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Winner-weaken-loser-strengthen rule leads to optimally cooperative interdependent networks

Nonlinear Dynamics volume 96, pages 49–56 (2019)Cite this article

Abstract

We introduce a winner-weaken-loser-strengthen rule and study its effects on how cooperation evolves on interdependent networks. The new rule lowers the learning ability of a player if its payoff is larger than the average payoff of its neighbors, thus enhancing its chance to hold onto its current strategy. Conversely, when a player gaining less than the average payoff of its neighborhood, its learning ability is increased, thus weakening the player by increasing the chance of strategy change. Furthermore, considering the nature of human pursue fairness, we let a loser, someone who has larger learning ability, can benefit from another network, whereas a winner cannot. Our results show that moderate values of the threshold lead to a high cooperation plateau, while too high or too small values of the threshold inhibit cooperation. At moderate thresholds, the flourishing cooperation is attributed to species diversity and equality, whereas a lacking of species diversity determines the vanishing of cooperation. We thus demonstrate that a simple winner-weaken-loser-strengthen rule significantly expands the scope of cooperation on structured populations.

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References

  1. Axelrod, R., Hamilton, W.D.: The evolution of cooperation. Science 211(4489), 1390–1396 (1981)

    MathSciNet  Article  MATH  Google Scholar 

  2. Nowak, M.A., May, R.M.: Evolutionary games and spatial chaos. Nature 359(6398), 826 (1992)

    Article  Google Scholar 

  3. Nowak, M.A., Sasaki, A., Taylor, C., Fudenberg, D.: Emergence of cooperation and evolutionary stability in finite populations. Nature 428(6983), 646 (2004)

    Article  Google Scholar 

  4. Hauert, C., De Monte, S., Hofbauer, J., Sigmund, K.: Volunteering as red queen mechanism for cooperation in public goods games. Science 296(5570), 1129–1132 (2002)

    Article  Google Scholar 

  5. Parker, G.A., Smith, J.M.: Optimality theory in evolutionary biology. Nature 348(6296), 27 (1990)

    Article  Google Scholar 

  6. Nowak, M.A.: Five rules for the evolution of cooperation. Science 314(5805), 1560–1563 (2006)

    Article  Google Scholar 

  7. Chen, X., Wang, L.: Promotion of cooperation induced by appropriate payoff aspirations in a small-world networked game. Phys. Rev. E 77(1), 017103 (2008)

    Article  Google Scholar 

  8. Macy, M.W., Flache, A.: Learning dynamics in social dilemmas. Proc. Natl Acad. Sci. 99(suppl 3), 7229–7236 (2002)

    Article  MATH  Google Scholar 

  9. Szolnoki, A., Perc, M.: Coevolution of teaching activity promotes cooperation. New J. Phys. 10(4), 043036 (2012)

    Article  Google Scholar 

  10. Tomassini, M., Luthi, L., Giacobini, M.: Hawks and doves on small-world networks. Phys. Rev. E 73(1), 016132 (2006)

    Article  Google Scholar 

  11. Santos, F.C., Santos, M.D., Pacheco, J.M.: Social diversity promotes the emergence of cooperation in public goods games. Nature 454(7201), 213 (2008)

    Article  Google Scholar 

  12. Santos, F.C., Pacheco, J.M.: Scale-free networks provide a unifying framework for the emergence of cooperation. Phys. Rev. Lett. 95(9), 098104 (2005)

    Article  Google Scholar 

  13. Van Segbroeck, S., Santos, F.C., Pacheco, J.M., Lenaerts, T.: Coevolution of cooperation, response to adverse social ties and network structure. Games 1(3), 317–337 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  14. Santos, F.C., Pacheco, J.M.: Risk of collective failure provides an escape from the tragedy of the commons. Proc. Natl Acad. Sci. 108(26), 10421–10425 (2011)

    Article  Google Scholar 

  15. Gömez-Gardeñes, J., Campillo, M., Floría, L.M., Moreno, Y.: Dynamical organization of cooperation in complex topologies. Phys. Rev. Lett. 98(10), 108103 (2007)

    Article  Google Scholar 

  16. Rong, Z., Li, X., Wang, X.: Roles of mixing patterns in cooperation on a scale-free networked game. Phys. Rev. E 76(2), 027101 (2007)

    Article  Google Scholar 

  17. Assenza, S., Gömez-Gardeñes, J., Latora, V.: Enhancement of cooperation in highly clustered scale-free networks. Phys. Rev. E 78(1), 017101 (2008)

    Article  Google Scholar 

  18. Poncela, J., Gömez-Gardeñes, J., Moreno, Y.: Cooperation in scale-free networks with limited associative capacities. Phys. Rev. E 83(5), 057101 (2011)

    Article  Google Scholar 

  19. Antonioni, A., Tomassini, M., Buesser, P.: Random diffusion and cooperation in continuous two-dimensional space. J. Theor. Biol. 344, 40–48 (2014)

    Article  Google Scholar 

  20. Wang, Z., Szolnoki, A., Perc, M.: If players are sparse social dilemmas are too: importance of percolation for evolution of cooperation. Sci. Rep. 2, 369 (2012)

    Article  Google Scholar 

  21. Ebel, H., Bornholdt, S.: Coevolutionary games on networks. Phys. Rev. E 66(5), 056118 (2002)

    Article  Google Scholar 

  22. Pacheco, J.M., Traulsen, A., Nowak, M.A.: Coevolution of strategy and structure in complex networks with dynamical linking. Phys. Rev. Lett. 97(25), 258103 (2006)

    Article  Google Scholar 

  23. Szabö, G., Fáth, G.: Evolutionary games on graphs. Phys. Rep. 446(4–6), 97–216 (2007)

    MathSciNet  Article  Google Scholar 

  24. Perc, M., Jordan, J.J., Rand, D.G., Wang, Z., Boccaletti, S., Szolnoki, A.: Statistical physics of human cooperation. Phys. Rep. 687, 1–51 (2017)

    MathSciNet  Article  MATH  Google Scholar 

  25. Parshani, R., Buldyrev, S.V., Havlin, S.: Reducing the coupling strength leads to a change from a first to second order percolation transition. Phys. Rev. Lett. 105(4), 048701 (2010)

    Article  Google Scholar 

  26. Huang, X., Gao, J., Buldyrev, S.V., Havlin, S., Stanley, H.E.: Robustness of interdependent networks under targeted attack. Phys. Rev. E 83(6), 065101 (2011)

    Article  Google Scholar 

  27. Gao, J., Buldyrev, S.V., Stanley, H.E., Havlin, S.: Networks formed from interdependent networks. Nat. Phys. 8(1), 40 (2011)

    Article  Google Scholar 

  28. Wang, B., Chen, X., Wang, L.: Probabilistic interconnection between interdependent networks promotes cooperation in the public goods game. J. Stat. Mech.: Theory Exp. 2012(11), P11017 (2012)

    Article  Google Scholar 

  29. Szolnoki, A., Perc, M.: Information sharing promotes prosocial behaviour. New J. Phys. 15(5), 053010 (2013)

    MathSciNet  Article  Google Scholar 

  30. Gömez-Gardeñes, J., Gracia-Lázaro, C., Floría, L.M., Moreno, Y.: Evolutionary dynamics on interdependent populations. Phys. Rev. E 86(5), 056113 (2012)

    Article  Google Scholar 

  31. Wang, Z., Szolnoki, A., Perc, M.: Interdependent network reciprocity in evolutionary games. Sci. Rep. 3, 1183 (2013)

    Article  Google Scholar 

  32. Zhou, D., Stanley, H.E., D’Agostino, G., Scala, A.: Assortativity decreases the robustness of interdependent networks. Phys. Rev. E 86(6), 066103 (2012)

    Article  Google Scholar 

  33. Helbing, D.: Globally networked risks and how to respond. Nature 497(7447), 51 (2013)

    Article  Google Scholar 

  34. Luo, C., Zhang, X.-L.: Effect of self-organized interdependence between populations on the evolution of cooperation. Commun. Nonlinear Sci. Numer. Simul. 42, 73 (2017)

    MathSciNet  Article  Google Scholar 

  35. Szabö, G., Tőke, C.: Evolutionary prisoner’s dilemma game on a square lattice. Phys. Rev. E 58(1), 69 (1997)

    Article  Google Scholar 

  36. Chen, X., et al.: Rise to modern levels of ocean oxygenation coincided with the Cambrian radiation of animals. Nat. Commun. 6, 7142 (2015)

    Article  Google Scholar 

  37. Tanaka, G., Hou, X., Ma, X., Edgecombe, G.D., Strausfeld, N.J.: Chelicerate neural ground pattern in a Cambrian great appendage arthropod. Nature 502(7471), 364 (2013)

    Article  Google Scholar 

  38. Wang, Z., Szolnoki, A., Perc, M.: Self-organization towards optimally interdependent networks by means of coevolution. New J. Phys. 16(3), 033041 (2014)

    Article  Google Scholar 

  39. Szolnoki, A., Perc, M.: Evolutionary dynamics of cooperation in neutral populations. New J. Phys. 20(1), 013031 (2018)

    Article  Google Scholar 

  40. Szolnoki, A., Mobilia, M., Jiang, L.-L., Szczesny, B., Rucklidge, A.M., Perc, M.: Cyclic dominance in evolutionary games: a review. J. R. Soc. Interface 11(100), 20140735 (2014)

    Article  Google Scholar 

  41. Weitz, J.-S., Eksin, C., Paarporn, K., Brown, S.-P., Ratcliff, W.-C.: An oscillating tragedy of the commons in replicator dynamics with game-environment feedback. Proc. Natl Acad. Sci. 113(47), E7518–E7525 (2016)

    Article  Google Scholar 

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Acknowledgements

We are grateful to Dr. Keke Huang for useful discussions. We acknowledge support from (i) National Natural Science Foundation of China (Grants No. U1803263), the National 1000 Young Talent Plan (No. W099102), the Fundamental Research Funds for the Central Universities (No. 3102017jc03007) and China Computer Federation - Tencent Open Fund (No. IAGR20170119) to Z.W, (ii) the National Natural Science Foundation of China (Grants No. 11671348) to L.S., (iii) the Yunnan Postgraduate Scholarship Award to C.S. and (iv) the Slovenian Research Agency (Grants J1-7009, J1-9112 and P1-0403) to M.P.

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    Authors and Affiliations

    1. School of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming, 650221, Yunnan, China

      Lei Shi, Chen Shen, Yini Geng & Chen Chu

    2. School of Software, Yunnan University, Kunming, 650504, Yunnan, China

      Haoran Meng

    3. Faculty of Natural Sciences and Mathematics, University of Maribor, Koroška cesta 160, 2000, Maribor, Slovenia

      Matjaž Perc

    4. Center for Applied Mathematics and Theoretical Physics, University of Maribor, Mladinska 3, 2000, Maribor, Slovenia

      Matjaž Perc

    5. Complexity Science Hub Vienna, Josefstädterstraße 39, 1080, Vienna, Austria

      Matjaž Perc

    6. CNR Institute for Complex Systems, Via Madonna del Piano 10, 50019, Florence, Italy

      Stefano Boccaletti

    7. Unmanned Systems Research Institute, Northwestern Polytechnical University, Xi’an, 710072, China

      Stefano Boccaletti

    8. School of Mechanical Engineering and Center for OPTical IMagery Analysis and Learning (OPTIMAL), Northwestern Polytechnical University, Xi’an, 710072, China

      Zhen Wang

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    1. Lei Shi
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    L. Shi and C. Shen contributed equally to this work.

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    Shi, L., Shen, C., Geng, Y. et al. Winner-weaken-loser-strengthen rule leads to optimally cooperative interdependent networks. Nonlinear Dyn 96, 49–56 (2019). https://doi.org/10.1007/s11071-019-04772-6

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    Keywords

    • Cooperation
    • Evolutionary Game Theory
    • Interdependent Network
    • Winner-Weaken-Loser-Strengthen Rule
    • Coevolution