Urban areas exhibit a wide range of population sizes, from small towns to megacities. Zipf's law, a Pareto distribution with a power-law exponent of one, has been the most prominent proposed model for the statistical distribution of city sizes. However, empirical evidence frequently deviates from Zipf's law, particularly for small and large cities. Existing theories often fail to derive city size distributions from fundamental demographic principles while simultaneously explaining observed variations. This research aims to address this gap by establishing a comprehensive framework rooted in demography. The authors hypothesize that the structure of migration flows between cities, coupled with variations in vital rates (birth and death rates), are the primary determinants of city size distributions. This framework will provide a robust mathematical methodology for deriving not only Zipf's law but also other size distributions under specific conditions, ultimately resolving inconsistencies and clarifying the significance of deviations from these idealized models in terms of individual choice, symmetry, information, and selective pressures.

The paper reviews the extensive literature surrounding Zipf's law and its application to urban systems. It highlights the law's frequent empirical violations, especially at the extremes of city size. Previous models, often based on proportional growth (Gibrat's Law) or preferential attachment, are critiqued for lacking a foundation in fundamental population dynamics. The authors argue that a more fundamental approach, grounded in a city's basic population dynamics, is necessary to resolve the empirical controversies and provide a meaningful interpretation of observed deviations from Zipf's law. This involves considering the interplay of selection and randomization processes within the context of population dynamics.

The authors begin by deriving the population size distribution of cities based on their demographic dynamics. The change in a city's population is modeled as a balance of births, deaths, and migration. The model incorporates migration flows between cities as population rates (Jᵢⱼ = mᵢⱼNᵢ(t)), where mᵢⱼ represents the probability of migration from city i to city j. This general formulation allows for the creation of new cities and the disappearance of others. The model is expressed in matrix form (N(t) = A(t)N(t-1)), where N(t) is the vector of city populations at time t and A is the environmental matrix, capturing vital rates and migration probabilities. Ergodic theorems from population dynamics are then applied to analyze the properties of the solutions under different conditions on the environment. The authors show that for a static environment, the population structure converges to the leading eigenvector of the environment, representing the eigenvector centrality of the urban system. This solution, however, doesn't resemble Zipf's law. To incorporate statistical fluctuations, stochasticity is introduced into the vital and migration rates. The migration rates are decomposed into symmetric (sᵢⱼ) and antisymmetric (δᵢⱼ) components, relating the model to the gravity model of geography. A self-consistent solution is sought for the population structure vector, considering the interplay of symmetric and antisymmetric migration components and their effects on the rotational symmetry of the system. The authors demonstrate that stochasticity restores broken symmetries, counteracting selective pressures on city growth rates. By assuming that, over long timescales, growth rates equalize, the dynamics simplifies into a form akin to geometric Brownian motion without drift. This leads to a Fokker-Planck equation that is solved analytically to derive the stationary distribution of city sizes. The solution reveals that Zipf's law emerges as a probability density when the average relative difference in vital rates vanishes due to fluctuations. This is confirmed through numerical simulations. The methodology also examines deviations from Zipf's law, relating them to non-zero growth rate fluctuations and the information content of these deviations from a neutral, Zipfian state, measured using Kullback-Leibler (KL) divergence.

The study's key findings demonstrate that the demographic dynamics of urban systems can lead to diverse city size distributions, with Zipf's law being one specific outcome under particular conditions. The model establishes that Zipf's law arises when stochastic fluctuations in vital rates and migration flows average out over long timescales, restoring rotational symmetry in the system's dynamics. Deviations from Zipf's law, therefore, indicate selection or preferences for certain cities, representing information embedded in the system. This is quantified using KL divergence, which measures the difference between an observed city size distribution and the Zipfian distribution. The analysis reveals that the convergence to Zipf's law is highly sensitive to boundary conditions, particularly those governing the lower bound (smallest cities). The model shows that a substantial, often unobserved, population of very small cities is implicitly required to maintain Zipf's law, highlighting limitations to the law's complete scale invariance. The time required for the stochastic dynamics to average out and yield Zipf's law is shown to be very long, in the range of centuries or even millennia. Empirical analysis of US city data from 1790 to 1990 demonstrates a gradual approach towards Zipf's law over time, with recent deviations suggesting the influence of growth preferences for certain regions. The framework also reveals that the integration of an urban system depends on strong intercity migration flows, not necessarily symmetric ones which might lead to Zipf's law. Deviations from Zipf's law can originate from various factors, including socioeconomic opportunities, natural disasters, and demographic changes.

The findings significantly advance our understanding of city size distributions by providing a demographic foundation for previously observed patterns. The model explains why Zipf's law is often approximated but rarely precisely observed in real urban systems. The emergence of Zipf's law as a long-term average emphasizes its status as a neutral state rather than a universal law. Deviations from the Zipfian distribution are interpreted as information reflecting various influences, ranging from economic opportunities to demographic shifts and policy interventions. The model's sensitivity to boundary conditions for smaller cities highlights the importance of considering the whole system's dynamics and not just isolated parts of it. The long timescales involved suggest that short-term observations might not accurately reflect the underlying demographic processes, underscoring the need for long-term perspectives in analyzing urban development. The framework also offers a theoretical basis for understanding the coherence and screening properties of Zipf's law, previously noted in the literature. The model has broader implications beyond urban systems, as its core principles of multiplicative growth, stochasticity, and symmetry restoration could potentially be extended to other complex systems exhibiting rank-size rules.

This research presents a robust demographic framework that successfully derives various city size distributions, including Zipf's law as a special case under conditions of long-term symmetry restoration. The model clarifies the meaning of deviations from Zipf's law, interpreting them as informative signals of specific choices and preferences. Future research should focus on incorporating more detailed demographic factors, exploring the specific reasons for deviations from Zipf's law in particular urban contexts, and validating the model's predictions using data from various urban systems around the world. The framework's generality suggests potential applicability to other complex systems with similar rank-size patterns.

The model simplifies certain aspects of urban dynamics, such as assuming homogeneous migration probabilities within city size classes. The assumption of long-term equalization of growth rates may not perfectly hold in all urban systems. The study focuses on long-term averages, potentially overlooking short-term fluctuations and transient phenomena that can significantly affect individual cities. The empirical analysis primarily relies on US data, limiting the model's immediate generalizability to other countries with differing historical and socio-economic contexts. Further research is required to investigate the model's robustness and applicability across diverse urban systems.