1 Introduction
A central issue in evolutionary game theory is describing conditions for the emergence of cooperation
[7, 26]. For games on evolutionary graphs with a single cooperator, the question of whether or not cooperation is favored over defection is generally settled for weak selection. The answer is given for regular graphs by a single quantity, the structure coefficients [1, 28, 38, 39], while for graphs in general a quantity related to coalescence times can be used [2]. If there are multiple cooperators, the situation is more complicated [18, 32]. Recently, an approach has been proposed that analyses a given number of cooperators and defectors on the evolutionary graph as configurations describing any possible arrangements of strategies over players [10, 11, 34, 35]. The approach also defines structure coefficients applicable for each configuration and for each network of interaction modeled by regular graphs [11].Game dynamics with multiple cooperators may be interesting for at least two reasons. A frequently studied question is the appearance of a single cooperator by mutation that may or may not fixate in a frequency–dependent strategy updating process [8, 10, 28, 24, 31, 38, 39]. If, starting from a single cooperator configuration, fixation of cooperation succeeds, then the evolutionary process may undergo a transition through configurations with multiple cooperators. Thus, studying multiple cooperator configurations can be seen as an attempt to describe how the transition process creates evolutionary trajectories. Moreover, in social networks, it may be interesting to actively design conditions positive for cooperation. Such a design could involve the strategic placement of more than one cooperator to enhance the likelihood that cooperation prevails.
In this paper results are presented suggesting that for regular graphs fixation properties for any number of cooperators and regular graphs can be approximated by the cooperator path length, which is the average shortest path length between cooperators on the evolutionary graph. In other words, we can link a property of the evolutionary game (emergence of cooperation) to a property of the graph (a certain path length). In Sec. 2 the methodological framework of configuration–based structure coefficients is briefly recalled, see [11, 35] for a more detailed description. The main results are given in Sec. 3, where we analyze and compare evolutionary graphs covering a wide range of graph–theoretical properties. It is shown how structure coefficients are distributed over configurations with the same number of cooperators and how they correlate to cooperator path length. These relationships between fixation properties and cooperator path length relate to previous results for evolutionary games on lattice grids that show clusters of cooperators have better chances to survive [16, 17, 21, 30]. The findings are supported by results for networks with a varying number of players and coplayers. Finally, the results are discussed and conclusion are drawn about finding or designing best conditions to favor cooperation in evolutionary games on regular graphs.
2 Methods
We consider coevolutionary games with players . Each pairwise interaction between 2 of these players, and , (who thus are mutual coplayers), provides them with payoff depending on which of 2 strategies, or and or , each player uses. A (possibly varying) network structure defines which player may interact with whom. Such games are specified by (i.) the payoff matrix, (ii.) the interaction network, and (iii.) the configuration of the game [11, 34]. The payoff matrix is
(1) 
The interaction network is represented by an evolutionary graph in which each player belongs to a vertex and an edge connecting vertex and vertex shows that the players and are mutual coplayers [2, 37]. Such an interaction graph is equivalent to the adjacency matrix . Set the element of to for an edge between vertex and vertex , while indicates that the players and are no coplayers. Finally, a configuration specifies the strategy that each player , (), uses. With 2 strategies ( and ) there are configurations. These configurations enumerate all possible arrangements of cooperators and defectors among the players. Additionally, the configurations describe any outcome of a player changing its strategy in a strategy updating process, for instance death–birth (DB) or birth–death (BD) updating [1, 31, 37]. It is convenient to binary code the strategies , thus having a binary string to specify the strategies of all players [11, 34]. Consider the example of players. There are configurations. For instance, the configuration shows that players and cooperate, while and defect. As cooperation is coded as , the Hamming weight of the string (the count of 1–elements in ) gives the number of cooperators of a configuration . Thus, has cooperators. For , there is more than one configuration with the same number of cooperators . For instance, there are configurations with a single cooperator (). The number of configurations with the same number of cooperators can be calculated by:
(2) 
with and .
Recently, it was shown by Chen et al. [11] that for games with players, payoff matrix (1), any configuration of cooperators and defectors and for any interaction network modeled by a simple, connected, –regular graph, in the case of weak selection strategy is favored over if
(3) 
We call the quantity the structure coefficient of the configuration implying that it may have different values for different arrangements of cooperators and defectors decribed by . The condition (3) generalizes the same condition with , which applies to a single cooperator, and is independent of where on the network the single cooperator is initally located [22, 38, 39]. Furthermore, is was shown that for weak selection, the network describing the replacement structure, and DB as well as BD updating, the structure coefficient can be calculated with time complexity . In particular, for DB updating we have
(4) 
with 4 local frequencies: , , and , see [11]. For these local frequencies, the following probabilistic interpretation has been suggested. Suppose on a given interaction network a random walk is carried out with the starting vertex chosen uniformly–at–random. The local frequency (or
) is the probability that for a configuration
the player at the first step of the walk is a cooperator (or defector). The local frequency is the probability that for a walk with 2 steps the player at the first step is a cooperator and at the second step it is a defector. Finally, if we carry out 2 random walks independent of each other, the local frequency is the probability that the player at the first step on the first walk is a cooperator, but a defector at the first step on the second walk.To define the cooperator path length we call the shortest path on the evolutionary graph between vertex (= player ) and vertex (= player ) with both players being cooperators . The quantity can be calculated by variants of Dijkstra’s algorithm with time complexity . If there are more than 2 cooperators, , there are pairs with individual path length . The cooperator path length is the mean over all pairs:
(5) 
Graph 




















3 Results
Studying relationships between configurations , structure coefficients and cooperator path length starts with 6 different interaction networks with players and coplayers that represent a variety of graph properties, see Tab. 1 and Figs. 1–5. We analyze the following graphs: Frucht [11, 14, 23], truncated tetrahedral [33], Franklin [12], Tietze [6, 23], Dürer and Möbius ladder [33]. Of these graphs, 3 are just regular, but differ by having no non–trivial symmetry (Frucht), possessing non–trivial automorphism and being Hamiltonian (Dürer) or not (Möbius ladder). The other 3 graphs are vertex–transitive, which is to say that some automorphism group acts transitively upon the graph’s vertices. Informally speaking, vertex–transitive means the graphs look the same from any two vertices and thus have a higher degree of symmetry as graphs that is just regular. Additionally, the truncated tetrahedral, Tietze and Dürer graph have triangles (cycles of length 3), but are square free, while the Franklin and Möbius ladder graphs have squares (cycles of length 4), but are triangle free.
For these graphs large values of the structure coefficients correspond to small values of the cooperator path length , and vice versa, suggesting the hypothesis that between the structure coefficients and the cooperator path length there is an inversely proportional relation. For example, the Frucht graph with 2 cooperators has the maximal structure coefficient for 9 configurations which all share that the 2 cooperators are distanced by the minimal and the 2 cooperators belong to one of the 3 triangles of the Frucht graph, Fig. 1(a), while for 3 cooperators the minimal is obtained for the configuration with the largest value with also all 3 individual path length , Fig. 1(f). However, the results also show that apart from , there are additional factors influencing . Particularly, for 4 cooperators, the largest or smallest value of alone does not guarantee the smallest or largest value of . For instance, for the Franklin graph and 4 cooperators, , not only the smallest value but also that the 4 cooperators form one of the squares, gives the largest structure coefficient , Fig. 4(c), while for the truncated tetrahedral graph, the minimal is obtained for the second largest value , Fig. 4(e). It can further be observed that the maximal and minimal value of is rarely unique for a configuration , even if the number of cooperators is constant.
In Tab. 1, the largest and smallest , denotes as and , are given for each . The table also gives the multiplicity of the structure coefficients , and the multiplicity of the maximal and minimal structure coefficient, and as well as the multiplicity of the maximal and minimal cooperator path length, and . The multiplicity of the structure coefficient is the number of different values of . For instance, for the Frucht graph and , means there are 25 different values of over the configurations with 6 cooperators. Similarly, the multiplicity of the maximal structure coefficient is the number of configurations that have . Thus, for the Franklin graph and , means 12 out of 792 configurations have the maximal structure coefficient . For , and this applies likewise.
Looking at the results in Tab. 1, we see some patterns over the 6 graphs considered, but also some differences. A first pattern is that the multiplicity of the structure coefficients does not scale to the number of configurations for each number of cooperators . Clearly, for and configurations with 2 cooperators, we also get the smallest in all cases given in Tab. 1, but as and increases, the multiplicity of the structure coefficient does not. The same property can also be found for the multiplicity of maximal and minimal and . Another interesting result is that generally the multiplicity of the minimal is larger than for the maximal , that is, mostly we have . A possible interpretation is that finding a configuration at random with the maximal structure coefficient is less likely than finding a configuration with . However, this interpretation should be viewed with respect to the number of configurations with the same number of cooperator . For instance, for the Frucht graph, we get for 2 cooperators () the values and . With configurations with 2 cooperators, the likelihood to find at random a configuration with thus is , while for it is . However, if we consider the same for 6 cooperators, we get and for and . In other words, looking at random for configurations with certain fixation properties might be promising for a small number of cooperators, but surely it is not for a larger .
Comparing the graphs the most interesting results is that for the vertex–transitive graphs (truncated tetrahedral, Franklin and Möbius ladder) the multiplicity of and more often than for the other graphs matches the multiplicity of and , see Tab. 1, where in the columns for , , and the matching values are underlined. This means for the vertex–transitive graphs considered it is more likely that the largest value of the structure coefficient is uniquely determined by the smallest value of the cooperator path length , and vice versa. While for the Frucht, Tietze and Dürer graph, there are 2, 4 and 2 matches, we get for the tetrahedral, Franklin and Möbius ladder graph 5, 6 and 7 matches. Thus, we can hypothesize that for vertex–transitive graphs the correlation between structure coefficients and cooperator path length might be stronger than for graphs that are just regular. The next set of numerical experiments deals with this and generally with the proposed inverse proportional relationships between structure coefficients and cooperator path length.
. (a) Pearson product–moment correlation coefficient
giving a measure on a linear function. (b) Spearman rank correlation coefficient giving a measure on a monotonic function.As shown above for different evolutionary graphs, the values of (including the largest and smallest, but also intermediate values) form a discrete distribution with different values. In other words, for every configuration we have a structure coefficients and a cooperator path length
. As the configurations can be classified according to the number of cooperators
, we may study how the structure coefficients are distributed over and also over the range of the cooperator path length , see Figs. 6–7. The distributions are depicted as histograms and shown as violin plots. The number of bins of the histograms is calculated according to the Freedman–Diaconis rule [13, 36], which is designed to minimize the error made by the partition. The central axis of each violin plot gives the range between largest and smallest value of .Fig. 6 shows some findings supporting the main results given with Fig. 7 by considering the Frucht graph and the Franklin graph. In the upper panels, Fig. 6 (a),(b), the distributions of the structure coefficients are given for each number of cooperators, , while the lower panels, Fig. 6 (c),(d), take the example of 6 cooperators () and depict the distributions for each unique value of the cooperator path length . The results for the distribution of over the number of cooperators , Fig. 6 (a),(b), show some characteristics that can similarly be found for other graphs as well. A first characteristics is that the distributions are symmetric with respect to the number of cooperators . A second is that for a single cooperator (or defector), or , there is the same single value of the structure coefficient for all configurations, see the blue line in Fig. 6. For and this value reproduces the well–known result , compare [22, 38, 39]. For cooperators we obtain structure coefficients that are both larger and smaller than , depending on the arrangement of cooperators and defectors on the graph, compare the discussion given with Figs. 1–5. Put differently, for multiple cooperators the arrangement of cooperators and defectors on the evolutionary graph may substantially affect whether or not cooperation is favored, even if the number of cooperators is the same. Generally, the largest range between maximal and minimal value of is not reached for but for or .
The distribution of over for the Frucht graph, Fig. 6(a), and the Franklin graph, Fig. 6(b), are rather similar with the range between maximal and minimal value of slightly larger for the Frucht graph. However, comparing the results with respect to how the structure coefficients are distributed over unique values of the cooperator path length shows substantial differences between the graphs, see Fig. 6(c),(d). Not only is the range of smaller for Franklin () than for Frucht (), there are also fewer unique values ( vs. ). Also, the range between and for each is smaller for Franklin than for Frucht. For other numbers of cooperators, and the other interaction graphs considered, we find similar results. This applies in particular to the general result that large values of coincide very frequently with small values of , and vice versa. To visualize how the histograms scale over the number of cooperators, Fig. 7 aggregates the violin plots over . Each violin is colored according to a color code for each , and on the axis below and above the graph the maximum and the minimum value of is given for each . By looking at this visual presentation of the relationships between structure coefficients and cooperator path length several observations can be made. A first is that for small and medium numbers of cooperators we see for all 6 graphs a clear reverse proportional relations between and . For the vertex–transitive graphs this remains for . Moreover, for these graphs the distribution of for each value of shows a certain symmetry with respect to the number of cooperators. This is not true for the other 3 graphs. Given the fact that the whole distribution of over is symmetric, see Fig. 6(a),(b), this symmetry found for the 3 vertex–transitive graphs is caused by the cooperator path length showing certain symmetry properties.
Compare the multiplicity of . For instance, for the Franklin graphs we get the multiplicities of as for the number of cooperators . As for the number of cooperators and the number of defectors there is , the multiplicities for matches that of . By contrast, for the Frucht graph we get , which does not show this kind of symmetry. However, note that this symmetry does not mean that for each and we obtain the same values of the cooperator path length . For instance, for the Franklin graph with we have the values , while for the values are . We have the same number of unique values but their range shrinks for increasing. This appears plausible as for a small number of cooperators, for instance , the cooperator path length can be as high as the diameter of the graph, while for larger the growing number of cooperators crowds the graph and makes average distances smaller. This symmetry property of the distribution of over , which can be found for the other 2 vertex transitives graphs as well, is related to the property that vertex transitivity implies the graph to be strongly distance balanced [19, 20].
We next analyze the relationships visualized in Fig. 7 by considering correlation coefficients. Fig. 8 gives the Pearson product–moment correlation coefficient and the Spearman rank correlation coefficient , each between the variables and . Pearson’s gives a measure of how well the relationship between the variables can be described by a linear function, Fig. 8(a), while Spearman’s is a measure of how well the relationship matches a monotonic function, Fig. 8(b). We notice that for small we obtain for both correlation coefficients and all 6 graphs values slightly smaller than , which indicates a strong inverse proportional relation. For the 3 vertex–transitive graphs this strong correlation remains for all , while for the 3 graphs that are just regular, the correlation weakens while increases. Comparing the two correlation coefficients shows that the relationships are better approximated by a linear than by a monotonic function.
Reporting numerical experiments is closed by a brief study of how the correlation between and behaves for other and . Therefore, we analyze coevolutionary games with players, each for all coplayers. Interaction networks are modeled as regular graphs, whose adjacency matrices are generated algorithmically [4, 5]
. We discard graphs that are not connected and vertex–transitive, which is based on analyzing the Fiedler eigenvalue (the second–smallest eigenvalue of the Laplacian matrix
) for connectivity [9], and symmetry properties of transition matrices describing random walks onwith Markov chain characteristics for vertex transitivity
[40]. For each and , we use a set of different graphs. For , all configurations are analyzed. As analyzing all configurations could not be realized with the computational resources available in this study, for we sampled configurations uniformly at random out of the whole set. Auxiliary experiments with different samples have shown that the error made in calculating the correlations by such a sampling is very small, which is plausible as there is an increasing number of configurations with the same and . Fig. 9 shows the correlations expressed as Pearson’s , which are averages over all 1000 graphs. We see that the results generally support the claims made for the 6 graphs with and . Particularity for coplayers the curves resemble those for the Frucht, the Dürer and the Tietze graph, see Fig. 8(a). For (cycle graphs), the strength of correlation increases with increasing (except for ), while for the correlations remain strong for an increasing number of cooperators, similar to the results for the vertex–transitive graphs. The results in Fig. 9 suggest that the correlations reported are not specific for the 6 graphs listed in Tab. 1 but apply more generally. Further works could study if this is also the case for players.4 Discussion
Whether or not cooperation is favored over defection in an evolutionary game on –regular graphs with vertices ( players) can be assigned in the case of weak selection by the structure coefficient for any arrangement of cooperators and defectors described by a configuration [11, 35]. As the emergence of cooperation is always opposed for BD updating, the focus here was on the apparently more interesting case of DB updating. For DB updating, can be calculated with polynomial time complexity by Eq. (4). It was illustrated using the example of 6 different interaction networks modeled as regular graphs that for 2 cooperators (and up to cooperators), the structure coefficients may vary over configurations, even if the number of cooperators is the same. Thus, if we interpret a single cooperator among defectors as a lone mutant, the results reported apply to configurations of multiple mutants as well. Furthermore, numerical results were given suggesting that the structure coefficients can be linked to the cooperator path length by an inverse linear relationship. Numerical experiments with a multitude of interaction graphs algorithmically generated with players and coplayers confirm this result. As this implies large values of for a low cooperator path length , and vice versa, the result provides another rationale for clusters of cooperators giving different fixation properties than cooperators that are sparsely distributed on the evolutionary graph. For evolutionary games on lattice grids clusters of cooperators have better chances to survive [16, 17, 21, 30]. Lattice grids can be understood as –regular graphs for Von Neumann neighborhoods and as –regular graphs for Moore neighborhoods and clusters of cooperators have smaller cooperator path length than cooperators that are isolated. Thus, the results for lattice grids are consistent with the findings for regular graphs in general.
The approach presented in this paper gives a methodology for analyzing how properties of interaction networks relate to arrangements of more than one cooperator and subsequently to fixation properties of the evolutionary game specified by the graph. It may add a new perspective at finding best conditions to favor cooperation. Within the framework and interpretation of evolutionary graph theory given in Sec. 2, this means looking for the best arrangement of cooperators on a given interaction graph, but possibly also for the best interaction graph with a given arrangement of cooperators, or the best combination of arrangement and graph. With the payoff matrix (1) and the structure coefficient cooperation is favored if
(6) 
Note that this condition indicates that the fixation probability of cooperation is higher than , but entails no statement about the value of the fixation probability itself. As known for the Frucht and for the Tietze graph, even for a single cooperator the same structure coefficient does not imply the same value of the fixation probability of cooperation as it may vary over initial configurations [23]. For 2 cooperators and more, the same appears plausible and some preliminary numerical tests confirm this.
To refocus on whether or not cooperation is favored, consider, for example, the parametrization of the payoff matrix (1) used in the seminal work of Axelrod [3], with later modifications by Nowak & Sigmund [27], for Prisoner’s dilemma (PD) and snowdrift (SD) games: for PD and for SD. We obtain that cooperation is favored over defection if for the PD and if for the SD game. In view of the findings given in Sec. 3 this means that 2 cooperators attain fixation on none of the 6 graphs considered for the PD game, but on all 6 graphs for the SD game. For , all , and all . For a single cooperator the same applied as . Consequently, for the SD game fixation of cooperation occurs for more than 2 cooperators as well. For the PD game and cooperators, the situation is different. As for all 6 graphs and all for the interaction networks considered, see Tab. 1, cooperation is never unconditionally favored, that is for all configurations and no matter how many cooperators there are. However, except for the Franklin graph there are configurations that favor cooperation with , see Fig. 6. For instance, for the Frucht graph and cooperators, the configuration shown in Fig. 1(d) has . The same applies to the other 2 configurations occupying the other 2 triangles of the Frucht graph. Similarly, for the other graphs in Figs. 2 and 5 as well as certain numbers of cooperators , there are some other configurations favoring cooperation. Comparing the graphs suggests again a difference between the vertex–transitive graphs and those just regular with the 3 vertex–transitive graphs considered more rarely favoring cooperation with the parameter of the payoff matrix given. Whether this result is generally true for vertex–transitive graphs or an anomaly due to special properties of the Franklin and the Möbius ladder graph should be addressed by future work. Thus, if we were just to consider the 6 interaction networks listed in Tab. 1, the Frucht graph with cooperators and configuration would be most favorable for cooperation as this combination of network and configuration yields the largest value . Due to the symmetry of with respect to another solution is the configuration with and the same . In other words, if we view the parameter of the payoff matrix of the PD game as the temptation to defect, then these 2 configurations would favor cooperation on the Frucht graph even if the temptation gets as high as . Moreover, for the PD game with the reward to cooperate and the punishment to defect, cooperation is Pareto–efficient as . Thus, the 2 configurations on the Frucht graph would favor cooperation even if the value of the difference between reward and punishment is as low as . In a similar manner the effect of configuration and network on relations between the parameters of the payoff matrix can be studied, for instance on risk–dominance .
Another interesting aspect and pointer to future work comes from analyzing properties of the configurations with largest . We notice that indicates that the 5 cooperators are situated on the overlap of the square and a triangle of the Frucht graph, while the 7 cooperators of are on the overlap of 2 triangles and the hexagon (cycle of length 6). These results relate to findings showing for 3 and 4 cooperators that largest structure coefficients are frequently found for the cooperators located on triangles or squares of the evolutionary graph, see for instance Fig. 1(d), or Fig. 2(d), or Fig. 4(c). Naturally, if the cooperators are located on these cycles, such overlaps of cycles with different length coincide with low values of the cooperator path length. These results suggest the hypothesis that particularly large values of can be obtained if the interaction graph possesses a certain abundance of cycles of length up to the number of cooperators . Large values of appear if the cooperators are located on these cycles, particularly if the cycles overlap, that is cycles of different length share vertices. Further work is needed for clarification.
This kind of discussion immediately leads to a related question, which is to design optimal conditions of cooperation. Freezing the values of the payoff matrix (1), the design space encompasses the initial configurations as well as the interaction networks. For games with 2 strategies and players the number of configurations is and just considering evolutionary graphs that are regular their number grows for with [34, 41]. Thus, while for a small number of players, such as as in the examples discussed, and pre–selected interaction networks, a calculation of all can be done, solving the design problem in general becomes numerically expensive for intermediate and infeasible for getting large. For designing configurations, there are, however, some further aspects. For , the structure coefficients uniformly tend to [11, 28]. Put differently, whether cooperation is favored ceases to depend on which configuration the game has initially and thus solving the design problem ceases to be particularity meaningful for the number of players getting large. Another aspect is the scaling of the number of configurations over the number of cooperators , which is governed by Eq. (2). Accordingly, grows polynomially with for all , for instance by for or for . Only for and , grows exponentially. Thus, from a numerical point of view designing an optimal configuration for a given number of cooperators should remain feasible, except for , at least for not as large as the structure coefficients becoming uniform with . The results also have the potential to inform finding interaction graphs that are better suited than a random selection to either promote or suppress cooperation. For instance, the Franklin graph completely suppresses cooperation in the PD game discussed above, while the Frank graph much more promotes cooperation. The analysis suggests than graphs with overlapping cycles of length up to the number of cooperators produces more favorable structure coefficients than graphs with few or separate cycles. Thus, we may potentially prescribe interaction networks with these properties and thus reduce the size of the design space.
The results given are for evolutionary game dynamics with weak selection and interaction networks modeled by regular graphs. The paper closes with brief comments on going beyond regular graphs and weak selection. Recently, an approach to assign fixation properties using coalescence times of random walks has been proposed that is suitable for a single cooperator and any graph structure [2]. If this technique can be extended to configurations with more than one cooperator, the approach given here would become applicable for graph that are not regular. It is known that the validity of extrapolating results from weak to intermediate and strong selection is not always possible and depends highly on game characteristics, population size and spatial heterogeneity of the network [15, 25, 42], but comparison between fixation probabilities has been shown to be fairly robust for varying intensity of selection and a single cooperator [15]. It remains to be seen if this is also valid for any arrangement of cooperators and defectors.
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