Simon's model does not produce Zipf's law: The fundamental rich-get-richer mechanism for any power-law size ranking
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866913032523743232 |
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| author | Rosillo-Rodes, Pablo Zimmerman, Julia Witte Hébert-Dufresne, Laurent Dodds, Peter Sheridan |
| author_facet | Rosillo-Rodes, Pablo Zimmerman, Julia Witte Hébert-Dufresne, Laurent Dodds, Peter Sheridan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_13184 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Simon's model does not produce Zipf's law: The fundamental rich-get-richer mechanism for any power-law size ranking Rosillo-Rodes, Pablo Zimmerman, Julia Witte Hébert-Dufresne, Laurent Dodds, Peter Sheridan Physics and Society 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. |
| title | Simon's model does not produce Zipf's law: The fundamental rich-get-richer mechanism for any power-law size ranking |
| topic | Physics and Society |
| url | https://arxiv.org/abs/2604.13184 |