Scaling laws and indications of self-organized criticality in urban systems

Abstract Evolution of urban systems has been considered to exhibit some form of self-organized criticality (SOC) in the literature. This paper provides further mathematical foundations and empirical evidences to support the supposition. The hierarchical structure of systems of cities can be formulated as three exponential functions: the number law, the population size law, and the area law. These laws are identical in form to the Horton–Strahler laws of rivers and Gutenberg–Richter laws of earthquakes. From the exponential functions, three indications of SOC are also derived: the frequency–spectrum relation indicting the 1/f noise, the power laws indicating the fractal structure, and the Zipf’s law indicating the rank-size distribution. These mathematical models form a set of scaling laws for urban systems, as demonstrated in the empirical study of the system of cities in China. The fact that the scaling laws of urban systems bear an analogy to those on rivers and earthquakes lends further support to the notion of possible SOC in urban systems.

[1]  S. Appleby.  Multifractal Characterization of the Distribution Pattern of the Human Population , 2010 .

[2]  Michael Batty,et al.  Urban Systems as Cellular Automata , 1997 .

[3]  Martin J. Beckmann,et al.  City Hierarchies and the Distribution of City Size , 1958, Economic Development and Cultural Change.

[4]  D. Zanette,et al.  ROLE OF INTERMITTENCY IN URBAN DEVELOPMENT : A MODEL OF LARGE-SCALE CITY FORMATION , 1997 .

[5]  R. Horton.  EROSIONAL DEVELOPMENT OF STREAMS AND THEIR DRAINAGE BASINS; HYDROPHYSICAL APPROACH TO QUANTITATIVE MORPHOLOGY , 1945 .

[6]  Yanguang Chen,et al.  The Rank-Size Rule and Fractal Hierarchies of Cities: Mathematical Models and Empirical Analyses , 2003 .

[7]  S. Schumm.  EVOLUTION OF DRAINAGE SYSTEMS AND SLOPES IN BADLANDS AT PERTH AMBOY, NEW JERSEY , 1956 .

[8]  Lucien Benguigui,et al.  When and Where is a City Fractal? , 2000 .

[9]  P. Allen.  Cities and Regions as Self-Organizing Systems: Models of Complexity , 1997 .

[10]  Y Lee,et al.  An Allometric Analysis of the US Urban System: 1960 – 80 , 1989, Environment & planning A.

[11]  Brian J. L. Berry,et al.  RIVERS AND CENTRAL PLACES: ANALOGOUS SYSTEMS? , 1967 .

[12]  K. Davis,et al.  World urbanization, 1950-1970 , 1969 .

[13]  A. Fotheringham,et al.  Urban systems as examples of bounded chaos: exploring the relationship between fractal dimension, rank-size, and rural-to- urban migration , 1990 .

[14]  Mark Buchanan.  Ubiquity: The Science of History . . . or Why the World Is Simpler Than We Think , 2000 .

[15]  P. Bak,et al.  Self-organized criticality in the 'Game of Life" , 1989, Nature.

[16]  A. N. Strahler.  Hypsometric (area-altitude) analysis of erosional topography. , 1952 .

[17]  Pierre Frankhauser,et al.  Using Fractal Dimensions for Characterizing Intra-urban Diversity: The Example of Brussels , 2003 .

[18]  Pierre Frankhauser,et al.  La fractalité des structures urbaines , 1994 .

[19]  Jacques-François Thisse,et al.  Cities and geography , 2004 .

[20]  B. Gutenberg,et al.  Seismicity of the Earth and associated phenomena , 1950, MAUSAM.

[21]  Juval Portugali,et al.  Self-Organization and the City , 2009, Encyclopedia of Complexity and Systems Science.

[22]  Michael Batty,et al.  Fractal Cities: A Geometry of Form and Function , 1996 .

[23]  Benoit B. Mandelbrot,et al.  Fractal Geometry of Nature , 1984 .

[24]  M. Jensen,et al.  Self-organized critical dynamics of fronts: Intermittency and multiscaling , 1995 .

[25]  Isao Orishimo,et al.  Systems of Cities , 1982 .

[26]  B. Berry,et al.  Central places in Southern Germany , 1967 .

[27]  Dimitrios S. Dendrinos,et al.  Nonlinearities, interdependent dynamics and interacting scales; Progress in urban and transportation analysis , 1994 .

[28]  Benoit B. Mandelbrot,et al.  Fractals and Chaos , 2004 .

[29]  Yannis M. Ioannides,et al.  The Evolution of City Size Distributions , 2004 .

[30]  P. Krugman.  Confronting the Mystery of Urban Hierarchy , 1996 .

[31]  M. Batty.  New ways of looking at cities , 1995, Nature.

[32]  Roger White,et al.  The Use of Constrained Cellular Automata for High-Resolution Modelling of Urban Land-Use Dynamics , 1997 .

[33]  Henrik Jeldtoft Jensen,et al.  Self-Organized Criticality: Emergent Complex Behavior in Physical and Biological Systems , 1998 .

[34]  Roger White,et al.  Cellular Automata and Fractal Urban Form: A Cellular Modelling Approach to the Evolution of Urban Land-Use Patterns , 1993 .

[35]  David Harvey.  Explanation in Geography , 1969 .

[36]  Michael Batty,et al.  Self-organized criticality and urban development , 1999 .

[37]  Roger White,et al.  Urban systems dynamics and cellular automata: Fractal structures between order and chaos , 1994 .

[38]  Per Bak,et al.  How Nature Works: The Science of Self‐Organized Criticality , 1997 .

[39]  Heinz-Otto Peitgen,et al.  The science of fractal images , 2011 .

[40]  Ikuo Matsuba,et al.  Scaling behavior in urban development process of Tokyo City and hierarchical dynamical structure , 2003 .

[41]  Yanguang Chen,et al.  Multi-fractal measures of city-size distributions based on the three-parameter Zipf model , 2004 .

[42]  Günter Haag,et al.  The rank-size distribution of settlements as a dynamic multifractal phenomenon , 1994 .

[43]  Michael Batty,et al.  Cities and fractals: simulating growth and form , 1991 .

[44]  Fractals, self-organized-criticality and the fixed scale transformation , 1995 .

[45]  I. Nakanishi,et al.  Scaling laws of earthquakes derived by renormalization group method , 2005 .

[46]  George Kingsley Zipf,et al.  Human behavior and the principle of least effort , 1949 .

[47]  Jisheng,et al.  Derivations of fractal models of city hierarchies using entropy-maximization principle , 2002 .

[48]  M Batty.  FRACTALS - NEW WAYS OF LOOKING AT CITIES , 1995 .

[49]  Yanguang Chen,et al.  Reinterpreting Central Place Networks Using Ideas from Fractals and Self-Organized Criticality , 2006 .

[50]  D. L. Anderson,et al.  Theoretical Basis of Some Empirical Relations in Seismology by Hiroo Kanamori And , 1975 .

[51]  Y. Aizawa,et al.  Self-organized criticality and partial synchronization in an evolving network , 2000 .

[52]  S. Hergarten.  Self-Organized Criticality in Earth Systems , 2002 .