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
Submitted (2026)
Submitted (2026)
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∝r−1 -- and its generalization S∝r−α for α≥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→∞, rather than α→1. Here, for pure rich-get-richer systems, we derive the time-dependent innovation rate ρt that correctly produces power-law size rankings across all α≥0. To produce Zipf's law, we uncover that ρt must decay as the inverse of the log of the number of types, 1/lnN. 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.
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