On Metropolis Growth

TL;DR

This paper models metropolis growth by analyzing energy consumption, spatial mobility, and entropy, revealing how population and area dynamics influence energy variance and economic scaling in complex urban systems.

Contribution

It introduces a theoretical framework linking city size, population, energy consumption, and entropy, providing insights into growth limits and economic scaling laws.

Findings

Variance and entropy of energy consumption peak and then decline with population growth.

City growth is limited when population density drops below a threshold.

Economic size scales linearly with city area under constant population density.

Abstract

We consider the scaling laws, second-order statistics and entropy of the consumed energy of metropolis cities which are hybrid complex systems comprising social networks, engineering systems, agricultural output, economic activity and energy components. We abstract a city in terms of two fundamental variables; $s$ resource cells (of unit area) that represent energy-consuming geographic or spatial zones (e.g. land, housing or infrastructure etc.) and a population comprising $n$ mobile units that can migrate between these cells. We show that with a constant metropolis area (fixed $s$), the variance and entropy of consumed energy initially increase with $n$, reach a maximum and then eventually diminish to zero as saturation is reached. These metrics are indicators of the spatial mobility of the population. Under certain situations, the variance is bounded as a quadratic function of the mean consumed energy of the metropolis. However, when population and metropolis area are endogenous, growth in the latter is arrested when $n\leq\frac{s}{2}\log(s)$ due to diminished population density. Conversely, the population growth reaches equilibrium when $n\geq {s}\log{n}$ or equivalently when the aggregate of both over-populated and under-populated areas is large. Moreover, we also draw the relationship between our approach and multi-scalar information, when economic dependency between a metropolis's sub-regions is based on the entropy of consumed energy. Finally, if the city's economic size (domestic product etc.) is proportional to the consumed energy, then for a constant population density, we show that the economy scales linearly with the surface area (or $s$).