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Is this scaling nonlinear?

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Leitão,  Jorge C.
Max Planck Institute for the Physics of Complex Systems, Max Planck Society;

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Miotto,  José María
Max Planck Institute for the Physics of Complex Systems, Max Planck Society;

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Gerlach,  Martin
Max Planck Institute for the Physics of Complex Systems, Max Planck Society;

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Altmann,  Eduardo G.
Max Planck Institute for the Physics of Complex Systems, Max Planck Society;

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Citation

Leitão, J. C., Miotto, J. M., Gerlach, M., & Altmann, E. G. (2016). Is this scaling nonlinear? Royal Society Open Science, 3(7): 150649. doi:10.1098/rsos.150649.


Cite as: http://hdl.handle.net/11858/00-001M-0000-002B-AD1B-6
Abstract
One of the most celebrated findings in complex systems in the last decade is that different indexes y (e. g. patents) scale nonlinearly with the population x of the cities in which they appear, i. e. y similar to x(beta), beta not equal 1. More recently, the generality of this finding has been questioned in studies that used new databases and different definitions of city boundaries. In this paper, we investigate the existence of nonlinear scaling, using a probabilistic framework in which fluctuations are accounted for explicitly. In particular, we show that this allows not only to (i) estimate beta and confidence intervals, but also to (ii) quantify the evidence in favour of beta not equal 1 and (iii) test the hypothesis that the observations are compatible with the nonlinear scaling. We employ this framework to compare five different models to 15 different datasets and we find that the answers to points (i)-(iii) crucially depend on the fluctuations contained in the data, on how they are modelled, and on the fact that the city sizes are heavy-tailed distributed.