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      Is this scaling nonlinear?

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          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. yx β , β≠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 β and confidence intervals, but also to (ii) quantify the evidence in favour of β≠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.

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          Urban characteristics attributable to density-driven tie formation

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            On Scaling of Scientific Knowledge Production in U.S. Metropolitan Areas

            Using data on all scientific publications from the Scopus database, we find a superlinear scaling effect for U.S. metropolitan areas as indicated by the increase of per capita publication output with city size. We also find that the variance of residuals is much higher for mid-sized cities (100,000 to 500,000 inhabitants) compared to larger cities. The latter result is indicative of the critical mass required to establish a scientific center in a particular discipline. Finally, we observe that the largest cities publish much less than the scaling law would predict, indicating that the largest cities are relatively unattractive locations for scientific research.
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              Constructing cities, deconstructing scaling laws

              Cities can be characterised and modelled through different urban measures. Consistency within these observables is crucial in order to advance towards a science of cities. Bettencourt et al have proposed that many of these urban measures can be predicted through universal scaling laws. We develop a framework to consistently define cities, using commuting to work and population density thresholds, and construct thousands of realisations of systems of cities with different boundaries for England and Wales. These serve as a laboratory for the scaling analysis of a large set of urban indicators. The analysis shows that population size alone does not provide enough information to describe or predict the state of a city as previously proposed, indicating that the expected scaling laws are not corroborated. We found that most urban indicators scale linearly with city size regardless of the definition of the urban boundaries. However, when non-linear correlations are present, the exponent fluctuates considerably.
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                Author and article information

                Journal
                R Soc Open Sci
                R Soc Open Sci
                RSOS
                royopensci
                Royal Society Open Science
                The Royal Society
                2054-5703
                July 2016
                13 July 2016
                13 July 2016
                : 3
                : 7
                : 150649
                Affiliations
                Max Planck Institute for the Physics of Complex Systems , Dresden, Germany
                Author notes
                Author for correspondence: E. G. Altmann e-mail: edugalt@ 123456pks.mpg.de

                One contribution to a special feature ‘City analytics: mathematical modelling and computational analytics for urban behaviour’.

                Author information
                http://orcid.org/0000-0003-1503-9242
                http://orcid.org/0000-0002-5850-3394
                http://orcid.org/0000-0002-0879-7865
                http://orcid.org/0000-0002-1932-3710
                Article
                rsos150649
                10.1098/rsos.150649
                4968456
                27493764
                e9363e6c-39ea-4034-9194-300fcf01431d
                © 2016 The Authors.

                Published by the Royal Society under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/4.0/, which permits unrestricted use, provided the original author and source are credited.

                History
                : 30 November 2015
                : 15 June 2016
                Funding
                Funded by: Portuguese Foundation for Science and Technology
                Award ID: SFRH/BD/90050/2012
                Categories
                1008
                175
                74
                Special Feature
                City Analytics
                Custom metadata
                July, 2016

                scaling laws,statistical inference,allometry
                scaling laws, statistical inference, allometry

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