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 configurationbased 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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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.

1.
The Frucht graph (Chen et al. 2016; Frucht 1949; McAvoy and Hauert 2015), which has no nontrivial 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 secondsmallest 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 \(\sigma (\pi )\).

2.
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 \(c(\pi )=3\).

3.
The Franklin graph (Franklin 1934), which is vertextransitive 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 secondsmallest value \(\sigma (\pi )=1.4546\), Fig. 9f. In total, there are \(\#_\sigma = 8\) different values of \(\sigma (\pi )\).

4.
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 secondlargest 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 secondsmallest 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/s12064019002933
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DOI: https://doi.org/10.1007/s12064019002933
Keywords
 Evolutionary graph theory
 Fixation
 Structure coefficients
 Regular graphs
 Multiple mutants