Skip to main content
SpringerLink
Log in

Evolutionary Stability and the Evolution of Cooperation on Heterogeneous Graphs

  • Published:
Dynamic Games and Applications Aims and scope Submit manuscript

Abstract

Recent studies have revealed that graph heterogeneity can considerably affect evolutionary processes and that it promotes the emergence and maintenance of cooperation in social dilemmas. In this paper, we analytically derive the evolutionary dynamics and the evolutionarily stable strategy (ESS) condition for \(2 \times 2\) games on heterogeneous graphs based on “pairwise comparison” updating. Using pair approximation, we introduce a new state variable to measure the evolutionary process. In the limit of weak selection, we show that the evolutionary dynamics can be approximated as a replicator equation with a transformed payoff matrix, and the ESS condition depends on both the mean value and the variance of the degree distribution. These results are subsequently applied to the Prisoner’s Dilemma game and the Stag Hunt game. In both games, we find that the variance plays a determinant role in the evolution of cooperation: Cooperative strategy cannot evolve in regular graphs, but it is favored by natural selection in strongly heterogeneous graphs.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

  1. Albert R, Barabási AL (2002) Statistical mechanics of complex networks. Rev Mod Phys 74:47–98

    Article  MathSciNet  MATH  Google Scholar 

  2. Albert R, Jeong H, Barabási AL (1999) Internet: diameter of the world-wide web. Nature 401:130–131

    Article  Google Scholar 

  3. Axelord R (1984) The evolution of cooperation. Basic Books, New York

    Google Scholar 

  4. Barabási AL, Albert R (1999) Emergence of scaling in random networks. Science 15:509–512

    MathSciNet  MATH  Google Scholar 

  5. Bergstrom TC (2003) The algebra of assortative encounters and the evolution of cooperation. Int Game Theory Rev 5:1–18

    Article  MathSciNet  MATH  Google Scholar 

  6. Dorogotsev SN, Mendes JFF (2003) Evolution of networks: from biological nets to the internet and www. Oxford University Press, Oxford

    Book  Google Scholar 

  7. Hofbauer J, Sigmund K (1998) Evolutionary games and population dynamics. Cambridge University Press, Cambridge

    Book  MATH  Google Scholar 

  8. Fu F, Nowak MA, Christakis NA, Fowler JH (2012) The evolution of homophily. Sci Rep 2:00845

    Google Scholar 

  9. Jeong H, Tombor B, Albert R, Ottval ZN, Barabási AL (2000) The large-scale organization of metabolic networks. Nature 407:651–654

    Article  Google Scholar 

  10. Li C, Zhang BY, Cressman R, Tao Y (2013) Evolution of cooperation in a heterogeneous graph: fixation probabilities under weak selection. PLoS One 8:e66560

    Article  Google Scholar 

  11. Maciejewski W, Fu F, Hauert C (2014) Evolutionary game dynamics in populations with heterogenous structures. PLoS Comput Biol 10:e1003567

    Article  Google Scholar 

  12. McPherson M, Smith-Lovin L, Cook JM (2001) Birds of a feather: homophily in social networks. Annu Rev Sociol 27:415–444

  13. Newman MEJ, Strogatz SH, Watts DJ (2001) Random graphs with arbitrary degree distribution and their applications. Phys Rev E 64:026118

    Article  Google Scholar 

  14. Nobari S, Lu XS, Karras P, Bressan S (2011) Fast random graph generation. In: Proceedings of the 14th international conference on extending database technology, pp 331–342

  15. Nowak MA, May R (1992) Evolutionary games and spatial chaos. Nature 359:826–829

    Article  Google Scholar 

  16. Nowak MA (2006) Evolutionary dynamics. Harvard University Press, Cambridge

    MATH  Google Scholar 

  17. Ohtsuki H, Hauert C, Lieberman E, Nowak MA (2006) A simple rule for the evolution of cooperation on graphs and social networks. Nature 441:502–505

    Article  Google Scholar 

  18. Ohtsuki H, Nowak MA (2006) The replicator equation on graphs. J Theor Biol 243:68–97

    Article  MathSciNet  Google Scholar 

  19. Ohtsuki H, Nowak MA (2008) Evolutionary stability on graphs. J Theor Biol 251:698–707

    Article  MathSciNet  Google Scholar 

  20. Perc M, Gómez-Gardeñes J, Szolnoki A, Floría LM, Moreno Y (2013) Evolutionary dynamics of group interactions on structured populations: a review. J R Soc Interface 10:20120997

    Article  Google Scholar 

  21. Pinheiro FL, Santos FC, Pacheco JM (2012) How selection pressure changes the nature of social dilemmas in structured populations. New J Phys 14:073035

    Article  Google Scholar 

  22. Pinheiro FL, Santos FC, Pacheco JM (2012) From local to global dilemmas in social networks. PLoS One 7:e32114

    Article  Google Scholar 

  23. Poncela J, Gómez-Gardeñes J, Floría LM, Moreno Y, Sánchez A (2009) Cooperative scale-free networks despite the presence of defector hubs. Europhys Lett 88:38003

    Article  Google Scholar 

  24. Poncela J, Gómez-Gardeñes J, Moreno Y, Floría LM (2010) Cooperation in the prisoner’s dilemma game in random scale-free graphs. Int J Bifurcat Chaos 20:849–857

    Article  MATH  Google Scholar 

  25. Rand DG, Arbesman S, Christakis NA (2011) Dynamic social networks promote cooperation in experiments with humans. Proc Natl Acad Sci USA 108:19193–19198

    Article  Google Scholar 

  26. Roca CP, Cuesta JA, Sánchez A (2009) Evolutionary game theory: temporal and spatial effects beyond replicator dynamics. Phys Life Rev 6:208–249

    Article  Google Scholar 

  27. Santos FC, Pacheco JM, Lenaerts T (2006) Evolutionary dynamics of social dilemmas in the structured heterogeneous populations. Proc Natl Acad Sci USA 103:3490–3494

    Article  Google Scholar 

  28. Santos FC, Santos MD, Pacheco JM (2008) Social diversity promotes the emergence of cooperation in the public goods game. Nature 454:213–216

    Article  Google Scholar 

  29. Santos FC, Pinheiro FL, Lenaerts T, Pacheco JM (2012) The role of diversity in the evolution of cooperation. J Theor Biol 299:88–96

    Article  MathSciNet  Google Scholar 

  30. Szabó G, Fáth G (2007) Evolutionary games on graphs. Phys Rep 446:97–216

    Article  MathSciNet  Google Scholar 

  31. Szolnoki A, Perc M, Danku Z (2008) Towards effective payoffs in the prisoner’s dilemma game on scale-free networks. Phys A 387:2075–2082

    Article  Google Scholar 

  32. Traulsen A, Claussen JC, Hauert C (2005) Coevolutionary dynamics: from finite to infinite populations. Phys Rev Lett 95:238701

    Article  Google Scholar 

Download references

Acknowledgments

This research is supported by the National Natural Science Foundation of China (Nos. 11301032 and 31270439) and “the Fundamental Research Funds for the Central Universities” of China.

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, China

    Boyu Zhang

  2. Département de mathématiques et de statistique, Université de Montréal, C.P. 6128, Succursale Centre-ville, Montreal, QC, H3C 3J7, Canada

    Cong Li

  3. Key Lab of Animal Ecology and Conservational Biology, Institute of Zoology, Chinese Academy of Sciences, Beijing, 100101, China

    Yi Tao

Authors
  1. Boyu Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Cong Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yi Tao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Cong Li.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, B., Li, C. & Tao, Y. Evolutionary Stability and the Evolution of Cooperation on Heterogeneous Graphs. Dyn Games Appl 6, 567–579 (2016). https://doi.org/10.1007/s13235-015-0146-2

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13235-015-0146-2

Keywords

Search

Navigation