’s law in the zero-innovation limit, and then introduces the minimal, mechanistically necessary reformation of the innovation rate to furnish the correct spectrum of power-law size rankings.

Analysis of Simon’s Model Breakdown

Simon’s 1955 model, foundational in accounts of size-rank power laws, posits that at every time step, one token is assigned to a type, with probability $p$ for innovation (a newly observed type) and $1-p$ for reinforcement, where types attract future tokens in direct proportion to their accumulated size. Simon’s analysis links the exponent $a$ of the asymptotic power-law $S(r) \sim r^{-a}$ to the innovation probability via $a = 1 - p$. Historically, this has motivated the understanding that setting $p \to 0$ (vanishing innovation) should yield Zipf’s law ($a = 1$).

The paper demonstrates that this is qualitatively incorrect. As $p \to 0$, Simon’s model actually collapses into a winner-take-all regime, where the first-mover’s size becomes a factor $1/p$ larger than all others, and for strictly $p = 0$ only one type exists. The asymptotic size distribution thus approaches a degenerate state, not Zipf's law. This result rigorously substantiates previous, but less widely recognized, demonstrations of the excessive first-mover advantage in the Simon process [28], showing that the standard rate equation analysis is inconsistent with the underlying stochastic process when $1-p$0. Consequently, the model does not cover the entire empirically relevant domain ($1-p$1). This finding invalidates the use of uncritical Simon-type RGR justifications for Zipf- and post-Zipf-ranked phenomena.

Generalizing the Rich-Get-Richer Mechanism: Time-Dependent Innovation Rate

The authors introduce a mechanistic correction to the innovation process. By focusing on the dynamics of the $1-p$2-th type’s initiation and growth times, they rigorously derive the time-dependent innovation rate $1-p$3 required to generate a stable power-law rank distribution for all $1-p$4, resolving the limitation of Simon’s static $1-p$5. The innovation rate must decrease as the system grows, in a manner contingent on both the target exponent and the current type-count. This is formalized as:

$1-p$6

where $1-p$7 is the Riemann zeta function and $1-p$8 is the number of distinct types at time $1-p$9. This function reproduces Simon’s result for $a$0, transitions smoothly across $a$1, and for $a$2 decays as a power-law in $a$3.

Theoretical Derivation for Zipf's Law

The key result is that in order to generate Zipf’s law ($a$4), the innovation rate must not vanish, but instead must decrease as the inverse logarithm of the number of types:

$a$5

This function decays much slower than any power law and never reaches zero for finite systems. This “Zipf innovation rate” is minimal and unique, as shown via both dynamic and static (“mechanism-free”) process analyses. Any system producing a true power-law size ranking must, in effect, conform to this dynamical law for type emergence.

Empirical Validation

The model is compared against word frequency rankings in novels across several languages. Only the generalized dynamic innovation mechanism accurately fits the observed rank-size statistics; Simon’s model systematically overrepresents the leading type (first-mover) and fails to reproduce the empirical slope and scaling. The precise matching of the new model to a diverse range of empirical texts underscores the inadequacy of fixed-innovation RGR mechanisms for describing linguistic and similar Zipfian data.

Implications and Theoretical Perspective

This work corrects a pervasive theoretical misconception regarding the generative process for Zipf’s law and more generally, for power-law size rankings. The results impose a fundamental constraint: any model proposing an RGR mechanism for the rank-size distribution of types in growing systems must employ a properly decaying dynamic innovation rate, which, for $a$6, is strictly $a$7. The framework provides a robust null model for assessing deviations from power-law universality and calibrating alternative mechanisms (e.g., fitness models, non-preferential attachment, or additional deterministic structure).

Practically, this model can also be employed as a reference for identifying and quantifying “innovation rates” in empirical sequence data—ranging from text, ecology, and network evolution to urban system growth—and separating mechanistically plausible RGR systems from those requiring more nuanced explanations.

Future lines of research may include investigating micro-scale processes that could produce such logarithmic innovation adaptation, extending the approach to non-preferential attachment domains, and integrating external covariates or higher-order memory.

Conclusion

The paper represents a comprehensive technical advancement in the understanding of RGR generative processes underlying scale-free phenomena. By exposing the failure of the canonical Simon model to produce Zipf’s law and rigorously deriving the necessary dynamic innovation rate, the work provides the first universal reference model capable of generating power-law rank distributions across the entire exponent spectrum. The model should become the standard theoretical baseline for rich-get-richer analyses in complex systems and for the interpretation of empirical power-law scaling.