Abstract
Whether or not cooperation is favored in evolutionary games on graphs depends on the population structure and spatial properties of the interaction network. The population structure can be expressed as configurations. Such configurations extend scenarios with a single cooperator among defectors to any number of cooperators and any arrangement of cooperators and defectors on the network. For interaction networks modeled as regular graphs and for weak selection, the emergence of cooperation can be assessed by structure coefficients, which can be specified for each configuration and each regular graph. Thus, as a single cooperator can be interpreted as a lone mutant, the configuration-based structure coefficients also describe fixation properties of multiple mutants. We analyze the structure coefficients and particularly show that under certain conditions, the coefficients strongly correlate to the average shortest path length between cooperators on the evolutionary graph. Thus, for multiple cooperators fixation properties on regular evolutionary graphs can be linked to cooperator path lengths.
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References
Allen B, Nowak MA (2014) Games on graphs. EMS Surv Math Sci 1:113–151
Allen B, Lippner G, Chen YT, Fotouhi B, Momeni N, Yau ST, Nowak MA (2017) Evolutionary dynamics on any population structure. Nature 544:227–230
Axelrod R (1980) Effective choice in the prisoners dilemma. J Confl Resol 24:325
Bayati M, Kim JH, Saberi A (2010) A sequential algorithm for generating random graphs. Algorithmica 58:860–910
Blitzstein J, Diaconis P (2011) A sequential importance sampling algorithm for generating random graphs with prescribed degrees. Internet Math 6:489–522
Bondy A, Murty MR (2008) Graph theory. Springer, London
Broom M, Rychtar J (2013) Game-theoretical models in biology. Chapman and Hall/CRC, Boca Raton
Broom M, Rychtar J, Stadler BT (2011) Evolutionary dynamics on graphs - the effect of graph structure and initial placement on mutant spread. J Stat Theory Pract 5:369–381
Brouwer AE, Haemers WH (2012) Spectra of graphs. Springer, New York
Chen YT (2013) Sharp benefit-to-cost rules for the evolution of cooperation on regular graphs. Ann Appl Probab 3:637–664
Chen YT, McAvoy A, Nowak MA (2016) Fixation probabilities for any configuration of two strategies on regular graphs. Sci Rep 6:39181
Franklin P (1934) A six color problem. J Math Phys 13:363–379
Freedman D, Diaconis P (1981) On the histogram as a density estimator: \(L_ 2\) Theory. Probab Theory Relat Fields 57:453–476
Frucht R (1949) Graphs of degree three with a given abstract group. Can J Math 1:365–378
Fu F, Wang L, Nowak MA, Hauert C (2009) Evolutionary dynamics on graphs: efficient method for weak selection. Phys Rev E 79:046707
Hauert C (2001) Fundamental clusters in spatial \(2 \times 2\) games. Proc R Soc B268:761–769
Hauert C, Doebeli M (2004) Spatial structure often inhibits the evolution of cooperation in the snowdrift game. Nature 428:643–646
Hindersin L, Traulsen A (2014) Counterintuitive properties of the fixation time in network-structured populations. J R Soc Interface 11:20140606
Jerebic J, Klavžar S, Rall DF (2008) Distance-balanced graphs. Ann Combin 12:71–79
Kutnar K, Malnič A, Marušič D, Miklavič Š (2006) Distance-balanced graphs: symmetry conditions. Discrete Math 306:1881–1894
Kutnar K, Malnič A, Marušič D, Miklavič Š (2009) The strongly distance-balanced property of the generalized Petersen graphs. Ars Math Contemp 2:41–47
Langer P, Nowak MA, Hauert C (2008) Spatial invasion of cooperation. J Theor Biol 250:634–641
Lehmann L, Keller DJ, Sumpter DJT (2007) The evolution of helping and harming on graphs: the return of the inclusive fitness effect. J Evol Biol 20:2284–2295
McAvoy A, Hauert C (2015) Structural symmetry in evolutionary games. J R Soc Interface 12:20150420
McAvoy A, Hauert C (2016) Structure coefficients and strategy selection in multiplayer games. J Math Biol 72:203–238
Mullon C, Lehmann L (2014) The robustness of the weak selection approximation for the evolution of altruism against strong selection. J Evol Biol 27:2272–2282
Nowak MA (2006) Evolutionary dynamics: exploring the equations of life. Harvard University Press, Cambridge
Nowak MA, Sigmund K (1993) A strategy of win-stay, lose-shift that outperforms tit-for-tat in the Prisoner’s Dilemma game. Nature 364:56–58
Nowak MA, Tarnita CE, Antal T (2010) Evolutionary dynamics in structured populations. Philos Trans R Soc B365:19–30
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
Ohtsuki H, Nowak MA (2006) Evolutionary games on cycles. Proc R Soc B373:2249–2256
Page KM, Nowak MA, Sigmund K (2000) The spatial ultimatum game. Proc Roy Soc B267:2177–2182
Pattni K, Broom M, Silvers L, Rychtar J (2015) Evolutionary graph theory revisited: when is an evolutionary process equivalent to the Moran process? Proc R Soc A 471:20150334
Paley C, Taraskin S, Elliott S (2010) The two-mutant problem: clonal interference in evolutionary graph theory. J Chem Biol 3:189–194
Read RC, Wilson RJ (1998) An atlas of graphs. Oxford University Press, Oxford
Richter H (2017) Dynamic landscape models of coevolutionary games. BioSystems 153–154:26–44
Richter H (2018) Properties of interaction networks, structure coefficients, and benefit–to–cost ratios. arXiv:1805.11359 [q-bio. PE]
Scott DW (1992) Multivariate density estimation: theory, practice, and visualization. Wiley, New York
Shakarian P, Roos P, Johnson A (2012) A review of evolutionary graph theory with applications to game theory. BioSystems 107:66–80
Tarnita CE, Ohtsuki H, Antal T, Fu F, Nowak MA (2009) Strategy selection in structured populations. J Theor Biol 259:570–581
Taylor PD, Day T, Wild G (2007) Evolution of cooperation in a finite homogeneous graph. Nature 447:469–472
Van Slijpe ARD (1986) Random walks on the triangular prism and other vertex-transitive graphs. J Comput Appl Math 15:383–394
Wormald NC (1999) Models of random regular graphs. In: Lamb JD, Preece DA (eds) Surveys in combinatorics, London mathematical society lecture note series, vol 267. Cambridge University Press, Cambridge, pp 239–298
Wu B, Garcia J, Hauert C, Traulsen A (2013) Extrapolating weak selection in evolutionary games. PLoS Comput Biol 9(12):e1003381. https://doi.org/10.1371/journal.pcbi.1003381
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Appendix 1
Appendix 1
A more detailed description of the relationships between configurations \(\pi\), structure coefficients \(\sigma (\pi )\) and cooperator path lengths \(l_c\) is presented for selected graphs.
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The Frucht graph (Chen et al. 2016; Frucht 1949; McAvoy and Hauert 2015), which has no non-trivial symmetry. Figure 7a–c: Configurations with two cooperators (\(c(\pi )=2\). There are \(\#_2=66\) configurations with two cooperators according to Eq. (2). Calculating the structure coefficients \(\sigma (\pi )\) by Eq. (4) yields \(\#_{\sigma _{max}}=9\) configurations out of these 66 that have the maximal value of \(\sigma _{max}=1.5846\). All these configurations have the two cooperators distanced by the minimal cooperator path length \(l_c=1\)and the two cooperators belonging to one of the three triangles of the Frucht graph \(({\mathcal {I}}_3,{\mathcal {I}}_4,{\mathcal {I}}_5)\), \(({\mathcal {I}}_6,{\mathcal {I}}_7,{\mathcal {I}}_8)\) and \(({\mathcal {I}}_9,{\mathcal {I}}_{10},{\mathcal {I}}_{11})\), see Fig. 7a showing the example of configuration \(\pi =(0011\,0000\,0000)\). The minimal cooperator path length \(l_c=1\) alone also yields the second largest \(\sigma (\pi )=1.5455\), but \(l_c=1\) is not sufficient for the maximal value, see Fig. 7b showing the configuration \(\pi =(1000\,0000\,0001)\). The smallest value of \(\sigma _{min}=1.4546\) corresponds with the \(\#_{\sigma _{min}}=23\) configurations with largest values of \(l_c\) (21 configurations with \(l_c=3\) and two configurations with \(l_c=4\)), see the example of \(\pi =(0010\,0000\,1000)\) with \(l_c=4\) in Fig. 7c. Over all configurations with \(c(\pi )=2\), there are \(\#_\sigma =5\) different values of \(\sigma (\pi )\). Fig. 7d–f: Configurations with three cooperators (\(c(\pi )=3\)). The largest value \(\sigma _{max}=1.6897\) is obtained for \(\#_{\sigma _{max}}=3\) configurations out of the \(\#_3=220\). Configurations maximizing \(\sigma (\pi )\) are characterized by the minimal \(l_c\) such that the cooperators are occupying the three triangles of Frucht graph, see Fig. 7d showing the example \(\pi =(0011\,1000\,0000)\) which has \(l_c=1\) according to Eq. (5) with individual path lengths \(d=(d_{34},d_{35},d_{45})=(1,1,1)\). Small values of \(\sigma\) are obtained for large values of \(l_c\). There are two configurations with the largest value \(l_c=3\) for which \(\sigma (\pi )=1.4270\) (the second-smallest value) is obtained for individual path lengths between cooperators \(d=(d_{57},d_{5 \,10},d_{7 \, 10})=(2,4,3)\), Fig.7e, while the smallest value of \(\sigma _{min}=1.4000\) belongs to the configuration \(\pi =(0001\,0010\,0100)\) with \(d=(d_{47},d_{4 \,10},d_{7 \, 10})=(3,3,3)\), Fig. 7f. Over all configurations with \(c(\pi )=3\), there are \(\#_\sigma =14\) different values of \(\sigma (\pi )\).
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The truncated tetrahedral graph (Read and Wilson 1998), which is vertex–transitive and square free. Figure 8a–c: Configurations with two cooperators (\(c(\pi )=2\)). There are \(\#_{\sigma _{max}}=12\) out of 66 configurations that have the maximal \(\sigma _{max}=1.5846\), all of which have the minimal \(l_c=1\)and belong to one of the four triangles of the truncated tetrahedral graph \(({\mathcal {I}}_1,{\mathcal {I}}_2,{\mathcal {I}}_3)\), \(({\mathcal {I}}_4,{\mathcal {I}}_5,{\mathcal {I}}_6)\), \(({\mathcal {I}}_7,{\mathcal {I}}_8,{\mathcal {I}}_9)\) and \(({\mathcal {I}}_{10},{\mathcal {I}}_{11},{\mathcal {I}}_{12})\), see Fig. 8a showing the example of configuration \(\pi =(0110\,0000\,0000)\). As for the Frucht graph (Fig. 7), the minimal \(l_c=1\) alone also yields the second highest value \(\sigma (\pi )=1.5455\), see Fig. 8b. There are 24 configurations with the maximal \(l_c=3\) which all give the smallest value of \(\sigma _{min}=1.4546\), see Fig. 8c. Over all configurations with \(c(\pi )=2\), there are \(\#_\sigma = 4\) different values of \(\sigma (\pi )\). Figure 8d–f: Configurations with three cooperators (\(c(\pi )=3\)). The largest value \(\sigma _{max}=1.6897\) is obtained for 4 out of 220 configurations, each configuration representing one of the four triangles of the graph, see the example \(\pi =(1110\,0000\,0000)\) in Fig. 8d. The smallest value \(\sigma _{min}=1.4000\) is obtained for the four configurations with \(l_c=3\), see Figs.8e, f with the examples \(\pi =(1000\,1000\,1000)\) and \(\pi =(0001\,0001\,0001)\). There are \(\#_\sigma =10\) different values of \(\sigma (\pi )\) with \(c(\pi )=3\).
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The Franklin graph (Franklin 1934), which is vertex-transitive and triangle free. Fig. 9a–b: Configurations with two cooperators (\(c(\pi )=2\)). There are \(\#_{\sigma _{max}}=18\) out of 66 configurations that have the maximal value \(\sigma _{max}=1.5455\), all of these (and only these) configurations have the minimal \(l_c=1\), see Fig. 9a. Note that the maximal structure coefficient \(\sigma (\pi )\) is smaller than for the Frucht and truncated tetrahedral graph (Figs. 7 and 8 ), which is \(\sigma _{max}=1.5846\) obtained for configurations within a triangle of the graph. The Franklin graph is triangle free and only the second highest value of \(\sigma (\pi )\) is obtained. The minimal value of \(\sigma _{min}=1.4546\) is obtained for all 18 configurations with the maximal \(l_c=3\), Fig. 9a with \(\pi =(1001\,0000\,0000)\). There are \(\#_\sigma =4\) different values of \(\sigma (\pi )\). Fig. 9c–f: Configurations with three cooperators. The maximal value of \(\sigma _{max})=1.5909\) is obtained for 12 out of 220 configurations, all have the minimal \(l_c=4/3\) and additionally belong to one of the three squares of the Franklin graph, \(({\mathcal {I}}_1,{\mathcal {I}}_2,{\mathcal {I}}_7,{\mathcal {I}}_8)\), \(({\mathcal {I}}_3,{\mathcal {I}}_4,{\mathcal {I}}_9,{\mathcal {I}}_{10})\), \(({\mathcal {I}}_5,{\mathcal {I}}_6,{\mathcal {I}}_{11},{\mathcal {I}}_{12})\), see Fig. 9c. The minimal \(l_c=4/3\) in itself only yields the second largest \(\sigma (\pi )=1.5618\), Fig. 9d. The minimal value of \(\sigma _{min}=1.4270\) is obtained for 24 configurations with the largest value of \(l_c=8/3\) and not two out of three cooperators belonging to the same square, Fig. 9e. However, if \(l_c=8/3\) and two of the cooperators belong to the same square of the graph, we get the second-smallest value \(\sigma (\pi )=1.4546\), Fig. 9f. In total, there are \(\#_\sigma = 8\) different values of \(\sigma (\pi )\).
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Configurations \(\pi\), structure coefficients \(\sigma (\pi )\) and cooperator path lengths \(l_c\) for four cooperators (\(c(\pi )=4\)). There are \(\#_4=495\) configurations with \(c(\pi )=4\). The maximal structure coefficient \(\sigma _{max}=1.7059\) for the Frucht graph is obtained for two configurations, both with the minimal value \(l_c=4/3\)and additionally overlapping a triangle and the square of the Frucht graph, \(({\mathcal {I}}_1,{\mathcal {I}}_{10},{\mathcal {I}}_{11},{\mathcal {I}}_{12})\), see Fig. 10a for the example \(\pi =(1000\,0000\,1110)\), the other configuration is \(\pi =(0000\,0000\,1111)\). For the truncated tetrahedral graph, the maximal \(\sigma _{max}=1.6796\) is obtained for all the 12 configurations with the minimal \(l_c=4/3\), Fig. 10b. The Franklin graph has three configurations with maximal \(\sigma _{max}=1.6539\), they have minimal \(l_c=4/3\) and each form one of the three squares of the graph, Fig. 10c. The smallest values of \(\sigma (\pi )\) are obtained as follows. For the Frucht graph, there are 23 configurations with the smallest value \(\sigma _{min}=1.4231\), all with large values of \(l_c\) (one configuration with \(l_c=8/3\), ten configurations with \(l_c=5/2\) and 12 configurations with \(l_c=7/3\)), see the example of \(\pi =(0000\,1010\,0101)\) with \(l_c=8/3\) in Fig. 10d. However, there are three more configurations with the highest values \(l_c=8/3\) that have larger values of \(\sigma (\pi )\), but for these the individual path lengths between cooperators are widely distributed, compared to Fig.7e, f. For the truncated tetrahedral graph, there are 30 configurations with the smallest value \(\sigma _{min}=1.4231\) with the second-largest value \(l_c=7/3\), see the example \(\pi =(1001\,0010\,0100)\) in Fig. 10e. However, there are also 24 configurations with the largest value \(l_c=5/2\) that have the second-smallest value \(\sigma (\pi )=1.4340\). For the Franklin graph, there are 15 configurations with the smallest value \(\sigma _{min}=1.4231\), three of them with the largest value \(l_c=8/3\) and the remaining with the second largest \(l_c=5/2\), see the example \(\pi (0100\,1001\,0010)\) in Fig. 10f.
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Richter, H. Fixation properties of multiple cooperator configurations on regular graphs. Theory Biosci. 138, 261–275 (2019). https://doi.org/10.1007/s12064-019-00293-3
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DOI: https://doi.org/10.1007/s12064-019-00293-3