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The role of self-loops and link removal in evolutionary games on networks

1 Department of Information Engineering and Mathematics, University of Siena, Italy
2 Instituto de Matemática e Estatística, Universidade Federal do Rio Grande do Sul, Brazil
3 Instituto Nacional de Matemática Pura e Aplicada, Rio de Janeiro, Brazil

Special Issues: Mathematical Methods in the Biosciences

Recently, a new mathematical formulation of evolutionary game dynamics [1] has been introduced accounting for a finite number of players organized over a network, where the players are located at the nodes of a graph and edges represent connections between them. Internal steady states are particularly interesting in control and consensus problems, especially in a networked context where they are related to the coexistence of different strategies. In this paper we consider this model including self-loops. Existence of internal steady states is studied for different graph topologies in two-strategy games. Results on the effect of removing links from central players are also presented.
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Keywords evolutionary game theory; games on graphs; games on networks; equilibrium states; competition; cooperation; self-loops on graphs; connectivity of networks

Citation: Dario Madeo, Chiara Mocenni, Jean Carlo Moraes, Jorge P. Zubelli. The role of self-loops and link removal in evolutionary games on networks. Mathematical Biosciences and Engineering, 2019, 16(5): 5287-5306. doi: 10.3934/mbe.2019264


  • 1. D. Madeo and C. Mocenni, Game Interactions and dynamics on networked populations, IEEE T. Automat. Contr., 60 (2015), 1801–1810.
  • 2. A. Barrat, M. Barthelemy and A. Vespignani, Dynamical Processes on Complex Networks. Cambridge University Press, UK, 2008.
  • 3. G. Ehrhardt, M. Marsili and F. Vega-Redondo, Diffusion and growth in an evolving network, Int. J. Game Theory, 334 (2006), 383–397.
  • 4. V. Colizza, A. Barrat, M. Barthélemy, et al., The role of the airline transportation network in the prediction and predictability of global epidemics, P. Natl. Acad. Sci. USA, 103 (2006), 2015–2020.
  • 5. V. Colizza and A. Vespignani, Epidemic modeling in metapopulation systems with heterogeneous coupling pattern: Theory and simulations, J. Theor. Biol., 251 (2008), 450–467.
  • 6. S. Tully, M. G. Cojocaru and C. T. Bauch, Multiplayer games and HIV transmission via casual encounters, Math. Biosci. Eng., 14 (2017), 359–376.
  • 7. M. D'Orsogna and M. Perc, Statistical physics of crime: A review, Phys. Life Rev., 12 (2015), 1–21.
  • 8. D. Madeo, L. R. Comolli and C. Mocenni, Emergence of microbial networks as response to hostile environments, Front. Microbiol., 5 (2014), 407.
  • 9. N. Quijano, C. Ocampo-Martinez, J. Barreiro-Gomez, et al., The role of population games and evolutionary dynamics in distributed control systems: The advantages of evolutionary game theory, IEEE Contr. Sys. Mag., 37 (2017), 70–97.
  • 10. R. Gray, A. Franci, V. Srivastava, et al., Multi-agent decision-making dynamics inspired by honeybees, IEEE T. Contr. Netw. Sys., 5 (2018), 793–806.
  • 11. F. C. Santos, J. M. Pacheco and T. Lenaerts, Evolutionary dynamics of social dilemmas in structured heterogeneous populations, P. Natl. Acad. Sci. USA, 103 (2006), 3490–3494.
  • 12. H. Ohtsuki and M. A. Nowak, The replicator equation on graphs, J. Theor. Biol., 243 (2006), 86–97.
  • 13. T. Konno, A condition for cooperation in a game on complex networks, J. Theor. Biol., 269 (2011), 224–233.
  • 14. J. Gómez-Gardenes, I. Reinares, A. Arenas, et al., Evolution of cooperation in multiplex networks, Sci. Rep., 2 (2012), 620.
  • 15. S. M. Cameron and A. Cintrón-Arias, Prisoner's Dilemma on real social networks: Revisited, Math. Biosci. Eng., 10 (2013), 1381–1398.
  • 16. D. G. Rand, M. A. Nowak, J. H. Fowler, et al., Static network structure can stabilize human cooperation, P. Natl. Acad. Sci. USA, 11 (2014), 17093–17098.
  • 17. B. Allen, G. Lippner, Y. Chen, et al., Evolutionary dynamics on any population structure, Nature, 544 (2017), 227.
  • 18. B. Fotouhi, N. Momeni, B. Allen, et al., Evolution of Cooperation on Stochastic Block Models, preprint, arXiv:1807.03093.
  • 19. J. Weibull, Evolutionary Game Theory, MIT Press, Cambridge, MA, 1995.
  • 20. J. Hofbauer and K. Sigmund, Evolutionary game dynamics, B. Am. Math. Soc, 40 (2003) 479–519.
  • 21. M. A. Nowak, Evolutionary Dynamics: Exploring the Equations of Life, Belknap Press of Harvard University Press, Harvard, MA, 2006.
  • 22. G. Iacobelli, D. Madeo and C. Mocenni, Lumping evolutionary game dynamics on networks, J. Theor. Biol., 407 (2016), 328–338.
  • 23. D. Pais, C. H. Caicedo-Nùñez and N. E. Leonard, Hopf bifurcations and limit cycles in evolutionary network dynamics, SIAM J. Appl. Dyn. Syst., 11 (2012), 1754–1884.
  • 24. W. Ren and R. Beard, Consensus seeking in multiagent systems under dynamically changing interaction topologies, IEEE T. Automat. Contr., 50 (2005), 655–661.
  • 25. R. Olfati-Saber, A. Fax and R. Murray, Consensus and cooperation in networked multi-agent systems, P. IEEE, 95 (2007), 215–233.
  • 26. B. Kozma and A. Barrat, Consensus formation on adaptive networks, Phys. Rev. E, 77 (2008), 016102.
  • 27. G. Punzo, G. F. Young, M Macdonald, et al., Using network dynamical influence to drive consensus, Sci. Rep., 6 (2016), 26318.
  • 28. A. Traulsen, F. C. Santos and J. M. Pacheco, Evolutionary Games in Self-Organizing Populations, in Adaptive networks: Theory, Models and Applications (eds. T. Gross and H. Sayama), Springer Berlin Heidelberg, Germany, (2009), 253–267.
  • 29. S. Boccaletti, V. Latora, Y. Moreno, et al., Complex networks: Structure and dynamics, Phys. Rep., 424 (2006), 175–308.
  • 30. Y. Bramoullé and R. Kranton, Games Played on Networks, in The Oxford Handbook of the Economics of Networks (eds. Y. Bramoullé, A. Galeotti and B. Rogers), Oxford University Press. Available from: http://www.oxfordhandbooks.com/view/10.1093/oxfordhb/9780199948277.001. 0001/oxfordhb-9780199948277.
  • 31. A. Banerjee, A. G. Chandrasekhar, E. Duflo, et al., Gossip: Identifying Central Individuals in a Social Network, preprint, arXiv:1406.2293v3.
  • 32. D. Madeo and C. Mocenni, Self-regulation promotes cooperation in social networks, preprint, arXiv:1807.07848.
  • 33. M. Newman, Network: An introduction, Oxford University Press, 2010.


This article has been cited by

  • 1. Dario Madeo, Somnath Mazumdar, Chiara Mocenni, Roberto Zingone, Evolutionary game for task mapping in resource constrained heterogeneous environments, Future Generation Computer Systems, 2020, 108, 762, 10.1016/j.future.2020.03.026

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