Evolution of Cooperation for Multiple Mutant Configurations on All Regular Graphs with $N \leq 14$ players

We study the emergence of cooperation in structured populations with any arrangement of cooperators and defectors on the evolutionary graph. Using structure coefficients defined for configurations describing such arrangements of any number of mutants, we provide results for weak selection to favor cooperation over defection on any regular graph with $N \leq 14$ vertices. Furthermore, the properties of graphs that particularly promote cooperation are analyzed. It is shown that the number of graph cycles of certain length is a good predictor for the values of the structure coefficient, and thus a tendency to favor cooperation. Another property of particularly cooperation-promoting regular graphs with a low degree is that they are structured to have blocks with clusters of mutants that are connected by cut vertices and/or hinge vertices.


Introduction
Describing conditions for the emergence of cooperation in structured populations is a fundamental problem in evolutionary game theory [4,19,20,35]. In structured populations the network describing which players interact with each other may be crucial for the fixation of a strategy. Recently, several attempts have been made to explore the universe of interaction graphs in order to link graph properties to fixation. For a single cooperator this question has been studied intensively and recently relationships have been mapped for a large variety of different interaction graphs connecting which strategy is favored with the fixation probabilities and the fixation times [2,18,25,32]. These results clarify for a single mutant the relationships between the graph structure, on the one hand, and fixation probability and fixation time, on the other. The main findings are that generally fixation probability and fixation time is correlated such that a higher fixation probability comes with a higher fixation time. Within this general rule, it has further been shown that generalized stars maximize fixation probability while minimizing fixation time, while comet-kites minimize fixation probability while maximizing fixation time [18]. Furthermore, if we allow self loops and weighted links, we may construct arbitrarily strong amplifiers of selection [25]. Compared with these findings, the problem of multiple cooperators (or more than one mutant) is far less studied. One approach uses configurations and contrast, σ(π, G) is valid for any arrangement of cooperators and defectors on the evolutionary graph and specifically for several cooperators (or multiple mutants).
As the structure coefficient varies over configurations π and graphs G, it is natural to ask about upper and lower bounds of σ(π, G). In this paper, we approach this question by checking all σ(π, G), which appears feasible for a small number of players N ≤ 14 and all regular graphs with up to 14 vertices. We classify the structure coefficients and graphs with respect to the number of players N . Furthermore, the configurations π are also grouped according to the number of cooperators c(π), 2 ≤ c(π) ≤ N − 2, while the graphs G are sorted according to the number of coplayers k (which equals the degree of the graph). As the structure coefficients σ(π, G) vary over configurations and graphs G, we may define two bounds. A first is over all 2 N − 2 non-absorbing configurations, which we call σ max i . Thus, we obtain for each graph G i , i = 1, 2, . . . L k (N ), the quantity σ max i = max π σ(π, G i ). A second bound, called σ max , is derived from the first bound and additionally collects over all L k (N ) regular graphs with a given N and k according to Tab. 2. Thus, we get σ max = max i σ max i . For the minimum, the bounds are defined like-wise. Fig. 1 shows the maximal structure coefficient σ max and the maximal difference ∆σ = σ max − σ min over players N and coplayers k. As discussed in Appendix 2 these results apply to any instance of a regular graph, for example to random regular graphs. It can be seen that the maximal structure coefficient σ max is largest for k = 3, which is cubic graphs. For k > 3, the values of σ max get gradually smaller. In other words, the more coplayers there are, the smaller is σ max . Also, for a constant number of coplayers, σ max increases with N , which is the number of players. The increase, however, gets gradually smaller and converges for N → ∞ to a constant, which is σ(π, G) → σ = (k + 1)/(k − 1) [6,21]. For instance, for k = 3, the structure coefficients converge to σ(π, G) → σ = 2. In other words, for the thermodynamic limit with an infinite population, prevalence of cooperation only depends on the number of coplayers k of a regular graph, but not on the graph structure or the number and arrangement of cooperators on the graph. The largest difference between maximal and minimal structure coefficient ∆σ = σ max − σ min we also get for k = 3. Here, ∆σ increase to a largest values (for instance for k = 3 this happens for N = 10) before falling for N getting even larger, converging to ∆σ = 0 for N → ∞.
We next analyze the maximal structure coefficients depending on the number of cooperators c(π). Thus, (a) N = 12 the maximum is over all # c(π) = N c(π) configurations with the same number of cooperators 2 ≤ c(π) ≤ N −2 and all regular graphs according to Tab. 2. The maximal values of σ max and ∆σ are obtained for c(π) = N/2 for N even and for both (N + 1)/2 and (N − 1)/2 for N odd. An exception is N = 12 and k = 3, where σ max is obtained for c(π) = 5 and c(π) = 7. Furthermore, we get the following results, see Fig. 2 as examples for N = 12 and N = 14. The value σ max and ∆σ are symmetric with the number of cooperators c(π) and generally higher for the number of cooperators and defectors exactly or approximately the same than for a small number of cooperators or a small number of defectors. For the number of coplayers k getting larger, the differences over the number of cooperators c(π) for both σ max and ∆σ are levelled.
Apart from the numerical values of the maximal structure coefficients σ max and their relations to the number of players N , coplayers k and cooperators c(π), it is also interesting to know for which of the L k (N ) graphs the maximal values occurs. We call the graphs for which this happens the σ max -graphs. Their number is # σmax . Tab. 1 give the number of σ max -graphs, # σmax , for all N and k considered here, see also Appendix 3 for some examples of σ max -graphs. If we compare these numbers with the total number L k (N ) of k-regular graphs on N vertices, see Tab. 2, we observe that L k (N ) grows much faster than # σmax . In other words, the σ max -graphs become rare as N increases. Fig. 3 shows the quantity # log = − 1 N 2 log #σ max (4k−1/4k 2 )L k (N ) over N and k (Fig. 3a), and over c(π) and k for N = 14 (Fig. 3b). We   may conclude that as a rough approximation the ratio #σ max L k (N ) falls exponentially in N and polynomially in k for k ≈ N/2 and N getting larger. Furthermore, observe from Fig. 3b that for small and large values of the number of cooperators c(π) there is a larger number of graphs that are σ max -graphs. The σ max -graphs become rarer for c(π) ≈ N/2, for which but σ max is largest.

Relationships between structure coefficients and graph cycles
Recently, Giscard et al. [7] proposed an algorithm to count efficiently the number of cycles with length in a graph: C (N, k) with 3 ≤ ≤ N . Thus, it is feasible to count C (N, k) for all L k (N ) regular graphs with N ≤ 14, as given in Tab. 2. As an example see Fig. 7 with the count C (6, 3), = {3, 4, 5, 6}, for the L 3 (6) = 2 graphs with N = 6 and k = 3. The following discussion is based on taking into account these numerical results.
In the previous section, it was shown that the maximal structure coefficients vary over interaction networks modelled as regular graphs, even if the number of players, coplayers and cooperators is constant. Thus, it appears reasonable to assume that some features of the graphs may be associated with these differences. In the following, results are presented in support for an approximately linear relationship between the number of graph cycles with certain length and the maximal structure coefficients. Two previous results can be interpreted as to point at the validity of such a relationship between the number of graph cycles and fixation properties. A first is from evolutionary games on lattice grids [8,9,14,23]. For these games, it has been shown that clusters of cooperators have a higher fixation probability than cooperators that are widely distributed on the grid. The location of the cluster on the grid does not matter. As lattice grids can be described by regular graphs (a Von Neumann neighborhood is a 4-regular graph, a Moore neighborhood a 8-regular graph) clusters imply short and closed paths between the nodes of the grid. Furthermore, the grid means an abundance of cycles with even cycle length. A second result is that between the structure coefficients and the path length between the cooperators there is a strong negative correlation [28]. Cooperator path length is defined as the path length averaged over all pairs of cooperators on the evolutionary graph. If there are more than two cooperators, the cooperator path length has particularly small values if the cooperators cluster next to each other and are linked by loops. Thus, small values of the cooperator path length correspond with the abundance of cycles of certain length. As there are L k (N ) regular graphs for a given N and k, we obtain L k (N ) maximal structure coefficients σ max i , i = 1, 2, . . . L k (N ) together with the same count of cycle length C i (N, k). Thus, we may assume for each i a linear relationship σ max i = C i (N, k)x + i for some variables x with an error term i . To test the validity of this linear relationship, we calculate the residual error where C comprises of all L k (N ) cycle length C i (N, k) and σ max contains all L k (N ) structure coefficients σ max i for a given N and k. The variable x * is the solution of the non-negative least square problem As the length of x * varies with varying L k (N ), the residual error in (1) is weighted by L k (N ) to make it comparable over all N and k. Note that the residual error (1) gives equivalent results to the root-meansquare deviation, which is also sometimes used to measure the accuracy of a (linear) model. The results are given in Fig. 4. We see that the residual error res is small for all 6 ≤ N ≤ 14, 3 ≤ k ≤ N − 3 and To conclude we can observe that the results for the residual error res are generally very small, which is equivalent to saying that the error term i in the assumed linear relationship Thus, there is some justification to observe that between the maximal structure coefficients σ max i and the cycle count C i (N, k) there is an approximately linear relationship. Finally, another aspect of the interplay between graph structure and fixation properties should be highlighted. To begin with, we analyze the cycle count C (N, k) of σ max -graphs, which are those graphs among the L k (N ) regular graphs that have maximal structure coefficients. Consider the example N = 12 and k = 3. There are L 3 (12) = 85 graphs of which # σmax = 4 are σ max -graphs, compare Tab. 1 with Tab. 2.
For these 4 graphs we analyze how the count C (12, 3) is distributed over = 3, 4, . . . , 12. A possible way to visualize such an analysis is based on schemaballs [13,27], see Fig. 5a. In such a schemaball we draw Bezier curves connecting the count C (N, k) in the upper half of the ball with the associated cycle length in the lower half. The actual values of both and C (N, k) are written on the ball. The curves are colored in such a way that equal values of the cycle length have the same (and specific) color, no matter to which cycle count C (N, k) they are belonging. The colors are selected equidistant from a RGB color wheel. If there are several σ max -graphs, as there are # σmax = 4 for N = 12, k = 3 in Fig. 5a, each graph has its own set of curves between and C . The schemaball thus contains all of them, which means there may be curves between the same value of and several C (and vice versa). For instance, in Fig. 5a showing the schemaball for N = 12 and k = 3, we see that for = 3, which is cycles of length 3, also known as triangle, we find connection to C 3 (13, 3) = (2,3,4,5). This means each of the # σmax = 4 graphs has triangle, one has 2 of them, another one has 3, still another one has 4 and the final one has 5 triangle.
From the visualization using a schemaball it can be immediately seen that for N = 12 and k = 3 small cycles lengths, that is = {3, 4, . . . , 7}, have generally a count C (12, 3) > 0. For large cycle lengths, that is = {8, 9, . . . , 12}, we have C (12, 3) = 0. For N = 14 and k = 3, see Fig. 5c, we get very similar results. By contrast, for larger k, not only the cycle count C (N, k) is much higher than for lower k, but also the distribution over cycles lengths is quite different, see the examples N = 12, k = 8, Fig. 5b and N = 14, k = 10, Fig. 5d. Here, small as well as large cycle lengths have a substantial count C (N, k). Moreover, every cycles length is connected to a distinct interval of C (N, k). This means that the σ max -graphs have very similar counts C (N, k) for each . These properties becomes even more clear if we additionally consider the schemaballs for σ min -graphs, which are the graphs with minimal structure coefficients see Fig. 6 for the same examples as Fig. 5. Not only there are more σ min -graphs than σ max -graphs, (for instance 77 vs. 4 for N = 12, k = 3, or 359 vs. 6 for N = 12, k = 8), the balls for small k look very different, compare Figs. 6a and 6c with Figs. 5a and 5c. For the σ min -graphs and small k even large cycle length have a substantial count C (N, k). The count is actually much higher, which means that σ min -graphs have generally more cycles of a given length than σ max -graphs. On the other hand, for large k the differences are rather marginal. The only difference is that the schemaballs are more dense, which means that σ min -graphs have more different counts for a given cycle length than σ max -graphs. For the other tested number of players N ≤ 14 similar results are obtained as shown in Figs. 5 and 6. We next discuss some implications of these results on the evolution of cooperation on regular evolutionary graphs.

Discussion and Conclusions
In this paper structure coefficients σ(π, G) introduced by Chen et al. [6] (see [27,28] for further analysis) are studied for all regular interaction graphs with N ≤ 14 players and 3 ≤ k ≤ N − 3 coplayers. These structure coefficients provide a simple condition connecting long-term prevalence of cooperation with the values of the payoff matrix (4), the structure of the evolutionary graph G and the arrangement of any number of cooperators and defectors on this graph, which is expressed by the configuration π. Cooperation is favored for weak selection and a configuration π on a graph G if For σ(π, G) < 1, the game favors the evolution of spite, which can be seen as a sharp opposite to cooperation. For σ(π, G) = 1, the condition (3) matches the standard condition of risk-dominance. For σ(π, G) > 1, the diagonal elements of the payoff matrix (4), a and d, are more critical than the off-diagonal elements, b and c, for determining which strategy is favored. For instance, cooperation can be favored in the Prisoner's Dilemma game, which is specified by c > a > d > b. The condition (3) implies that a larger value of σ(π, G) still allows cooperation to emerge if a − d is small (or c − b is large). For the Stag Hunt game (Coordination game), characterized by a > c ≥ d > b, the condition σ(π, G) > 1 means to favor a Pareto-efficient strategy (a > d) over a risk-dominant strategy (a + b < c + d). Again, a larger value of σ(π, G) tolerates a smaller Pareto-efficiency a − d. Put differently, cooperation is favored even if the difference between reward and punishment is rather low. A generalization of these discussions can be achieved by the universal scaling approach for payoff matrices that facilitates studying a continuum of social dilemmas [34]. According to this approach a larger value of σ(π, G) implies a larger section of the parameter space spanned by gambleintending and risk-averting dilemma strength [29]. Based on this interpretation of the structure coefficient σ(π, G), we review the following major results of the numerical experiments presented in Sec. 2.
a. There is an approximately linear relationship between maximal structure coefficients and the count of cycles of the interaction graph with certain length. Moreover, the number of σ max -graphs grows much slower for a rising number of players than the number of k-regular graphs on N vertices. Thus, graphs with maximal structure coefficients get rare for the number of players N getting large.
b. The values of the structure coefficients are larger for a small number of coplayers, that is for graphs with a small degree, and maximal for k = 3, which is cubic graphs, than for larger numbers of coplayers. This is also the case for the largest differance between maximal and minimal structure coefficients. Thus, for regular evolutionary graphs describing the interactions between players, the results for N ≤ 14 players suggest that a smaller number of coplayers is particularly prone to promote cooperation if a favorable graph is selected. The selection of graphs does matter less for a larger number of coplayers. The σ max -graphs with small numbers of coplayers k not only have largest maximal structure coefficients, they are also characterized by the absence of cycles with a length above a certain limit, see examples in the collection of σ max -graphs in Appendix 3.
c. There are not only no long cycles in σ max -graphs with small k. The graphs are also structured into blocks that are connected by cut vertices and/or hinge vertices. A cut vertex is a vertex whose removal disconnects the graph, while a hinge vertex is a vertex whose removal makes the distance longer between at least two other vertices of the graphs [5,11]. For instance, for N = 12 and k = 3, the vertices occupied by the players I 3 and I 9 , see Fig. 11, are cut vertices, while for N = 10 and k = 4, see Fig. 10b, the vertices occupied by the players I 5 and I 6 are hinge vertices as their removal would make the distance between I 4 and I 7 longer. The blocks are occupied by clusters of cooperators. The clusters can be seen as to serve as a mutant family that invades the remaining graph. As vertices with players of opposing strategies are connected by cut and/or hinge vertices there is only a small number of (or even just a single) migration path for the cooperators and/or defectors. A similar observation has been reported for evolutionary games on lattices grids [8,14], see also the discussion in Sec. 2.2.
To summarize: the results suggest that σ max -graphs for small numbers of coplayers have some distinct graph-theoretical properties. Searching for these properties in a given graph may inform the design of interactions graphs that are either particularly prone to cooperation or particularly opposed to it.
d. The property of missing long cycles is also a possible explanation as to why regular graphs with small degree differ substantially from graphs with larger degree in terms of promoting cooperation in evolutionary games. A larger degree makes it impossible to have blocks that are connected by only a few edges. As the number of edges increases linearly with the degree by kN/2 and each vertex has the same number of edges, there is an ample supply on connections. These results imply that connectivity properties of the interaction graph play an important role in the emergence of cooperation. It may be interesting to see if these connectivity issues may possibly also show in algebraic graph measures, for instance algebraic connectivity expressed by the Fiedler vector.
The results given above show a clear dependency between the long-term prevalence of cooperation in evolutionary games on regular graphs and some of their graph-theoretical properties, which generally confirm previous findings on clusters of cooperators in games on lattice grids [8,9,14,23], on pairs of mutants on a circle graph (k = 2) [36], and on short cooperator path lengths on some selected regular graphs with N = 12 and k = 3, among them the Frucht, the Tietze and the Franklin graph [28]. However, apart from statements about the prevalence of cooperation there are also other quantifiers of evolutionary dynamics that are highly relevant. In other words, some of the difficulty in the given approach for evaluating the emergence of cooperation in evolutionary games on graphs arises from structure coefficients merely treating a comparison of fixation probabilities. The condition indicates that the fixation probability of cooperation is higher than the fixation probability of defection. This, however, does not entail the values of these probabilities. However, structure coefficients can be calculated with polynomial time complexity [6], while computing fixation probabilities is generally intractable due to an exponential time complexity [10,12,33]. In other words, by using the approach involving structure coefficients, we exchange computational tractability by obtaining just a comparison of fixation probabilities instead of their exact values. Moreover, apart from the difference in the information obtained, the variety in the descriptive power of the structure coefficients as compared to the fixation probabilities is salient in another way. Most likely, there is a rather complex relationship between structure coefficients and fixation probability, which is illustrated by the example of a single cooperator for which the structure coefficient does precisely not imply unique values of the fixation probability of cooperation. For a single cooperator we get a single value of the structure coefficient, but fixation probabilities vary over initial configurations as shown for the Frucht and for the Tietze graph [16]. All these considerations show that calculating fixation probabilities and fixation times for multiple mutant configurations is not only computationally expensive, but also has a huge number of possible setups, for instance, which one of the considerable number of graphs to analyze, or where to place cooperators on the evolutionary graph and how many. There are various experimental parameters to be taken into account, which might be why so far systematically conducted numerical studies are sparse. In this sense, another contribution of this paper might be seen in pointing at settings for numerical experiments calculating fixation probabilities and fixation times. The results given in this paper show that among all the regular interaction graphs with N ≤ 14 players and 3 ≤ k ≤ N − 3 coplayers, there is a comparably small number of graphs (as given in Tab. 1) which favor cooperation more than others. It may be interesting to see if these graphs also stand out in terms of fixation probability and fixation time as compared to a graph randomly drawn from the other ones.

Acknowledgments:
I wish to thank Markus Meringer for making available the genreg software [17] used for generating the regular graphs according to Tab. 2 and for helpful discussions.

Appendix A Configurations, regular graphs and structure coefficients
The co-evolutionary games we consider here have N players I = {I i }, i = 1, 2, . . . , N , that each uses either of two strategies π i ∈ {C, D}, which we may interpret as cooperating or defecting. Each player I i , which interacts with a coplayer I j , receives payoff according to the 2 × 2 payoff matrix Which player interacts with whom is described by the interaction graph G = (V, E), where the vertices v i ∈ V represent the players and the edges e ij ∈ E indicate that the players I i and I j interact as mutual coplayers [15,22,26]. Which strategy is used by which player at a given point of time is specified by a configuration π = (π 1 , π 2 , . . . , π N ) with π i ∈ {C, D}. If we represent the two strategies by a binary code {C, D} → {1, 0}, a configuration appears as a binary string the Hamming weight of which denotes the number of cooperators c(π). For games with N players, there are 2 N configurations with 2 configurations (π = (00 . . . 0) and π = (11 . . . 1)) absorbing. Players may update their strategies in an updating process, for instance death-birth (DB) or birth-death (BD) updating [1,24]. Recently, it was shown by Chen et al. [6] that strategy π i = 1 = C is favored over This results applies to weak selection and 2 × 2 games with N players, payoff matrix (4), any configuration π of cooperators and defectors and for any interaction network modeled by a simple, connected, k-regular graph.
The quantity σ(π, G) in Eq. (5) is the structure coefficient of the configuration π and the graph G. It may not have the same value for different arrangements of cooperators and defectors described by the configuration π and also for different interaction networks modeled by a regular graph G. In particular, it was shown that for weak selection and the graph G describing interaction as well as replacement graph, the structure coefficient σ(π, G) can be calculated with time complexity O(k 2 N ) for DB and BD updating [6]. For DB updating there is with 4 local frequencies (ω 1 , ω 0 , ω 10 and ω 1 ω 0 ), which depend on π and G, see [6,27,28] for a probabilistic interpretation of these frequencies. Our focus here is on DB updating as it has been shown that BD updating never favors cooperation [6].

Appendix B Isomorphic graphs, isomorphic configurations and cycle counts
The structure coefficient σ(π, G), as for instance defined for DB updating by Eq. (6), may vary over configurations π and graphs G. This suggests the question of upper and lower bounds of σ(π, G). For a rather low number of players it appears feasible to check all σ(π, G), as demonstrated in the paper for N ≤ 14 and all regular graphs with up to 14 vertices. For a 2 × 2 game with N players, there are 2 N − 2 nonabsorbing configurations π. These configurations can be grouped according to the number of cooperators Table 2: The numbers L k (N ) of simple connected k-regular graphs on N vertices, [17], which corresponds to the number of regular interaction graphs with N players and k coplayers for 6 ≤ N ≤ 14 and 3 ≤ k ≤ N − 1.
Note that there is more than one graph, L k (N ) > 1, only for k ≤ N − 3.
The number of simple, connected regular graphs is known for small numbers of vertices, e.g. [17], see Tab. 2. Note that these numbers apply to graphs that are all not isomorphic with each isomorphism class being represented by exactly one graph. In other words, Tab. 2 also gives the number of isomorphism classes for all 6 ≤ N ≤ 14 and 3 ≤ k ≤ N − 1. Isomorphism refers to the property that two graphs are structurally alike and merely differ in how the vertices and edges are named. More precisely, two graphs are isomorphic if there is a bijective mapping θ between their vertices which preserves adjacency [3], pp. 12-14.
Consider, for example, the L 3 (6) = 2 interaction graphs with N = 6 players, each with k = 3 coplayers, see Fig. 7. For the graph in Fig. 7a we get the maximal structure coefficient σ max = 1.1818 for 2 configurations, π = (111000) as shown in Fig. 7a and π = (000111). By the isomorphism θ = , we obtain an isomorphic graph as shown in Fig. 7b. For this graph, the configuration π = (111000) has σ = 1.0000, but π = (110001) and π = (001110) have σ max = 1.1818. Note that between the configurations with σ max the same isomorphic mapping θ applies. In other words, the structure coefficients are invariant under isomorphic mappings. For each pair of isomorphic graphs, there are isomorphic configurations that have the same value of the structure coefficient. For the graph in Fig. 7c, we obtain the result that the structure coefficient is constant over all configurations (except the absorbing configurations). Thus, isomorphic transformations do not alter the values of σ(π, G).
These results apply generally to structure coefficients σ(π, G) of regular graphs. The local frequencies in Eq. (6) solely depend on counting two types of paths on the interaction graph [6,27,28]. The quantities ω 1 , ω 0 and ω 1 ω 0 relate to the number of paths with length 1 that connect any vertex with adjacent vertices that hold a cooperator (or defector). The quantity ω 10 relates to the number of paths with length 2 from any vertex to adjacent vertices on which the first vertex of the path holds a cooperator and the second vertex holds a defector. As an isomorphic reshuffling of vertices preserves adjacency, these numbers stay the same if the isomorphism acts on both the vertices and the configurations. Thus, suppose two graphs G i and G j are isomorphic with isomorphism θ. Then, it follows σ(π, G i ) = σ(θ(π), G j ). Furthermore, the maximal structure coefficient is invariant as well, that is for isomorphic graphs G i and G j there is . Any regular graph belongs to one of the isomorphism classes and can be obtained by isomorphic transformations by any member of this class. Regular interaction graphs that are isomorphic have the same distribution of structure coefficients σ(π, G) over the number of cooperators c(π). Thus, by considering one representative of each isomorphism class, we can make statements about structure coefficients for all regular graphs.

Appendix C Collection of σ max -graphs with N ≤ 14
We here give a collection of selected σ max -graphs with N ≤ 14. The graphs are shown to illustrate some graph-theoretical properties associated with prevalence of cooperation. The single σ max -graph with N = 6 is already shown in Fig. 7a. For N = 7, there are L 4 (7) = 2 regular graph, which both have the same maximal structure coefficients. In other words, the count of graphs equals the count of σ max -graph, which is why they are not included in the collection.
Figs. 8-10 shown all σ max -graphs for N = 8, 9, 10 and 3 ≤ k ≤ N − 3 together with σ max and the associated configurations. For N = 12 and N = 14, only some examples of σ max -graphs are given in Figs. 11-13 due to brevity. A full list of all σ max -graphs for 11 ≤ N ≤ 14 and 3 ≤ k ≤ N − 3 is made available here [30]. It is particularly noticeable that the σ max -graphs are structured to have blocks with clusters of mutants. For instance, we see such a block with (I 1 , I 2 , I 3 , I 4 ) for the graph with N = 8 and k = 3 in In addition, the same structure coefficient is obtained also for the configuration π = (0000 1111), and only for (d) additionally for π = (1100 0011) and π = (0011 1100). Figure 9: The σ max -graphs for N = 9 and k = 4, 6. We get σ max = 1.3206 for k = 4 (a) and the configuration π = (11110 0000), but also for π = (11111 0000), π = (00000 1111) and π = (00001 1111). For k = 6, there are 3 σ max -graphs, (b),(c),(d), each with σ max = 0.9115 for the configuration π = (11110 0000). There are several more configurations that have the same σ max due to the symmetry properties of these 3 graphs. Fig. 8a and for N = 9 and k = 4 in Fig. 9a, or for N = 10 and k = 3, Fig. 10a and for the cubic graphs (k = 3) with N = 12 and N = 14 as well, see Figs. 11 and 13. The σ max -graphs with larger degree (= coplayers) still somewhat retains such a "blockish" appearance (for instance (I 1 , I 2 , I 3 , I 4 , I 5 ) in Fig. 10c) but to a far lesser degree. In addition, σ max -graphs with larger degree are frequently vertex-transitive (for instance Figs. 9d, 10e and 10g) which is not the case for cubic (k = 3) and quartic (k = 4) σ max -graphs with N ≤ 14, with the exception of N = 6 and k = 3, see Fig. 7a. Furthermore, it can be observed that the blocks are occupied by clusters of cooperators which are frequently connected by cut vertices and/or hinge vertices. For instance, for N = 12 and k = 3, the vertices occupied by the players I 3 and I 9 , see Fig. 11, are cut vertices, while for N = 10 and k = 4, see Fig. 10b, the vertices occupied by the players I 5 and I 6 are hinge vertices as their removal would make the distance between I 4 and I 7 longer. As discussed above, the clusters can be seen as to serve as a mutant family that invades the remaining graph. As vertices with players of opposing strategies are connected by cut and/or hinge vertices there is only a small number of (or even just a single) migration path for the cooperators and/or defectors.