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Emergence of self-similarity in football dynamics

An Erratum to this article was published on 02 April 2014


The multiplayer dynamics of a football game is analyzed to unveil self-similarities in the time evolution of player and ball positioning. Temporal fluctuations in both the team-turf boundary and the ball location are uncovered to follow the rules of fractional Brownian motion with a Hurst exponent of H ~ 0.7. The persistence time below which self-similarity holds is found to be several tens of seconds, implying a characteristic time scale that governs far-from-equilibrium motion on a playing field.


  1. R.S. Mendes, L.C. Malacarne, C. Anteneodo, Eur. Phys. J. B 57, 357 (2007)

    ADS  Article  Google Scholar 

  2. A. Grunz, D. Memmert, J. Perl, Hum. Mov. Sci. 31, 334 (2012)

    Article  Google Scholar 

  3. K. Yokoyama, Y. Yamamoto, PLoS Comput. Biol. 7, e1002181 (2011)

    Article  MathSciNet  Google Scholar 

  4. Y. Yamamoto, K. Yokoyama, PLoS ONE 6, e29638 (2011)

    ADS  Article  Google Scholar 

  5. C. Castellano, S. Fortunato, V. Loreto, Rev. Mod. Phys. 81, 591 (2009)

    ADS  Article  Google Scholar 

  6. S. Wang, P.G. Wolynes, Proc. Natl. Acad. Sci. USA 108, 15189 (2011)

    ADS  Google Scholar 

  7. E. Bittner, A. Nußbaumer, W. Janke, M. Weigel, Europhys. Lett. 78, 58002 (2007)

    ADS  Article  Google Scholar 

  8. E. Bittner, A. Nußbaumer, W. Janke, M. Weigel, Eur. Phys. J. B 67, 459 (2009)

    ADS  Article  Google Scholar 

  9. A. Heuer, A.C. Müller, O. Rubner, Europhys. Lett. 89, 38007 (2010)

    ADS  Article  Google Scholar 

  10. E. Ben-Naim, N.W. Hengartner, S. Redner, F. Vazquez, J. Stat. Phys. 151, 458 (2013)

    ADS  Article  MATH  MathSciNet  Google Scholar 

  11. R. da Silva, M.H. Vainstein, L.C. Lambb, S.D. Prado, Comput. Phys. Commun. 184, 661 (2013)

    ADS  Article  Google Scholar 

  12. R.N. Onody, P.A. de Castro, Phys. Rev. E 70, 037103 (2004)

    ADS  Article  Google Scholar 

  13. J. Duch, J.S. Waitzman, L.A.N. Amaral, PLoS ONE 5, e10937 (2010)

    ADS  Article  Google Scholar 

  14. C. Carlos, M.M. Antonio, M.J. Julián, C. Merelo-Molina, J. Syst. Sci. Complex 26, 21 (2013)

    Article  MATH  Google Scholar 

  15. D.A. Winter, Biomechanics of human movement (John Wiley & Sons, New York, 1979)

  16. J.P. Bouchaud, A. Georges, Phys. Rep. 195, 127 (1990)

    ADS  Article  MathSciNet  Google Scholar 

  17. S. Kim, Fractals 14, 71 (2006)

    Article  Google Scholar 

  18. H. Shima, T. Nomura, T. Nakayama, Phys. Rev. B 70, 075116 (2004)

    ADS  Article  Google Scholar 

  19. N.P. Greis, H.S. Greenside, Phys. Rev. A 44, 2324 (1991)

    ADS  Article  Google Scholar 

  20. C. Carling, M.A. Williams, T. Reilly, Handbook of soccer match analysis (Routledge, Oxford, 2005)

  21. C.K. Peng, S.V. Buldyrev, S. Havlin, M. Simons, H.E. Stanley, A.L. Goldberger, Phys. Rev. E 49, 1685 (1994)

    ADS  Article  Google Scholar 

  22. K. Hu, P.Ch. Ivanov, Z. Chen, P. Carpena, H.E. Stanley, Phys. Rev. E 64, 011114 (2001)

    ADS  Article  Google Scholar 

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Correspondence to Hiroyuki Shima.

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Kijima, A., Yokoyama, K., Shima, H. et al. Emergence of self-similarity in football dynamics. Eur. Phys. J. B 87, 41 (2014).

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  • Statistical and Nonlinear Physics