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Power laws in cities population, financial markets and internet sites (scaling in systems with a variable number of components)

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  • Blank, Aharon
  • Solomon, Sorin

Abstract

We study a few dynamical systems composed of many components whose sizes evolve according to multiplicative stochastic rules. We compare them with respect to the emergence of power laws in the size distribution of their components. We show that the details specifying and enforcing the smallest size of the components are crucial as well as the rules for creating new components. In particular, a growing system with a fixed number of components and a fixed smallest component size does not converge to a power law. We present a new model with variable number of components that converges to a power law for a very wide range of parameters. In a very large subset of this range, one obtains for the exponent α the special value 1 specific for the city populations distribution. We discuss the conditions in which α can take different values. In the case of the stock market, the distribution of the investors’ wealth is related to the ratio between the new capital invested in stock and the rate of increase of the stock index.

Suggested Citation

  • Blank, Aharon & Solomon, Sorin, 2000. "Power laws in cities population, financial markets and internet sites (scaling in systems with a variable number of components)," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 287(1), pages 279-288.
  • Handle: RePEc:eee:phsmap:v:287:y:2000:i:1:p:279-288
    DOI: 10.1016/S0378-4371(00)00464-7
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    Cited by:

    1. Itzhack, Royi & Muchnik, Lev & Erez, Tom & Tsaban, Lea & Goldenberg, Jacob & Solomon, Sorin & Louzoun, Yoram, 2010. "Empirical extraction of mechanisms underlying real world network generation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(22), pages 5308-5318.
    2. Sorin Solomon & Nataša Golo, 2015. "Microeconomic structure determines macroeconomic dynamics: Aoki defeats the representative agent," Journal of Economic Interaction and Coordination, Springer;Society for Economic Science with Heterogeneous Interacting Agents, vol. 10(1), pages 5-30, April.
    3. Guido Fioretti, 2005. "The Production Function," Papers physics/0511191, arXiv.org.
    4. Pushkin, Dmitri O & Aref, Hassan, 2004. "Bank mergers as scale-free coagulation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 336(3), pages 571-584.
    5. Navarro-Barrientos, Jesús Emeterio & Cantero-Álvarez, Rubén & Matias Rodrigues, João F. & Schweitzer, Frank, 2008. "Investments in random environments," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(8), pages 2035-2046.
    6. Segarra, Agustí & Teruel, Mercedes, 2012. "An appraisal of firm size distribution: Does sample size matter?," Journal of Economic Behavior & Organization, Elsevier, vol. 82(1), pages 314-328.
    7. repec:ebl:ecbull:v:15:y:2003:i:6:p:1-7 is not listed on IDEAS
    8. Fioretti, Guido, 2007. "The production function," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 374(2), pages 707-714.
    9. Esteban Rossi-Hansberg & Mark L. J. Wright, 2007. "Urban Structure and Growth," Review of Economic Studies, Oxford University Press, vol. 74(2), pages 597-624.
    10. de Wit, Gerrit, 2005. "Firm size distributions: An overview of steady-state distributions resulting from firm dynamics models," International Journal of Industrial Organization, Elsevier, vol. 23(5-6), pages 423-450, June.
    11. Jan Eeckhout, 2004. "Gibrat's Law for (All) Cities," American Economic Review, American Economic Association, vol. 94(5), pages 1429-1451, December.
    12. Amaral, L.A.N. & Gopikrishnan, P. & Plerou, V. & Stanley, H.E., 2001. "A model for the growth dynamics of economic organizations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 299(1), pages 127-136.
    13. Wang, Cheng-Jun & Wu, Lingfei, 2016. "The scaling of attention networks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 448(C), pages 196-204.
    14. Andrea Bonaccorsi & Maurizio Martinelli & Cristina Rossi & Irma Serrecchia, 2002. "Measuring and modelling Internet diffusion using second level domains: the case of Italy," LEM Papers Series 2002/17, Laboratory of Economics and Management (LEM), Sant'Anna School of Advanced Studies, Pisa, Italy.
    15. Bucsa, G. & Jovanovic, F. & Schinckus, C., 2011. "A unified model for price return distributions used in econophysics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 390(20), pages 3435-3443.
    16. Malevergne, Y. & Saichev, A. & Sornette, D., 2013. "Zipf's law and maximum sustainable growth," Journal of Economic Dynamics and Control, Elsevier, vol. 37(6), pages 1195-1212.
    17. Zengwang Xu & Robert Harriss, 2010. "A Spatial and Temporal Autocorrelated Growth Model for City Rank—Size Distribution," Urban Studies, Urban Studies Journal Limited, vol. 47(2), pages 321-335, February.
    18. Corrado Di Guilmi & Mauro Gallegati & Edoardo Gaffeo, 2003. "Power Law Scaling in the World Income Distribution," Economics Bulletin, AccessEcon, vol. 15(6), pages 1-7.
    19. Ramos, Arturo & Sanz-Gracia, Fernando, 2015. "US city size distribution revisited: Theory and empirical evidence," MPRA Paper 64051, University Library of Munich, Germany.
    20. repec:eee:ecomod:v:221:y:2010:i:1:p:85-89 is not listed on IDEAS
    21. Jovanovic, Franck & Schinckus, Christophe, 2017. "Econophysics and Financial Economics: An Emerging Dialogue," OUP Catalogue, Oxford University Press, number 9780190205034.
    22. V. Gontis, 2002. "Multiplicative Stochastic Model of the Time Interval between Trades in Financial Markets," Papers cond-mat/0211317, arXiv.org.

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