Zipf's Law Emergence

Rank-frequency power law: f(r) ∝ r^{-α}. Emerges in language, cities, wealth, earthquakes. Simon model shows preferential attachment generates it from scratch.

Zipf's law (Zipf 1949): rank × frequency ≈ constant in natural language. The 1000th word is used 1000× less than the most common. Simon (1955) derived it from preferential attachment: a new token picks an existing word proportional to its current count (rich-get-richer), or with probability q invents a new word. This generates α≈1 exactly. Cities follow Zipf (Gabaix 1999 — random growth), earthquakes follow Gutenberg-Richter (seismic moment), wealth follows Pareto — all the same power law with different exponents.