The workings of the Maximum Entropy Principle in collective human behavior

We exhibit compelling evidence regarding how well does the MaxEnt principle describe the rank-distribution of city-populations via an exhaustive study of the 50 Spanish provinces (more than 8000 cities) in a time-window of 15 years (1996-2010). We show that the dynamics that governs the population-growth is the deciding factor that originates the observed distributions. The connection between dynamics and distributions is unravelled via MaxEnt.

💡 Research Summary

The paper presents a comprehensive empirical and theoretical investigation of how the Maximum Entropy (MaxEnt) principle can account for the rank‑size distribution of city populations, using an exhaustive dataset from Spain. The authors compiled population figures for more than 8,000 municipalities across all 50 Spanish provinces, covering a fifteen‑year period from 1996 to 2010. For each year they ordered the cities by size and constructed the classic rank‑size plot, which traditionally is compared to Zipf’s law (a pure 1/r relationship). The observed curves, however, deviate markedly from a simple power law: the tail (largest cities) is heavier than Zipf predicts, while the lower ranks fall off faster than a pure power law would suggest.

To explain this systematic departure, the authors turn to the underlying dynamics of population growth. They compute annual growth rates for each city and find two robust statistical regularities. First, the mean growth rate is essentially zero, indicating that, on average, cities neither systematically expand nor shrink over the studied interval. Second, the variance of the growth rate is roughly constant across the entire range of city sizes. This constancy is the hallmark of Gibrat’s law, which posits that proportional growth is size‑independent. In other words, the stochastic process governing city size is a multiplicative random walk with stationary variance.

Armed with this dynamical insight, the authors formulate a MaxEnt model in which the entropy (S = -\sum_i p_i \ln p_i) is maximized subject to two constraints derived directly from the observed dynamics: (i) the average of (\ln N) (the logarithmic mean city size) and (ii) the average of ((\Delta N/N)^2) (the variance of the proportional growth rate). Introducing Lagrange multipliers for these constraints yields a probability density of the form

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