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No Access Submitted: 22 February 2015 Accepted: 14 April 2015 Published Online: 27 April 2015

The expansion of neighborhood and pattern formation on spatial prisoner's dilemma

Chaos 25, 043115 (2015); https://doi.org/10.1063/1.4919080
Xiaolan Qian1, a),b), Fangqian Xu1, Junzhong Yang2, and Jürgen Kurths3, c)
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  • 1School of Electronics and Information, Zhejiang University of Media and Communications, Hangzhou 310018, China
  • 2School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, China
  • 3Institute of Physics, Humboldt University Berlin, Berlin D-12489, Germany

  • a)Author to whom correspondence should be addressed. Electronic mail:

    b)Also at Institute of Physics, Humboldt University Berlin, Berlin D-12489, Germany and Potsdam Institute for Climate Impact Research, Potsdam D-14415, Germany.

    c)Also at Potsdam Institute for Climate Impact Research, Potsdam D-14415, Germany and Institute for Complex Systems and Mathematical Biology, University of Aberdeen, Aberdeen AB24 3FX, United Kingdom.

ABSTRACT
The prisoner's dilemma (PD), in which players can either cooperate or defect, is considered a paradigm for studying the evolution of cooperation in spatially structured populations. There the compact cooperator cluster is identified as a characteristic pattern and the probability of forming such pattern in turn depends on the features of the networks. In this paper, we investigate the influence of expansion of neighborhood on pattern formation by taking a weak PD game with one free parameter T, the temptation to defect. Two different expansion methods of neighborhood are considered. One is based on a square lattice and expanses along four directions generating networks with degree increasing with K=4m. The other is based on a lattice with Moore neighborhood and expanses along eight directions, generating networks with degree of K=8m. Individuals are placed on the nodes of the networks, interact with their neighbors and learn from the better one. We find that cooperator can survive for a broad degree 4≤K≤70 by taking a loose type of cooperator clusters. The former simple corresponding relationship between macroscopic patterns and the microscopic PD interactions is broken. Under a condition that is unfavorable for cooperators such as large T and K, systems prefer to evolve to a loose type of cooperator clusters to support cooperation. However, compared to the well-known compact pattern, it is a suboptimal strategy because it cannot help cooperators dominating the population and always corresponding to a low cooperation level.

ACKNOWLEDGMENTS

This work was supported by the National Natural Science Foundation of China under Grant No. 11374254 and Zhejiang Provincial Natural Science Foundation of China under Grant No. LY12A04003.

REFERENCES

  1. 1. A. L. Barabási and R. Albert, “ Emergence of scaling in random networks,” Science 286, 509–512 (1999). https://doi.org/10.1126/science.286.5439.509, Google ScholarCrossref
  2. 2. S. N. Dorogovtsev and J. F. F. Mendes, Evolution of Networks: From Biological Nets to the Internet and WWW ( Oxford University Press, Oxford, 2003). Google ScholarCrossref
  3. 3. S. Boccaletti, V. Latora, Y. Moreno, M. Chavez, and D. Hwang, “ Complex networks: Structure and dynamics,” Phys. Rep. 424, 175–308 (2006). https://doi.org/10.1016/j.physrep.2005.10.009, Google ScholarCrossref
  4. 4. A. Barrat, M. Barthlemy, and A. Vespignani, Dynamical Processes on Complex Networks ( Cambridge University Press, Cambridge, England, 2008). Google ScholarCrossref
  5. 5. M. Nowak and R. May, “ Evolutionary games and spatial chaos,” Nature (London) 359, 826–829 (1992). https://doi.org/10.1038/359826a0, Google ScholarCrossref
  6. 6. M. Nowak and R. May, “ The spatial dilemmas of evolution,” Int. J. Bifurcation Chaos 3, 35–78 (1993). https://doi.org/10.1142/S0218127493000040, Google ScholarCrossref
  7. 7. F. C. Santos and J. M. Pacheco, “ Scale-free network provide a unifying framework for the emergence of cooperation,” Phys. Rev. Lett. 95, 098104 (2005). https://doi.org/10.1103/PhysRevLett.95.098104, Google ScholarCrossref
  8. 8. C. Hauert and M. Doebeli, “ Spatial structure often inhibits the evolution of cooperation in the snowdrift game,” Nature 428, 643–646 (2004). https://doi.org/10.1038/nature02360, Google ScholarCrossref
  9. 9. P. D. Taylor and T. Day, “ Behavioural evolution: Cooperate with thy neighbour?,” Nature 428, 611–612 (2004). https://doi.org/10.1038/428611a, Google ScholarCrossref
  10. 10. M. G. Zimmermann and V. M. Eguíluz, “ Cooperation, social networks, and the emergence of leadership in a prisoners dilemma with adaptive local interactions,” Phys. Rev. E 72, 056118 (2005). https://doi.org/10.1103/PhysRevE.72.056118, Google ScholarCrossref
  11. 11. F. C. Santos, J. M. Pacheco, and T. Lenaerts, “ Evolutionary dynamics of social dilemmas in structured heterogeneous populations,” Proc. Natl. Acad. Sci. U.S.A. 103, 3490–3494 (2006). https://doi.org/10.1073/pnas.0508201103, Google ScholarCrossref
  12. 12. J. Vukov, G. Szabó, and A. Szolnoki, “ Cooperation in the noisy case: Prisoners dilemma game on two types of regular random graphs,” Phys. Rev. E 73, 067103 (2006). https://doi.org/10.1103/PhysRevE.73.067103, Google ScholarCrossref
  13. 13. G. Szabó and G. Fáth, “ Evolutionary games on graphs,” Phys. Rep. 446, 97–216 (2007). https://doi.org/10.1016/j.physrep.2007.04.004, Google ScholarCrossref
  14. 14. C. Darwin, The Origin of Species ( Harvard University Press, MA, 1964). Google Scholar
  15. 15. R. Axelrod, The Evolution of Cooperation ( Basic Book, New York, 1984). Google Scholar
  16. 16. J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics ( Cambridge University Press, Cambridge, England, 1998). Google ScholarCrossref
  17. 17. C.-L. Tang, W.-X. wang, X. Wu, and B.-H. Wang, “ Effects of average degree on cooperation in networked evolutionary game,” Eur. Phys. J. B 53, 411–415 (2006). https://doi.org/10.1140/epjb/e2006-00395-2, Google ScholarCrossref
  18. 18. A. Szolnoki, M. Perc, and Z. Danku, “ Towards effective payoffs in the prisoner's dilemma game on scale-free networks,” Physica A 387, 2075–2082 (2008). https://doi.org/10.1016/j.physa.2007.11.021, Google ScholarCrossref
  19. 19. A. Albert and A. L. Brarabási, “ Statistical mechanics of complex networks,” Rev. Mod. Phys. 74, 47–97 (2002). https://doi.org/10.1103/RevModPhys.74.47, Google ScholarCrossref
  20. 20. R. A. Hill and R. I. M. Dunbar, “ Social network size in humans,” Human Nat. 14, 53–72 (2003). https://doi.org/10.1007/s12110-003-1016-y, Google ScholarCrossref
  21. 21. C. Hauert, S. De Monte, J. Hofbauer, and K. Sigmund, “ Volunteering as red queen mechanism for cooperation in public goods games,” Science 296, 1129–1132 (2002). https://doi.org/10.1126/science.1070582, Google ScholarCrossref
  22. 22. G. Szabó and C. Hauert, “ Evolutionary prisoners dilemma games with voluntary participation,” Phys. Rev. E 66, 062903 (2002). https://doi.org/10.1103/PhysRevE.66.062903, Google ScholarCrossref
  23. 23. A. Szolnoki and G. Szabó, “ Cooperation enhanced by inhomogeneous activity of teaching for evolutionary Prisoner's Dilemma games,” Europhys. Lett. 77, 30004 (2007). https://doi.org/10.1209/0295-5075/77/30004, Google ScholarCrossref
  24. 24. M. H. Vainstein and J. J. Arenzon, “ Disordered environments in spatial games,” Phys. Rev. E 64, 051905 (2001). https://doi.org/10.1103/PhysRevE.64.051905, Google ScholarCrossref
  25. 25. F. Fu, M. Nowak, and C. Hauert, “ Invasion and expansion of cooperators in lattice populations: Prisoner's dilemma vs. snowdrift games,” J. Theor. Biol. 266, 358–366 (2010). https://doi.org/10.1016/j.jtbi.2010.06.042, Google ScholarCrossref
  26. 26. M. Ifti, T. Killingback, and M. Doebeli, “ Effects of neighbourhood size and connectivity on the spatial continuous prisoner's dilemma,” J. Theor. Biol. 231, 97 (2004). https://doi.org/10.1016/j.jtbi.2004.06.003, Google ScholarCrossref
  27. 27. Z. Wang, S. Kokubo, J. Tanimoto, E. Fukuda, and K. Shigaki, “ Insight into the so–called spatial reciprocity,” Phys. Rev. E 88, 042145 (2013). https://doi.org/10.1103/PhysRevE.88.042145, Google ScholarCrossref
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