Why Does Zipfs Law Break Down in Rank-Size Distribution of Cities?

We study rank-size distribution of cities in Japan on the basis of data analysis. From the census data after World War II, we find that the rank-size distribution of cities is composed of two parts, each of which has independent power exponent. In addition, the power exponent of the head part of the distribution changes in time and Zipf’s law holds only in a restricted period. We show that Zipf’s law broke down due to both of Showa and Heisei great mergers and recovered due to population growth in middle-sized cities after the great Showa merger.

💡 Research Summary

The paper investigates why Zipf’s law, which predicts a rank‑size exponent of b ≈ 1 for city populations, breaks down and later recovers in Japan. Using Japanese census data from 1950 to 2006, the authors first show that the rank‑size distribution of cities is not a single power law but consists of two distinct regimes: a “head” (the largest ~20–260 cities) and a “tail”. Each regime follows a power‑law R(x)=a·x⁻ᵇ, but the exponent b differs between them. Over time the exponent of the head regime changes dramatically: it is about 1.2 in the early 1950s, drops below 1 during the 1960s, stabilizes near 1 from the 1970s to 2000, and rises again after 2000.

The authors link these fluctuations to two major administrative reorganizations: the “Great Showa merger” (1955‑1960) and the “Great Heisei merger” (starting in 2000). Both mergers dramatically increased the number of municipalities, causing a sudden rise in the head‑region exponent because many new, relatively small cities entered the top ranks, diluting the average size of the largest cities. After each merger, the exponent gradually returns toward unity as mid‑size cities grow and internal migration from large metropolitan areas to surrounding towns (the “turn” phenomena) intensifies.

To explain the observed dynamics, the paper introduces an agent‑based model with 3,500 sites representing municipalities. Each site starts with a random population (0–1). The simulation alternates between (1) a migration step and (2) a merger step. In the migration step, a source site m is chosen at random; then, with probability α, the destination is selected from sites with smaller populations (G < Nₘ), otherwise from larger sites (G > Nₘ). A random fraction (0–20 %) of Nₘ moves to the destination. Repeating this 10⁶ times yields a stationary distribution with a power‑law exponent b≈1.12.

In the merger step, two sites are randomly paired. If neither is a city (population ≥ 0.95), they merge into a new city if the combined population exceeds the threshold; otherwise they form a town. If at least one is a city, the pair merges into a city with probability β = 0.5, reflecting the empirical fact that town‑town mergers are more common than city‑city mergers. The merger process is repeated until the number of cities rises by about 77 % relative to the pre‑migration state.

Simulation results reproduce the empirical pattern: after a merger the exponent b jumps upward, then, as migration proceeds, b declines back toward 1. The authors find a linear relationship between the migration bias α and the final exponent (b ≈ 3.7 α − 0.09), indicating that stronger preferential movement toward smaller municipalities drives the system toward Zipf’s law (α ≈ 0.3).

Overall, the study demonstrates that (i) Japan’s city rank‑size distribution is composed of two power‑law regimes, (ii) large‑scale administrative mergers temporarily disrupt Zipf’s law by inflating the number of small cities, (iii) subsequent population growth in mid‑size cities and internal migration restore the exponent to unity, and (iv) a simple agent‑based model can capture these dynamics. The findings highlight the importance of policy‑driven structural changes and migration patterns in shaping urban size distributions, offering quantitative insight for urban planners and scholars of complex systems.