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

TL;DR

This paper challenges Herbert Simon's classic rich-get-richer model, deriving a time-dependent innovation rate that accurately produces Zipf's law across various systems, and demonstrates its superiority over Simon's model.

Contribution

The authors derive a correct, time-dependent innovation rate that generates Zipf's law in rich-get-richer systems, correcting flaws in Simon's original analysis.

Findings

The correct innovation rate must decay as 1/ln(N) to produce Zipf's law.

Simon’s model fails to produce Zipf's law in the zero innovation limit.

The proposed dynamic innovation rate accurately models word rankings in literature.

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.