Skip to main content
SpringerLink
Log in

Evolutionary games on multilayer networks: a colloquium

  • Colloquium
  • Published:
The European Physical Journal B Aims and scope Submit manuscript

Abstract

Networks form the backbone of many complex systems, ranging from the Internet to human societies. Accordingly, not only is the range of our interactions limited and thus best described and modeled by networks, it is also a fact that the networks that are an integral part of such models are often interdependent or even interconnected. Networks of networks or multilayer networks are therefore a more apt description of social systems. This colloquium is devoted to evolutionary games on multilayer networks, and in particular to the evolution of cooperation as one of the main pillars of modern human societies. We first give an overview of the most significant conceptual differences between single-layer and multilayer networks, and we provide basic definitions and a classification of the most commonly used terms. Subsequently, we review fascinating and counterintuitive evolutionary outcomes that emerge due to different types of interdependencies between otherwise independent populations. The focus is on coupling through the utilities of players, through the flow of information, as well as through the popularity of different strategies on different network layers. The colloquium highlights the importance of pattern formation and collective behavior for the promotion of cooperation under adverse conditions, as well as the synergies between network science and evolutionary game theory.

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

References

  1. M. Gell-Mann, Eng. Sci. 57, 2 (1988)

    Google Scholar 

  2. A.L. Barabási, Nat. Phys. 8, 14 (2012)

    Google Scholar 

  3. S. Boccaletti, V. Latora, Y. Moreno, M. Chavez, D. Hwang, Phys. Rep. 424, 175 (2006)

    ADS  MathSciNet  Google Scholar 

  4. D.J. Watts, S.H. Strogatz, Nature 393, 440 (1998)

    ADS  Google Scholar 

  5. A.L. Barabási, R. Albert, Science 286, 509 (1999)

    ADS  MathSciNet  Google Scholar 

  6. R. Albert, A.L. Barabási, Rev. Mod. Phys. 74, 47 (2002)

    ADS  MATH  Google Scholar 

  7. M.E.J. Newman, SIAM Rev. 45, 167 (2003)

    ADS  MATH  MathSciNet  Google Scholar 

  8. J. Maynard Smith, Evolution and the Theory of Games (Cambridge University Press, 1982)

  9. J. Hofbauer, K. Sigmund, Evolutionary Games and Population Dynamics (Cambridge University Press, 1998)

  10. M.A. Nowak, Evolutionary Dynamics (Harvard University Press, 2006)

  11. K. Sigmund, The Calculus of Selfishness (Princeton University Press, 2010)

  12. J.W. Weibull, Evolutionary Game Theory (MIT Press, 1995)

  13. G. Szabó, G. Fáth, Phys. Rep. 446, 97 (2007)

    ADS  MathSciNet  Google Scholar 

  14. S. Schuster, J.U. Kreft, A. Schroeter, T. Pfeiffer, J. Biol. Phys. 34, 1 (2008)

    Google Scholar 

  15. M. Perc, A. Szolnoki, BioSystems 99, 109 (2010)

    Google Scholar 

  16. E. Frey, Physica A 389, 4265 (2010)

    ADS  MATH  MathSciNet  Google Scholar 

  17. C.P. Roca, J.A. Cuesta, A. Sánchez, Phys. Life Rev. 6, 208 (2009)

    ADS  Google Scholar 

  18. F.C. Santos, F. Pinheiro, T. Lenaerts, J.M. Pacheco, J. Theor. Biol. 299, 88 (2012)

    MathSciNet  Google Scholar 

  19. D.A. Rand, M.A. Nowak, Trends Cogn. Sci. 17, 413 (2013)

    Google Scholar 

  20. M. Perc, J. Gómez-Gardeñes, A. Szolnoki, L.M. Floría, Y. Moreno, J. R. Soc. Interface 10, 20120997 (2013)

    Google Scholar 

  21. A. Szolnoki, M. Mobilia, L.L. Jiang, B. Szczesny, A.M. Rucklidge, M. Perc, J. R. Soc. Interface 11, 20140735 (2014)

    Google Scholar 

  22. G. Ichinose, M. Saito, H. Sayama, D.S. Wilson, Sci. Rep. 3, 2509 (2013)

    ADS  Google Scholar 

  23. F. Fu, T. Wu, L. Wang, Phys. Rev. E 79, 036101 (2009)

    ADS  MathSciNet  Google Scholar 

  24. X. Chen, F. Fu, L. Wang, Phys. Rev. E 78, 051120 (2008)

    ADS  Google Scholar 

  25. G. Ichinose, Y. Tenguishi, T. Tanizawa, Phys. Rev. E 88, 052808 (2013)

    ADS  Google Scholar 

  26. Y. Li, X. Lan, X. Deng, R. Sadiq, Y. Deng, Physica A 403, 284 (2014)

    ADS  MathSciNet  Google Scholar 

  27. X. Chen, F. Fu, L. Wang, Physica A 378, 512 (2007)

    ADS  Google Scholar 

  28. C. Xia, S. Meloni, Y. Moreno, Adv. Complex Syst. 15, 1250056 (2012)

    MathSciNet  Google Scholar 

  29. C. Xia, Z. Ma, Y. Wang, J. Wang, Z. Chen, Physica A 390, 4602 (2011)

    ADS  MathSciNet  Google Scholar 

  30. W.X. Wang, J. Lü, G. Chen, P. Hui, Phys. Rev. E 77, 046109 (2008)

    ADS  Google Scholar 

  31. Z.X. Wu, P. Holme, Phys. Rev. E 80, 026108 (2009)

    ADS  Google Scholar 

  32. Z.X. Wu, Z. Rong, P. Holme, Phys. Rev. E 80, 036106 (2009)

    ADS  Google Scholar 

  33. S. Lee, P. Holme, Z.X. Wu, Phys. Rev. Lett. 106, 028702 (2011)

    ADS  Google Scholar 

  34. P. Cui, Z.X. Wu, J. Theor. Biol. 361, 111 (2014)

    MATH  Google Scholar 

  35. Z. Rong, Z.X. Wu, W.X. Wang, Phys. Rev. E 82, 026101 (2010)

    ADS  Google Scholar 

  36. T. Reichenbach, M. Mobilia, E. Frey, Nature 448, 1046 (2007)

    ADS  Google Scholar 

  37. M. Milinski, R.D. Sommerfeld, H.J. Krambeck, F.A. Reed, J. Marotzke, Proc. Natl. Acad. Sci. USA 105, 2291 (2008)

    ADS  Google Scholar 

  38. J.M. Pacheco, V.V. Vasconcelos, F.C. Santos, Phys. Life Rev. 11, 573 (2014)

    ADS  Google Scholar 

  39. C.T. Bauch, D.J. Earn, Proc. Natl. Acad. Sci. USA 101, 13391 (2004)

    ADS  MATH  MathSciNet  Google Scholar 

  40. Z.X. Wu, H.F. Zhang, Europhys. Lett. 104, 10002 (2013)

    ADS  Google Scholar 

  41. J. Tanimoto, T. Fujiki, Z. Wang, A. Hagishima, N. Ikegaya, J. Stat. Mech. 2014, P11027 (2014)

    MathSciNet  Google Scholar 

  42. R. Axelrod, The Evolution of Cooperation (Basic Books, New York, 1984)

  43. M.A. Nowak, R. Highfield, SuperCooperators: Altruism, Evolution, and Why We Need Each Other to Succeed (Free Press, New York, 2011)

  44. E.O. Wilson, The Insect Societies (Harvard University Press, 1971)

  45. R.W. Wang, L. Shi, S.M. Ai, Q. Zheng, J. Animal Ecol. 77, 616 (2008)

    Google Scholar 

  46. A.F. Skutch, Condor 63, 198 (1961)

    Google Scholar 

  47. J. Maynard Smith, E. Szathmáry, The Major Transitions in Evolution (Oxford University Press, 1995)

  48. E. Pennisi, Science 309, 93 (2005)

    Google Scholar 

  49. M.A. Nowak, Science 314, 1560 (2006)

    ADS  Google Scholar 

  50. M.A. Nowak, R.M. May, Nature 359, 826 (1992)

    ADS  Google Scholar 

  51. D.G. Rand, M.A. Nowak, J.H. Fowler, N.A. Christakis, Proc. Natl. Acad. Sci. USA 111, 17093 (2014)

    ADS  Google Scholar 

  52. F.C. Santos, J.M. Pacheco, Phys. Rev. Lett. 95, 098104 (2005)

    ADS  Google Scholar 

  53. F.C. Santos, J.M. Pacheco, T. Lenaerts, Proc. Natl. Acad. Sci. USA 103, 3490 (2006)

    ADS  Google Scholar 

  54. J. Gómez-Gardeñes, M. Campillo, L. Floría, Y. Moreno, Phys. Rev. Lett. 98, 108103 (2007)

    ADS  Google Scholar 

  55. A. Szolnoki, M. Perc, Z. Danku, Physica A 387, 2075 (2008)

    ADS  Google Scholar 

  56. S.V. Buldyrev, R. Parshani, G. Paul, H.E. Stanley, S. Havlin, Nature 464, 1025 (2010)

    ADS  Google Scholar 

  57. J. Gao, D. Li, S. Havlin, Nat. Sci. Rev. 1, 346 (2014)

    Google Scholar 

  58. D. Helbing, Nature 497, 51 (2013)

    ADS  Google Scholar 

  59. M. Kivelä, A. Arenas, M. Barthelemy, J.P. Gleeson, Y. Moreno, M.A. Porter, J. Complex Netw. 2, 203 (2014)

    Google Scholar 

  60. S. Boccaletti, G. Bianconi, R. Criado, C. del Genio, J. Gómez-Gardeñes, M. Romance, I. Sendiña-Nadal, Z. Wang, M. Zanin, Phys. Rep. 544, 1 (2014)

    ADS  MathSciNet  Google Scholar 

  61. G. D’Agostino, A. Scala, Networks of Networks: The Last Frontier of Complexity (Springer, 2014)

  62. S.N. Dorogovtsev, J.F. Mendes, Adv. Phys. 51, 1079 (2002)

    ADS  Google Scholar 

  63. R. Albert, H. Jeong, A.L. Barabási, Nature 401, 130 (1999)

    ADS  Google Scholar 

  64. S. Bornholdt, H.G. Schuster, in Handbook of Graphs and Networks (Wiley Online Library, 2003), Vol. 2

  65. P. Erdős, A. Rényi, Publ. Math. Inst. Hungar. Acad. Sci 5, 17 (1960)

    Google Scholar 

  66. S. Wasserman, in Social Network Analysis: Methods and Applications (Cambridge University Press, 1994), Vol. 8

  67. M. Barthélemy, Phys. Rep. 499, 1 (2011)

    ADS  MathSciNet  Google Scholar 

  68. P. Holme, J. Saramäki, Phys. Rep. 519, 97 (2012)

    ADS  Google Scholar 

  69. M. De Domenico, A. Solé-Ribalta, E. Cozzo, M. Kivelä, Y. Moreno, M.A. Porter, S. Gómez, A. Arenas, Phys. Rev. X 3, 041022 (2013)

    Google Scholar 

  70. B. Min, K.I. Goh, Phys. Rev. E 89, 040802(R) (2014)

    ADS  Google Scholar 

  71. H. Wang, Q. Li, G. D’Agostino, S. Havlin, H.E. Stanley, P. Van Mieghem, Phys. Rev. E 88, 022801 (2013)

    ADS  Google Scholar 

  72. J. Aguirre, R. Sevilla-Escoboza, R. Gutiérrez, D. Papo, J. Buldú, Phys. Rev. Lett. 112, 248701 (2014)

    ADS  Google Scholar 

  73. J.F. Donges, H.C. Schultz, N. Marwan, Y. Zou, J. Kurths, Eur. Phys. J. B 84, 635 (2011)

    ADS  Google Scholar 

  74. A. Saumell-Mendiola, M.Á. Serrano, M. Boguñá, Phys. Rev. E 86, 026106 (2012)

    ADS  Google Scholar 

  75. R. Parshani, S.V. Buldyrev, S. Havlin, Phys. Rev. Lett. 105, 048701 (2010)

    ADS  Google Scholar 

  76. X. Huang, J. Gao, S.V. Buldyrev, S. Havlin, H.E. Stanley, Phys. Rev. E 83, 065101(R) (2011)

    ADS  Google Scholar 

  77. J. Gao, S.V. Buldyrev, H.E. Stanley, S. Havlin, Nat. Phys. 8, 40 (2012)

    Google Scholar 

  78. W. Li, A. Bashan, S.V. Buldyrev, Phys. Rev. Lett. 108, 228702 (2012)

    ADS  Google Scholar 

  79. G. Dong, L. Tian, D. Zhou, R. Du, J. Xiao, H. Stanley, Europhys. Lett. 102, 68004 (2013)

    ADS  Google Scholar 

  80. S. Gómez, A. Díaz-Guilera, J. Gómez-Gardeñes, C. Pérez-Vicente, Y. Moreno, A. Arenas, Phys. Rev. Lett. 110, 028701 (2013)

    ADS  Google Scholar 

  81. M.J. Pocock, D.M. Evans, J. Memmott, Science 335, 973 (2012)

    ADS  Google Scholar 

  82. P. Pattison, S. Wasserman, Br. J. Math. Stat. Psych. 52, 169 (1999)

    Google Scholar 

  83. J. Gao, S.V. Buldyrev, S. Havlin, H.E. Stanley, Phys. Rev. Lett. 107, 195701 (2011)

    ADS  Google Scholar 

  84. S.W. Son, P. Grassberger, M. Paczuski, Phys. Rev. Lett. 107, 195702 (2011)

    ADS  Google Scholar 

  85. D. Zhou, J. Gao, H. Stanley, S. Havlin, Phys. Rev. E. 87, 052812 (2013)

    ADS  Google Scholar 

  86. T.P. Peixoto, S. Bornholdt, Phys. Rev. Lett. 109, 118703 (2012)

    ADS  Google Scholar 

  87. G. Bianconi, S.N. Dorogovtsev, Phys. Rev. E 89, 062814 (2014)

    ADS  Google Scholar 

  88. S. Gomez, A. Diaz-Guilera, J. Gomez-Gardeñes, C.J. Perez-Vicente, Y. Moreno, A. Arenas, Phys. Rev. Lett. 110, 028701 (2013)

    ADS  Google Scholar 

  89. Y. Wang, G. Xiao, J. Liu, New J. Phys. 14, 013015 (2012)

    ADS  Google Scholar 

  90. D. Zhao, L. Wang, S. Li, Z. Wang, L. Wang, B. Gao, PLoS ONE 9, e112018 (2014)

    ADS  Google Scholar 

  91. C. Buono, L.G. Alvarez-Zuzek, P.A. Macri, L.A. Braunstein, PLoS ONE 9, e92200 (2014)

    ADS  Google Scholar 

  92. Z. Wang, A. Szolnoki, M. Perc, Europhys. Lett. 97, 48001 (2012)

    ADS  Google Scholar 

  93. M. Santos, S. Dorogovtsev, J. Mendes, Sci. Rep. 4, 4436 (2014)

    ADS  Google Scholar 

  94. J. Gómez-Gardeñes, I. Reinares, A. Arenas, L.M. Floría, Sci. Rep. 2, 620 (2012)

    Google Scholar 

  95. Q. Jin, L. Wang, C. Xia, Z. Wang, Sci. Rep. 4, 4095 (2012)

    Google Scholar 

  96. B. Wang, X. Chen, L. Wang, J. Stat. Mech. 2012, P11017 (2012)

    Google Scholar 

  97. C. Tang, Z. Wang, X. Li, PLoS ONE 9, e88412 (2014)

    ADS  Google Scholar 

  98. M. Diakonova, M. San Miguel, V.M. Eguiluz, Phys. Rev. E 89, 062818 (2014)

    ADS  Google Scholar 

  99. F. Tan, J. Wu, Y. Xia, K.T. Chi, Phys. Rev. E 89, 062813 (2014)

    ADS  Google Scholar 

  100. B. Bollobás, Modern Graph Theory (Springer, 1998)

  101. G. Menichetti, D. Remondini, P. Panzarasa, R.J. Mondragón, G. Bianconi, PLoS ONE 9, e97857 (2014)

    Google Scholar 

  102. S. Shai, S. Dobson, Phys. Rev. E 87, 042812 (2013)

    ADS  Google Scholar 

  103. F. Battiston, V. Nicosia, V. Latora, Phys. Rev. E 89, 032804 (2014)

    ADS  Google Scholar 

  104. G. Bianconi, Phys. Rev. E 87, 062806 (2013)

    ADS  Google Scholar 

  105. L.D.F. Costa, F.A. Rodrigues, G. Travieso, P.R. Villas Boas, Adv. Phys. 56, 167 (2007)

    ADS  Google Scholar 

  106. V. Nicosia, G. Bianconi, V. Latora, M. Barthelemy, Phys. Rev. Lett. 111, 058701 (2013)

    ADS  Google Scholar 

  107. V. Latora, M. Marchiori, Eur. Phys. J. B 32, 249 (2003)

    ADS  Google Scholar 

  108. E. Cozzo, M. Kivelä, M. De Domenico, A. Solé, A. Arenas, S. Gómez, M.A. Porter, Y. Moreno, arXiv:1307.6780 (2013)

  109. F. Battiston, V. Nicosia, V. Latora, Phys. Rev. E 89, 032804 (2013)

    ADS  Google Scholar 

  110. R. Criado, J. Flores, A. García del Amo, J. Gómez-Gardeñes, M. Romance, Int. J. Comput. Math. 89, 291 (2012)

    MATH  MathSciNet  Google Scholar 

  111. M.E.J. Newman, Phys. Rev. Lett. 89, 208701 (2002)

    ADS  Google Scholar 

  112. R. Xulvi-Brunet, I. Sokolov, Phys. Rev. E 70, 066102 (2004)

    ADS  Google Scholar 

  113. B. Min, S. Do Yi, K.M. Lee, K.I. Goh, Phys. Rev. E 89, 042811 (2014)

    ADS  Google Scholar 

  114. S. Watanabe, Y. Kabashima, Phys. Rev. E 89, 012808 (2014)

    ADS  Google Scholar 

  115. S. Melnik, M.A. Porter, P.J. Mucha, J.P. Gleeson, Chaos 24, 023106 (2014)

    ADS  Google Scholar 

  116. M. De Domenico, A. Solé-Ribalta, E. Omodei, S. Gómez, A. Arenas, Nat. Commun. 6, 6868 (2015)

    ADS  Google Scholar 

  117. R. Parshani, C. Rozenblat, D. Ietri, C. Ducruet, S. Havlin, Europhys. Lett. 92, 68002 (2010)

    ADS  Google Scholar 

  118. G. Baxter, S. Dorogovtsev, A. Goltsev, J. Mendes, Phys. Rev. Lett. 109, 248701 (2012)

    ADS  Google Scholar 

  119. L. Solá, M. Romance, R. Criado, J. Flores, A.J.G. del Amo, S. Boccaletti, Chaos 23, 033131 (2013)

    ADS  Google Scholar 

  120. V. Nicosia, V. Latora, arXiv:1403.1546 (2014)

  121. N. Masuda, P. Holme, F1000Prime Rep. 5, 6 (2013)

    Google Scholar 

  122. N. Perra, B. Gonçalves, R. Pastor-Satorras, A. Vespignani, Sci. Rep. 2, 469 (2012)

    ADS  Google Scholar 

  123. M. Berlingerio, M. Coscia, F. Giannotti, A. Monreale, D. Pedreschi, World Wide Web 16, 567 (2013)

    Google Scholar 

  124. D. Krackhardt, Social Netw. 9, 109 (1987)

    MathSciNet  Google Scholar 

  125. X. Li, M.K. Ng, Y. Ye, in Proceedings of the 12th SIAM International Conference on Data Mining, SDM’12 (SIAM, 2012), pp. 141–152

  126. A. Cardillo, M. Zanin, J. Gómez-Gardeñes, M. Romance, A.J.G. del Amo, S. Boccaletti, Eur. Phys. J. Special Topics 215, 23 (2013)

    ADS  Google Scholar 

  127. A. Halu, R.J. Mondragon, P. Panzarasa, G. Bianconi, PLoS ONE 8, e78293 (2013)

    ADS  Google Scholar 

  128. W. Li, C.C. Liu, T. Zhang, H. Li, M.S. Waterman, X.J. Zhou, PLoS Comput. Biol. 7, e1001106 (2011)

    ADS  MathSciNet  Google Scholar 

  129. M.P. Rombach, M.A. Porter, J.H. Fowler, P.J. Mucha, SIAM J. Appl. Math. 74, 167 (2014)

    MATH  MathSciNet  Google Scholar 

  130. X. Huang, I. Vodenska, S. Havlin, H.E. Stanley, Sci. Rep. 3, 1219 (2013)

    ADS  Google Scholar 

  131. J. Yang, J. Leskovec, Proc. IEEE 12, 1892 (2014)

    Google Scholar 

  132. Z. Wang, L. Wang, M. Perc, Phys. Rev. E 89, 052813 (2014)

    ADS  Google Scholar 

  133. J. Gómez-Gardeñes, I. Reinares, A. Arenas, L.M. Floría, Sci. Rep. 2, 620 (2012)

    Google Scholar 

  134. Z. Wang, A. Szolnoki, M. Perc, Sci. Rep. 3, 2470 (2013)

    ADS  Google Scholar 

  135. Z. Wang, A. Szolnoki, M. Perc, Sci. Rep. 3, 1183 (2013)

    ADS  Google Scholar 

  136. A. Szolnoki, M. Perc, New J. Phys. 15, 053010 (2013)

    ADS  MathSciNet  Google Scholar 

  137. Z. Wang, A. Szolnoki, M. Perc, New J. Phys. 16, 033041 (2014)

    ADS  Google Scholar 

  138. Z. Wang, A. Szolnoki, M. Perc, J. Theor. Biol. 349, 50 (2014)

    MathSciNet  Google Scholar 

  139. J. Gómez-Gardeñes, C. Gracia-Lázaro, L.M. Floría, Y. Moreno, Phys. Rev. E 86, 056113 (2012)

    ADS  Google Scholar 

  140. L.L. Jiang, M. Perc, Sci. Rep. 3, 2483 (2013)

    ADS  Google Scholar 

  141. S. Fortunato, Phys. Rep. 486, 75 (2010)

    ADS  MathSciNet  Google Scholar 

  142. K.M. Lee, J.Y. Kim, W.K. Cho, K. Goh, I. Kim, New J. Phys. 14, 033027 (2012)

    ADS  Google Scholar 

  143. S. Funk, V.A. Jansen, Phys. Rev. E 81, 036118 (2010)

    ADS  Google Scholar 

  144. A. Cardillo, J. Gómez-Gardeñes, M. Zanin, M. Romance, D. Papo, F. del Pozo, S. Boccaletti, Sci. Rep. 3, 1344 (2013)

    ADS  Google Scholar 

  145. B. Min, S.D. Yi, K.M. Lee, K.I. Goh, Phys. Rev. E 89, 042811 (2014)

    ADS  Google Scholar 

  146. V. Marceau, P.A. Noël, L. Hébert-Dufresne, A. Allard, L.J. Dubé, Phys. Rev. E 84, 026105 (2011)

    ADS  Google Scholar 

  147. J.Y. Kim, K.I. Goh, Phys. Rev. Lett. 111, 058702 (2013)

    ADS  Google Scholar 

  148. V. Nicosia, G. Bianconi, V. Latora, M. Barthelemy, Phys. Rev. E 90, 042807 (2010)

    ADS  Google Scholar 

  149. B. Fotouhi, N. Momeni, arXiv:1411.0344 (2014)

  150. P. Wang, G. Robins, P. Pattison, E. Lazega, Social Netw. 35, 96 (2013)

    Google Scholar 

  151. M. Santos, S.N. Dorogovtsev, J.F.F. Mendes, Sci. Rep. 4, 4436 (2014)

    ADS  Google Scholar 

  152. A. Szolnoki, G. Szabó, Europhys. Lett. 77, 30004 (2007)

    ADS  Google Scholar 

  153. M. Perc, A. Szolnoki, Phys. Rev. E 77, 011904 (2008)

    ADS  Google Scholar 

  154. A. Szolnoki, M. Perc, New J. Phys. 10, 043036 (2008)

    ADS  Google Scholar 

  155. Z. Wang, A. Szolnoki, M. Perc, Phys. Rev. E 85, 037101 (2012)

    ADS  Google Scholar 

  156. Z. Wang, A. Szolnoki, M. Perc, Sci. Rep. 2, 369 (2012)

    ADS  Google Scholar 

  157. H. Lugo, M. San Miguel, Sci. Rep. 5, 7776 (2015)

    ADS  Google Scholar 

  158. A. Szolnoki, Z. Wang, M. Perc, Sci. Rep. 2, 576 (2012)

    ADS  Google Scholar 

  159. A. Szolnoki, M. Perc, J. R. Soc. Interface 12, 20141299 (2015)

    Google Scholar 

  160. J. Gómez-Gardeñes, M. Romance, R. Criado, D. Vilone, A. Sánchez, Chaos 21, 016113 (2011)

    ADS  MathSciNet  Google Scholar 

  161. J. Gómez-Gardeñes, D. Vilone, A. Sánchez, Europhys. Lett. 95, 68003 (2011)

    ADS  Google Scholar 

  162. E.T. Lofgren et al., Proc. Natl. Acad. Sci. USA 111, 18095 (2014)

    ADS  Google Scholar 

  163. M.R. Servedio, Y. Brandvain, S. Dhole, C.L. Fitzpatrick, E.E. Goldberg, C.A. Stern, J. Van Cleve, D.J. Yeh, PLoS Biol. 12, e1002017 (2014)

    Google Scholar 

  164. W. Güth, R. Schmittberger, B. Schwarze, J. Econ. Behav. Org. 3, 367 (1982)

    Google Scholar 

  165. K.M. Page, M.A. Nowak, K. Sigmund, Proc. R. Soc. B 267, 2177 (2000)

    Google Scholar 

  166. M.N. Kuperman, S. Risau-Gusman, Eur. Phys. J. B 62, 233 (2008)

    ADS  MATH  MathSciNet  Google Scholar 

  167. J. Gao, Z. Li, T. Wu, L. Wang, Europhys. Lett. 93, 48003 (2011)

    ADS  Google Scholar 

  168. A. Szolnoki, M. Perc, G. Szabó, Phys. Rev. Lett. 109, 078701 (2012)

    ADS  Google Scholar 

  169. T. Wu, F. Fu, Y. Zhang, L. Wang, Sci. Rep. 3, 1550 (2013)

    ADS  Google Scholar 

  170. A. Baronchelli, L. Dall’Asta, A. Barrat, V. Loreto, Phys. Rev. E 73, 015102 (2006)

    ADS  MathSciNet  Google Scholar 

  171. F.C. Santos, J.M. Pacheco, Proc. Natl. Acad. Sci. USA 108, 10421 (2011)

    ADS  Google Scholar 

  172. X. Chen, A. Szolnoki, M. Perc, Europhys. Lett. 99, 68003 (2012)

    ADS  Google Scholar 

  173. V.V. Vasconcelos, F.C. Santos, J.M. Pacheco, Nat. Climate Change 3, 797 (2013)

    ADS  Google Scholar 

  174. A. Szolnoki, Phys. Life Rev. 11, 589 (2014)

    ADS  Google Scholar 

  175. C.T. Bauch, A.P. Galvani, Science 342, 47 (2013)

    ADS  Google Scholar 

  176. C.T. Bauch, D.J.D. Earn, Proc. Natl. Acad. Sci. USA 101, 13391 (2004)

    ADS  MATH  MathSciNet  Google Scholar 

  177. A. d’Onofrio, P. Manfredi, J. Theor. Biol. 264, 237 (2010)

    MathSciNet  Google Scholar 

  178. F. Fu, D.I. Rosenbloom, L. Wang, M.A. Nowak, Proc. R. Soc. B 278, 42 (2011)

    Google Scholar 

  179. C.G. Gu, S.R. Zou, X.L. Xu, Y.Q. Qu, Y.M. Jiang, D.R. He, H.K. Liu, T. Zhou, Phys. Rev. E 84, 026101 (2011)

    ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Computer and Information Science, Southwest University, Chongqing, 400715, P.R. China

    Zhen Wang

  2. Department of Physics, Hong Kong Baptist University, Kowloon Tong, Hong Kong, P.R. China

    Zhen Wang

  3. Centre for Chaos and Complex Networks, City University of Hong Kong, Kowloon Tong, Hong Kong, P.R. China

    Lin Wang

  4. Institute of Technical Physics and Materials Science, Centre for Energy Research, Hungarian Academy of Sciences, P.O. Box 49, 1525, Budapest, Hungary

    Attila Szolnoki

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

    Matjaž Perc

  6. Department of Physics, Faculty of Sciences, King Abdulaziz University, Jeddah, Saudi Arabia

    Matjaž Perc

  7. CAMTP — Center for Applied Mathematics and Theoretical Physics, University of Maribor, Krekova 2, 2000, Maribor, Slovenia

    Matjaž Perc

Authors
  1. Zhen Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Lin Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Attila Szolnoki
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Matjaž Perc
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Matjaž Perc.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, Z., Wang, L., Szolnoki, A. et al. Evolutionary games on multilayer networks: a colloquium. Eur. Phys. J. B 88, 124 (2015). https://doi.org/10.1140/epjb/e2015-60270-7

Download citation

  • Received:

  • Revised:

  • Published:

  • DOI: https://doi.org/10.1140/epjb/e2015-60270-7

Keywords

Search

Navigation