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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<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1712.02937v2">arXiv:1712.02937v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/i3b2tyfqqnbmtaowooitkw5e2e">fatcat:i3b2tyfqqnbmtaowooitkw5e2e</a> </span>
more »... 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≤s/2(s) due to diminished population density. Conversely, the population growth reaches equilibrium when n≥sn 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).
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