# Network Reciprocity

## Abstract

In the previous chapter, we discussed Nowak’s five fundamental reciprocity mechanisms for adding social viscosity: kin selection, direct reciprocity, indirect reciprocity, network reciprocity, and group selection. In this chapter, we focus specifically on network reciprocity, as this mechanism has received the most attention in communities of statistical physicists and theoretical biologists who specialize in evolutionary game theory. Since 1992, when the first study of the spatial prisoner’s dilemma (SPD) was conducted by Nowak and May (1992), the number of papers dealing with network reciprocity has increased to several thousand. The main reason for this is that network reciprocity is regarded as the most important and interesting of the mechanisms from an application point of view. In fact, we can observe a lot of evidence in real life of network reciprocity working to establish mutual cooperation not only in human social systems but also in those of other animal species. The network reciprocity mechanism relies on two effects. The first is limiting the number of game opponents (that is, “depressing anonymity,” rather than having an infinite and well-mixed population), and the second is a local adaptation mechanism, in which an agent copies a strategy from a neighbor linked by a network. These two effects explain how cooperators survive in a social dilemma system, even though it requires agents to use only the simplest strategy—either cooperation (C) or defection (D), and this has attracted biologists who guess that network reciprocity may explain the evolution of cooperation even among primitive organisms without any sophisticated intelligence.

## Keywords

Strategy Game Cooperation Level Focal Agent Discrete Game Roulette Selection## References

- Barabasi, A.L., and R. Albert. 1999. Emergence of scaling in random networks.
*Science*286: 509–512.MathSciNetCrossRefPubMedADSGoogle Scholar - Berde, M. 2011. Playing against the fittest: A simple strategy that promotes the emergence of cooperation.
*EPL*94: 30003.CrossRefADSGoogle Scholar - Bollobás, B. 1985.
*Random graphs*. London: Academic.zbMATHGoogle Scholar - Brede, M. 2011. Playing against the fittest A simple strategy that promotes the emergence of cooperation.
*EPL*94: 30003.CrossRefADSGoogle Scholar - Chen, X., F. Fu, and L. Wang. 2009. Social tolerance allows cooperation to prevail in an adaptive environment.
*Physical Review E*80: 051104.CrossRefADSGoogle Scholar - Cong, R., Y–.Y. Qiu, X–.J. Chen, and L. Wang. 2010. Robustness of cooperation on highly clustered scale-free networks.
*Chinese Physical Letters*27(3): 030203.CrossRefADSGoogle Scholar - Dai, Q., H. Li, H. Cheng, Y. Li, and J. Yang. 2010. Double-dealing behavior potentially promotes cooperation in evolutionary prisoner’s dilemma games.
*New Journal of Physics*12: 113015.CrossRefADSGoogle Scholar - Day, T., and P.D. Taylor. 2003. Evolutionary dynamics and stability in discrete and continuous games.
*Evolutionary Ecology Research*5: 605–613.Google Scholar - Fu, F., T. Wu, and L. Wang. 2009. Partner switching stabilizes cooperation in coevolutionary prisoner’s dilemma.
*Physical Review E*79: 036101.MathSciNetCrossRefADSGoogle Scholar - Fu, F., M.A. Nowak, and C. Hauert. 2010. Invasion and expansion of cooperators in lattice populations: Prisoner’s dilemma vs. snowdrift games.
*Journal of Theoretical Biology*266: 358–386.PubMedCentralMathSciNetCrossRefPubMedGoogle Scholar - Gomez-Gardenes, J., M. Campillo, L.M. Floria, and T. Moreno. 2007. Dynamical organization of cooperation in complex topologies.
*Physical Review Letters*98: 108103.CrossRefPubMedADSGoogle Scholar - Grilo, C., and L. Correia. 2007. What makes spatial prisoner’s dilemma game sensitive to asynchronism?.
*Proceedings of the 11th international conference on the simulation and synthesis of living systems.*Google Scholar - Hamilton, W.D. 1964. The genetical evolution of social behavior. 1.
*Journal of Theoretical Biology*7: 1–16.CrossRefPubMedGoogle Scholar - Kirchkamp, O. 1999. Simultaneous evolution of learning rules and strategies.
*Journal of Economic Behavior & Organization*40: 295–312.CrossRefGoogle Scholar - Li, W., X. Zhang, and G. Hu. 2007. How scale-free networks and large-scale collective cooperation emerge in complex homogeneous social systems.
*Physical Review E*76: 045102.CrossRefADSGoogle Scholar - Moyano, L.G., and A. Sanchez. 2009. Evolving learning rules and emergence of cooperation in spatial prisoner’s dilemma.
*Journal of Theoretical Biology*259: 84–95.MathSciNetCrossRefPubMedGoogle Scholar - Newman, M.E.J. 2002. Assortative mixing in networks.
*Physical Review Letters*89: 208701.CrossRefPubMedADSGoogle Scholar - Nowak, M.A. 2006. Five rules for the evolution of cooperation.
*Science*314: 1560–1563.PubMedCentralCrossRefPubMedADSGoogle Scholar - Nowak, M.A., and R.M. May. 1992. Evolutionary games and spatial chaos.
*Nature*359: 826–829.CrossRefADSGoogle Scholar - Ogasawara, T., Tanimoto, J., Fukuda, E., and N. Ikegaya. 2014. Effect of a large gaming neighborhood and a strategy adaptation neighborhood for bolstering network reciprocity in a prisoner’s dilemma game.
*Journal of Statistical Mechanics: Theory and Experiment*2014: P12024.Google Scholar - Ohtsuki, H., C. Hauert, E. Lieberman, and M.A. Nowak. 2006. A simple rule for the evolution of cooperation on graphs and social networks.
*Nature*441: 502–505.PubMedCentralCrossRefPubMedADSGoogle Scholar - Pacheco, J.M., A. Traulsen, and M.A. Nowak. 2006. Coevolution of strategy and structure in complex networks with dynamical linking.
*Physical Review Letters*97: 258103.PubMedCentralCrossRefPubMedADSGoogle Scholar - Pan, Q., S. Shi, Y. Zhang, and M. He. 2013. Cooperation in spatial prisoner’s dilemma game with delayed decisions.
*Chaos, Solitons & Fractals*56: 166–174.CrossRefADSGoogle Scholar - Perc, M. 2006a. Coherence resonance in a spatial prisoner’s dilemma game.
*New Journal of Physics*8: 22.CrossRefADSGoogle Scholar - Perc, M. 2006b. Chaos promotes cooperation in the spatial prisoner’s dilemma game.
*Europhysics Letters*75(6): 841–846.MathSciNetCrossRefADSGoogle Scholar - Perc, M. 2007. Transition from Gaussian to Levy distributions of stochastic payoff variations in the spatial prisoner’s dilemma game.
*Physical Review E*75: 022101.CrossRefADSGoogle Scholar - Perc, M., and M. Marhl. 2006. Evolutionary and dynamical coherence resonances in the pair approximated prisoner’s dilemma game.
*New Journal of Physics*8: 101016.Google Scholar - Perc, M., and A. Szolnoki. 2010. Coevolutionary games – A mini review.
*Biosystems*99: 109–125.CrossRefPubMedGoogle Scholar - Perc, M., and Z. Wang. 2010. Heterogeneous aspiration promotes cooperation in the prisoner’s dilemma game.
*PLoS ONE*5(12): e15117.PubMedCentralCrossRefPubMedADSGoogle Scholar - Pestelacci, E., M. Tomassini, and L. Luthi. 2008. Evolution of cooperation and coordination in a dynamically networked society.
*Biological Theory*3: 139–153.CrossRefGoogle Scholar - Poncela, J., J. Gomez-Gardenes, L.M. Flora, and Y. Moreno. 2007. Robustness of cooperation in the evolutionary prisoner’s dilemma on complex networks.
*New Journal of Physics*9: 101088.CrossRefGoogle Scholar - Poncela, J., J. Gomez-Gardenes, L.M. Floria, A. Sanchez, and Y. Moreno. 2008. Complex cooperative networks from evolutionary preferential attachment.
*PLoS ONE*3: e2449.PubMedCentralCrossRefPubMedADSGoogle Scholar - Ren, G., and X. Wang. 2014. Robustness of cooperation in memory-based prisoner’s dilemma game on a square lattice.
*Physica A*408: 40–46.MathSciNetCrossRefADSGoogle Scholar - Roca, C.P., J.A. Cuesta, and A. Sanchez. 2006. Time scales in evolutionary dynamics.
*Physical Review Letters*97: 158701.CrossRefPubMedADSGoogle Scholar - Roca, C.P., J.A. Cuesta, and A. S’anchez. 2009. Effect of spatial structure on the evolution of cooperation.
*Physical Review E*80: 046106.CrossRefADSGoogle Scholar - Rong, Z., X. Li, and X. Wang. 2007. Roles of mixing patterns in cooperation on a scale-free networked game.
*Physical Review E*76: 027101.CrossRefADSGoogle Scholar - Santos, F.C., J.M. Pacheco, and T. Lenaerts. 2006a. Evolutionary dynamics of social dilemmas in structured heterogeneous populations.
*Proceedings of the National Academy of Science of the United States of America*103(9): 3490–3494.CrossRefADSGoogle Scholar - Santos, F.C., J.M. Pacheco, and T. Lenaerts. 2006b. Cooperation prevails when individuals adjust their social ties.
*PLoS Computational Biology*2(10): 1284–1291.CrossRefGoogle Scholar - Schuessler, R. 1989. Exit threats and cooperation under anonymity.
*Journal of Conflict Resolution*33: 728–749.CrossRefGoogle Scholar - Shigaki, K., J. Tanimoto, Z. Wang, and E. Fukuda. 2013. Effect of initial fraction of cooperators on cooperative behavior in evolutionary prisoner’s dilemma.
*PLoS ONE*8(11): e76942.PubMedCentralCrossRefPubMedADSGoogle Scholar - Szabo, G., and G. Fath. 2007. Evolutionary games on graphs.
*Physics Reports*446: 97–216.MathSciNetCrossRefADSGoogle Scholar - Szolnoki, A., and M. Perc. 2009a. Resolving social dilemmas on evolving random networks.
*EPL*86: 30007.CrossRefADSGoogle Scholar - Szolnoki, A., and M. Perc. 2009b. Emergence of multilevel selection in the prisoner’s dilemma game on coevolving random networks.
*New Journal of Physics*11: 093033.CrossRefADSGoogle Scholar - Szolnoki, A., M. Perc, and M. Mobilia. 2014. Facilitators on networks reveal optimal interplay between information exchange and reciprocity.
*Physical Review E*89: 042802.CrossRefADSGoogle Scholar - Tang, C.-L., W.-X. Wang, X. We, and B.-H. Wang. 2006. Effects of average degree on cooperation in networked evolutionary game.
*European Physical Journal B*53: 411–415.zbMATHCrossRefADSGoogle Scholar - Tanimoto, J. 2007a. Dilemma-solving effects by the coevolution of both networks and strategy in a 2 × 2 game.
*Physical Review E*76: 021126.CrossRefADSGoogle Scholar - Tanimoto, J. 2007b. Promotion of cooperation by payoff noise in a 2 × 2 game.
*Physical Review E*76: 041130.CrossRefADSGoogle Scholar - Tanimoto, J. 2009. Promotion of cooperation through co-evolution of networks and strategy in a 2 × 2 game.
*Physica A*388(6): 953–960.CrossRefADSGoogle Scholar - Tanimoto, J. 2010. Effect of assortativity by degree on emerging cooperation in a 2 × 2 dilemma game played on an evolutionary network.
*Physica A*389: 3325–3335.CrossRefADSGoogle Scholar - Tanimoto, J. 2011. A study of a quadruple co-evolutionary model and its reciprocity phase for various Prisoner’s Dilemma games.
*International Journal of Modern Physics C*22(4): 401–407.CrossRefADSGoogle Scholar - Tanimoto, J. 2014. Simultaneously selecting appropriate partners for gaming and strategy adaptation to enhance network reciprocity in the prisoner’s dilemma.
*Physical Review E*89: 012106.CrossRefADSGoogle Scholar - Tanimoto, J., M. Nakata, A. Hagishima, and N. Ikegaya. 2011. Spatially correlated heterogeneous aspirations to enhance network reciprocity.
*Physica A*391(3): 680–685.CrossRefADSGoogle Scholar - Tomassini, M., L. Luthi, and E. Pestelacci. 2007. Social dilemmas and cooperation in complex networks.
*International Journal of Modern Physics C*18(07): 1173–1185.zbMATHCrossRefADSGoogle Scholar - Tomochi, M. 2004. Defectors’ niches: Prisoner’s dilemma game on disordered networks.
*Social Networks*26(4): 309–321.CrossRefGoogle Scholar - Vainstein, M.H., and J.J. Arenzon. 2001. Disordered environments in spatial games.
*Physical Review E*64: 051905.CrossRefADSGoogle Scholar - Van Segbroeck, S., F.C. Santos, T. Lenaerts, and J.M. Pacheco. 2009. Reacting differently to adverse ties promotes cooperation in social networks.
*Physical Review Letters*102: 058105.CrossRefPubMedADSGoogle Scholar - Vincent, T.L., and R. Cressman. 2000. An ESS maximum principle for matrix games.
*Theoretical Population Biology*58: 173–186.zbMATHCrossRefPubMedGoogle Scholar - Vukov, J., G. Szabo, and A. Szolnoki. 2006. Cooperation in noisy case: Prisoner’s dilemma game on two types of regular random graphs.
*Physical Review E*73: 067103.CrossRefADSGoogle Scholar - Wang, Z., and M. Perc. 2010. Aspiring to the fittest and promoted of cooperation in the prisoner’s dilemma game.
*Physical Review E*82: 021115.CrossRefADSGoogle Scholar - Watts, D.J., and S.H. Strogatz. 1998. Collective dynamics of ‘small-world’ networks.
*Nature*393: 440–442.CrossRefPubMedADSGoogle Scholar - Xia, C., Q. Miao, and J. Zhang. 2013. Impact of neighborhood separation on the spatial reciprocity in the prisoner’s dilemma game.
*Chaos, Solitons & Fractals*51: 22–30.zbMATHMathSciNetCrossRefADSGoogle Scholar - Xulvi-Brunet, R., and I.M. Sokolov. 2004. Reshuffling scale-free networks: From random to assortative.
*Physical Review E*70: 066102.CrossRefADSGoogle Scholar - Yamauchi, A., J. Tanimoto, and A. Hagishima. 2010. What controls network reciprocity in the prisoner’s dilemma game?
*Biosystems*102(2–3): 82–87.CrossRefPubMedGoogle Scholar - Yamauchi, A., J. Tanimoto, and A. Hagishima. 2011. An analysis of network reciprocity in prisoner’s dilemma games using full factorial designs of experiment.
*Biosystems*103: 85–92.CrossRefPubMedGoogle Scholar - Zhong, W., S. Kokubo, and J. Tanimoto. 2012. How is the equilibrium of continuous strategy game different from that of discrete strategy game?
*Biosystems*107(2): 89–94.CrossRefGoogle Scholar - Zimmermann, M., and V. Eguiluz. 2005. Cooperation, social networks, and the emergence of leadership in a prisoner’s dilemma with adaptive local interactions.
*Physical Review E*72: 056118.MathSciNetCrossRefADSGoogle Scholar