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      The Statistics of Urban Scaling and Their Connection to Zipf’s Law

      1 , * , 2 , 2

      PLoS ONE

      Public Library of Science

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          Abstract

          Urban scaling relations characterizing how diverse properties of cities vary on average with their population size have recently been shown to be a general quantitative property of many urban systems around the world. However, in previous studies the statistics of urban indicators were not analyzed in detail, raising important questions about the full characterization of urban properties and how scaling relations may emerge in these larger contexts. Here, we build a self-consistent statistical framework that characterizes the joint probability distributions of urban indicators and city population sizes across an urban system. To develop this framework empirically we use one of the most granular and stochastic urban indicators available, specifically measuring homicides in cities of Brazil, Colombia and Mexico, three nations with high and fast changing rates of violent crime. We use these data to derive the conditional probability of the number of homicides per year given the population size of a city. To do this we use Bayes’ rule together with the estimated conditional probability of city size given their number of homicides and the distribution of total homicides. We then show that scaling laws emerge as expectation values of these conditional statistics. Knowledge of these distributions implies, in turn, a relationship between scaling and population size distribution exponents that can be used to predict Zipf’s exponent from urban indicator statistics. Our results also suggest how a general statistical theory of urban indicators may be constructed from the stochastic dynamics of social interaction processes in cities.

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          Most cited references 19

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          Power laws, Pareto distributions and Zipf's law

           MEJ Newman (2005)
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            Growth, innovation, scaling, and the pace of life in cities.

            Humanity has just crossed a major landmark in its history with the majority of people now living in cities. Cities have long been known to be society's predominant engine of innovation and wealth creation, yet they are also its main source of crime, pollution, and disease. The inexorable trend toward urbanization worldwide presents an urgent challenge for developing a predictive, quantitative theory of urban organization and sustainable development. Here we present empirical evidence indicating that the processes relating urbanization to economic development and knowledge creation are very general, being shared by all cities belonging to the same urban system and sustained across different nations and times. Many diverse properties of cities from patent production and personal income to electrical cable length are shown to be power law functions of population size with scaling exponents, beta, that fall into distinct universality classes. Quantities reflecting wealth creation and innovation have beta approximately 1.2 >1 (increasing returns), whereas those accounting for infrastructure display beta approximately 0.8 <1 (economies of scale). We predict that the pace of social life in the city increases with population size, in quantitative agreement with data, and we discuss how cities are similar to, and differ from, biological organisms, for which beta<1. Finally, we explore possible consequences of these scaling relations by deriving growth equations, which quantify the dramatic difference between growth fueled by innovation versus that driven by economies of scale. This difference suggests that, as population grows, major innovation cycles must be generated at a continually accelerating rate to sustain growth and avoid stagnation or collapse.
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              A unified theory of urban living.

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                Author and article information

                Affiliations
                [1 ]School of Human Evolution and Social Change, Arizona State University, Tempe, Arizona, United States of America
                [2 ]Santa Fe Institute, Santa Fe, New Mexico, United States of America
                University of Namur, Belgium
                Author notes

                Conceived and designed the experiments: AGL LMB. Analyzed the data: AGL. Contributed reagents/materials/analysis tools: AGL HY LMB. Wrote the paper: AGL LMB.

                Contributors
                Role: Editor
                Journal
                PLoS One
                PLoS ONE
                plos
                plosone
                PLoS ONE
                Public Library of Science (San Francisco, USA )
                1932-6203
                2012
                18 July 2012
                : 7
                : 7
                3399879
                22815745
                PONE-D-12-04809
                10.1371/journal.pone.0040393
                (Editor)
                Gomez-Lievano et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
                Counts
                Pages: 11
                Categories
                Research Article
                Mathematics
                Applied Mathematics
                Complex Systems
                Probability Theory
                Statistical Distributions
                Distribution Curves
                Bayes Theorem
                Probability Density
                Probability Distribution
                Statistics
                Statistical Theories
                Scaling Theory
                Statistical Methods
                Physics
                Interdisciplinary Physics
                Statistical Mechanics
                Social and Behavioral Sciences
                Economics
                Microeconomics
                Urban Economics
                Sociology
                Crime and Criminology
                Homicide
                Social Networks
                Social Systems

                Uncategorized

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