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Analogies between urban hierarchies and river networks: Fractals, symmetry, and self-organized criticality

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  • Chen, Yanguang

Abstract

A pair of nonlinear programming models is built to explain the fractal structure of systems of cities and those of rivers. The hierarchies of cities can be characterized by a set of exponential functions, which is identical in form to the Horton–Strahler’s laws of the river networks. Four power laws can be derived from these exponential functions. The evolution of both systems of cities and rivers are then represented as nonlinear dual programming models: to maximize information entropy subject to a certain energy use or to minimize energy dissipation subject to certain information capacity. The optimal solutions of the programming problems are just the exponential equations associated with scaling relations. By doing so, fractals and the self-organized criticality marked by the power laws are interpreted using the idea from the entropy-maximization principle, which gives further weight to the suggestion that optimality of the system as a whole defines the dynamical origin of fractal forms in both nature and society.

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  • Chen, Yanguang, 2009. "Analogies between urban hierarchies and river networks: Fractals, symmetry, and self-organized criticality," Chaos, Solitons & Fractals, Elsevier, vol. 40(4), pages 1766-1778.
    • Handle: RePEc:eee:chsofr:v:40:y:2009:i:4:p:1766-1778
    DOI: 10.1016/j.chaos.2007.09.059
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    1. R White & G Engelen, 1993. "Cellular Automata and Fractal Urban Form: A Cellular Modelling Approach to the Evolution of Urban Land-Use Patterns," Environment and Planning A, , vol. 25(8), pages 1175-1199, August.
    2. Lucien Benguigui & Daniel Czamanski & Maria Marinov & Yuval Portugali, 2000. "When and Where is a City Fractal?," Environment and Planning B, , vol. 27(4), pages 507-519, August.
    3. Chen, Yanguang & Zhou, Yixing, 2008. "Scaling laws and indications of self-organized criticality in urban systems," Chaos, Solitons & Fractals, Elsevier, vol. 35(1), pages 85-98.
    4. Claes Andersson & Steen Rasmussen & Roger White, 2002. "Urban Settlement Transitions," Environment and Planning B, , vol. 29(6), pages 841-865, December.
    5. Krugman, Paul, 1996. "Confronting the Mystery of Urban Hierarchy," Journal of the Japanese and International Economies, Elsevier, vol. 10(4), pages 399-418, December.
    6. Michael Batty & Yichun Xie, 1999. "Self-organized criticality and urban development," Discrete Dynamics in Nature and Society, Hindawi, vol. 3, pages 1-16, January.
    7. Guida, Michele & Maria, Funaro, 2007. "Topology of the Italian airport network: A scale-free small-world network with a fractal structure?," Chaos, Solitons & Fractals, Elsevier, vol. 31(3), pages 527-536.
    8. A. Stewart Fotheringham & Michael Batty & Paul A. Longley, 1989. "Diffusion‐Limited Aggregation And The Fractal Nature Of Urban Growth," Papers in Regional Science, Wiley Blackwell, vol. 67(1), pages 55-69, January.
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    Cited by:

    1. Chen, Yanguang, 2011. "Fractal systems of central places based on intermittency of space-filling," Chaos, Solitons & Fractals, Elsevier, vol. 44(8), pages 619-632.
    2. Chen, Yanguang, 2014. "An allometric scaling relation based on logistic growth of cities," Chaos, Solitons & Fractals, Elsevier, vol. 65(C), pages 65-77.
    3. Chen, Yanguang, 2012. "The rank-size scaling law and entropy-maximizing principle," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(3), pages 767-778.
    4. Moreno-Pulido, Soledad & Pavón-Domínguez, Pablo & Burgos-Pintos, Pedro, 2021. "Temporal evolution of multifractality in the Madrid Metro subway network," Chaos, Solitons & Fractals, Elsevier, vol. 142(C).
    5. Chen, Yanguang, 2012. "The mathematical relationship between Zipf’s law and the hierarchical scaling law," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(11), pages 3285-3299.

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