A 70-year-old math pillar used to explain everything from city sizes to wealth gaps is actually wrong.
April 16, 2026
Original Paper
Simon's model does not produce Zipf's law: The fundamental rich-get-richer mechanism for any power-law size ranking
arXiv · 2604.13184
The Takeaway
Since 1955, the 'Simon model' has been the go-to explanation for the 'rich-get-richer' effect, which results in a few huge cities and many tiny ones (Zipf’s Law). This paper proves that the math behind this pillar is fundamentally flawed and doesn't actually produce the patterns we see in the real world. Instead, the authors show that for these patterns to exist, the rate of innovation must decay in a very specific, inverse logarithmic way. This isn't just an academic spat; it means our fundamental understanding of how social and economic systems grow is based on a broken premise. It changes how we view the 'inevitable' dominance of certain companies or cities, suggesting the mechanism of their growth is far more complex than a simple feedback loop.
From the 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 me