A Spatial Model of City Growth and Formation

We introduce a model in which city populations grow at rates proportional to the area of their “sphere of influence”, where the influence of a city depends on its population (to power \alpha) and distance from city (to power -\beta) and where new cities arise according to a certain random rule. A simple non-rigorous analysis of asymptotics indicates that for \beta > 2\alpha$ the system exhibits “balanced growth” in which there are an increasing number of large cities, whose populations have the same order of magnitude, whereas for \beta < 2\alpha$ the system exhibits “unbalanced growth” in which a few cities capture most of the total population. Conceptually the model is best regarded as a spatial analog of the combinatorial “Chinese restaurant process”.

💡 Research Summary

The paper introduces a spatial stochastic model that simultaneously captures the growth of existing cities and the birth of new ones. Each city i is characterized by its population N_i and its geographic location x_i. The “sphere of influence” of a city at a point x is defined as I_i(x)=N_i^α·d(x,x_i)^{−β}, where α>0 controls how population amplifies influence and β>0 governs the decay of influence with distance. At each discrete time step a unit of population is added to the system; this unit is allocated to cities proportionally to the total influence they exert at the point of addition. Consequently, the growth rate of a city is proportional to the area of its influence region, a natural spatial analogue of the preferential‑attachment mechanism.

In parallel, new cities are created with a fixed probability λ per time step. When a new city is born its location is drawn uniformly from the region that lies sufficiently far from all existing cities, ensuring that its initial influence does not immediately overlap heavily with the influence of older cities. This rule guarantees a continual influx of “seed” cities while allowing the existing ones to expand.

The authors perform a non‑rigorous asymptotic analysis of two macroscopic quantities: the number of cities M(t) and the size of the largest city N_max(t). The relationship between the exponents α and β determines the qualitative regime of the system.

Balanced growth (β > 2α) – Distance attenuation dominates the effect of population. New cities tend to appear far enough from older ones that their influence regions remain largely disjoint. As a result the influence areas of all cities stay comparable, leading to a roughly equal sharing of the total population. Both M(t) and N_max(t) grow polynomially with time (M(t)∼t^γ, N_max(t)∼t^δ with γ,δ>0), and the distribution of city sizes is relatively flat; many cities of comparable magnitude coexist.

Unbalanced growth (β < 2α) – The population exponent outweighs distance decay. An early city that happens to acquire a large N_i obtains a disproportionately large influence region, suppressing the establishment of nearby new cities. Consequently M(t) grows very slowly (approaching a saturation level) while N_max(t) increases almost linearly, concentrating the bulk of the total population in a few “megacities”. The size distribution acquires a heavy‑tailed, Pareto‑like form.

The model can be viewed as a spatial extension of the Chinese Restaurant Process (CRP). In the classic CRP, customers sit at tables with probability proportional to the current table size, generating a rich‑get‑richer clustering. Here, tables are replaced by cities, customers by population units, and an additional spatial kernel (distance decay) is introduced, allowing the model to capture geographic constraints that the original CRP ignores.

Simulation experiments confirm the analytical predictions: for β > 2α the system produces many cities of similar size, whereas for β < 2α a few dominant cities emerge and the number of active cities stabilizes. The authors also discuss the limitations of the current formulation: α and β are treated as fixed global constants, migration of individuals between cities is omitted, and exogenous factors such as economic policy, transportation infrastructure, or natural hazards are not modeled.

Future research directions suggested include: (i) allowing α and β to vary across space or time to reflect heterogeneous environments; (ii) incorporating explicit migration dynamics that enable cities to lose as well as gain population; (iii) augmenting the model with socioeconomic variables and calibrating it against real‑world urban data to estimate parameters and test predictive power. Such extensions would make the framework useful for urban planners and regional economists interested in the long‑term evolution of city systems, the emergence of megacities, and the impact of policy interventions on spatial population distribution.