On Metropolis Growth
Abstract
We consider the scaling laws, secondorder 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 energyconsuming 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 overpopulated and underpopulated areas is large. Moreover, we also draw the relationship between our approach and multiscalar information, when economic dependency between a metropolis's subregions 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 $s$).
 Publication:

arXiv eprints
 Pub Date:
 December 2017
 DOI:
 10.48550/arXiv.1712.02937
 arXiv:
 arXiv:1712.02937
 Bibcode:
 2017arXiv171202937A
 Keywords:

 Physics  Physics and Society;
 Economics  Econometrics
 EPrint:
 19 pages, 16 figures