Cooperation percolation in spatial prisoner's dilemma game

The paradox of cooperation among selfish individuals still puzzles scientific communities. Although a large amount of evidence has demonstrated that cooperator clusters in spatial games are effective to protect cooperators against the invasion of defectors, we continue to lack the condition for the formation of a giant cooperator cluster that assures the prevalence of cooperation in a system. Here, we study the dynamical organization of cooperator clusters in spatial prisoner's dilemma game to offer the condition for the dominance of cooperation, finding that a phase transition characterized by the emergence of a large spanning cooperator cluster occurs when the initial fraction of cooperators exceeds a certain threshold. Interestingly, the phase transition belongs to different universality classes of percolation determined by the temptation to defect $b$. Specifically, on square lattices, $1<b<4/3$ leads to a phase transition pertaining to the class of regular site percolation, whereas $3/2<b<2$ gives rise to a phase transition subject to invasion percolation with trapping. Our findings offer deeper understanding of the cooperative behaviors in nature and society.


Introduction
Cooperation is ubiquitous in biological and social systems [1].
Understanding the emergence and persistence of cooperation among selfish individuals remains an outstanding problem. The development of evolutionary game theory has offered a powerful mathematical framework to address this problem [2]. In order to capture the interaction pattern among greedy individuals, various models have been introduced, among which the prisoner's dilemma game (PDG) has been a prevailing paradigm [3].
It has been known that the formation of cooperator clusters plays an important role in promoting cooperation in spatial games [52]. A cooperator cluster is defined as a connected component (subgraph) fully occupied by cooperators. Within clusters, cooperators can assist each other and the benefits of mutual cooperation outweigh the losses against defector. It is of paramount importance to explore the dynamical organization of cooperator clusters so as to understand the emergence of cooperation among selfish individuals. Previous studies focused on the size distribution of cooperator clusters [53,54], but pay little attention to the conditions under which a giant cooperator cluster of the order of the system size arises. In this paper, we attempt to address this issue from the perspective of the percolation theory [55,56]. In recent years, the percolation transition characterized by a large spanning cluster arising at the critical point, has been widely found in various dynamical processes, such as the spreading of disease [57], the formation of public opinion [58] and cascading failures [59]. In particular, percolation phenomena with respect to optimal cooperation level in diluted system were discovered in Ref. [60,61]. However, percolation phenomena pertaining to the emergence of cooperator clusters have not yet been studied in the framework of evolutionary games. Here, we report a percolation transition behavior characterized by the emergence of a large spanning cooperator cluster when the initial fraction of cooperators in the system exceeds a certain threshold. The phase transition can be classified to either regular site percolation or invasion percolation with trapping, depending on the temptation to defect. We substantiate our findings by systematic numerical simulations and analysis of phase transition.
The paper is organized as follows. In Sec. 2, we formalize the problem by introducing the PDG on square lattices. In Sec. 3, we study the cooperation percolation in terms of different values of the temptation to defect. Finally, conclusions and discussions are presented in Sec. 4.

Model and Methods
Individuals are located on a L × L square lattice with periodic boundary conditions. Each individual x can follow one of two strategies: cooperation or defection , described by respectively. At each time step, each individual plays the PDG with its four nearest neighbors. The accumulated payoff of individual x can be expressed as where the sum runs over the nearest neighbor set Ω x of node x and M is the rescaled payoff matrix with According to the setting in Ref. [62], initially cooperation and defection strategies are randomly assigned to all individuals in terms of some probabilities: with probability f a node is occupied by a cooperator and with probability 1 − f a node is occupied by a defector. Individuals asynchronously update their strategies in a random sequential order [63,64,65,66]. A randomly selected player compares its payoff with its nearest neighbors and changes strategy by following the one (including itself) with the highest payoff. After a period of transient time, the system will enter a steady state.
We focus on the formation of cooperator clusters in the steady state. In order to study the critical behavior regarding to the giant cooperator cluster, we employ the normalized size of the largest cooperator cluster s 1 , the susceptibility χ and the Binder's fourth-order cumulant U. These quantities are defined as follows [68]: where S 1 the size of the largest cooperator cluster, N = L × L is the system size and · · · stands for configurational averages. According to the standard finite-size scaling approach [68], the phase transition point can be identified by the Binder's fourth-order cumulant U and there are the following power-law relationships at the critical value: We can use the tool to identify phase transitions pertaining to the formation of cooperator clusters and explore the class to which the phase transition belongs.

Main results
According to our analysis of phase transition, we can separate the values of b into four regions: b ∈ (1, 4/3), (4/3, 3/2), (3/2, 2) and (2, ∞). We find that in the same region, the cooperator clusters with respect to percolation transition keep unchanged, irrespective of the value of b. In the following, we present the results in the four regions respectively.
We first explore the characteristics of cooperator clusters when 1 < b < 4/3. Figure 1 shows the snapshots of the spatial distributions of cooperators and defectors on a 200 × 200 square lattice. We can observe that the size of the largest cooperator cluster (denoted by red) expands as the initial fraction of cooperators f increases. In contrast, the size of the second largest cooperator cluster (denoted by blue) becomes larger firstly but then shrinks as f continuously increases. The non-monotonic relationship between the size of the second largest cooperator cluster and f implies the existence of a secondorder phase transition [67]. Figure 2 shows the normalized size of the largest cooperator cluster s 1 and the fraction F c of cooperators in the population, as a function of f . We see that both F c and s 1 increase toward 1 as f increases. Over a wide range of f , F c is larger than   Fig. 2, we can also find that there exists a critical value f c , below which s 1 approaches 0, while above which s 1 continuously increases as f increases. The fact that F c > 0.5 at f c suggests that it is possible for cooperators to form a giant cluster comparative to the size of the system, insofar as sufficient number of cooperators survive during the evolution. Figure 3 shows the normalized size of the largest cooperator cluster s 1 , the susceptibility χ and the reduced Binder's fourth-order cumulant U as a function of f for different lattice sizes L. Figure 3(a) shows that s 1 is almost the same for different  values of L for large f . However, when f is blow some value, s 1 decreases as L increases. Figure 3(b) shows that the susceptibility χ reaches a maximum at some value of f for different values of L. Moreover, one can see that the value of f that corresponds to the peak of χ increases with L. The critical value f c can then be identified in Fig. 3(c), where the curves of the reduced forth-order cumulant U for different L intersect with each other [69,70]. The intersection point gives f c ≃ 0.7551 for 1 < b < 4/3. and (b), we obtain the critical exponents β/ν ≃ 0.051 and γ/ν ≃ 0.908, which are very close to the critical exponents of regular site percolation (β/ν ≃ 0.052 and γ/ν ≃ 0.896) [67]. These results indicate that the cooperation percolation belongs to the same universality class as the regular site percolation when 1 < b < 4/3.

The case of 3/2 < b < 2
We explore the cooperation percolation when 3/2 < b < 2. Figure 5 shows the normalized size of the largest cooperator cluster s 1 , the susceptibility χ and the reduced Binder's fourth-order cumulant U as a function of f for different lattice sizes L. As shown in Fig. 5(a), s 1 decreases as the system size increases when f is below some value. The susceptibility χ reaches the maximal value at some value of f [see Fig. 5(b)] and the reduced fourth-order cumulants cross at the critical point f c ≃ 0.9553 [see Fig. 5(c)]. Figure 6 shows s 1 and χ as a function of the system size N at the critical value f c . In Figs. 6(a) and (b), we obtain the critical exponents β/ν ≃ 0.082 and γ/ν ≃ 0.856, which are close to the critical exponents of invasion percolation with trapping (β/ν ≃ 0.084 and γ/ν ≃ 0.832) [71]. These results demonstrate that the cooperation percolation belongs to the same universality class as invasion percolation with trapping in the region of 3/2 < b < 2.  One can observe that F c is much lower than 1 and s 1 ≈ 0 even when f is very close to 1. Figure 7(b) shows s 1 as a function of N when initially there is only one defector in the system. We can find that s 1 decreases as N increases. Figure 7 demonstrates that the percolation threshold for 4/3 understand why the threshold is 1 in the parameter region, we explore the evolution of the spatial distributions of cooperators and defectors on a square lattice with given a single defector at the center initially (t = 0). Figure 8 shows that eventually a giant reticular defector cluster is formed and many small cooperator clusters are separated by defectors. This result is in agreement with that reported in Ref. [72], i.e., defectors can spread over the system even initially there is only one defector in the region of 4/3 < b < 3/2. Hence only the absence of defector can lead to a large cooperator cluster that covers the whole lattice, accounting for the phase transition at f c = 1.
In the case of b > 2, cooperators cannot survive when f < 1 (results are not shown here), implying that even one defector can lead to the extinction of cooperators in the ocean of cooperators when b > 2. We can thus infer that the percolation threshold for b > 2 is also f c = 1.

Conclusions and Discussions
In conclusion, we have explored the formation of cooperator clusters in the prisoner's dilemma game, finding that the process of establishing a giant cluster pertains to the percolation behavior. In particular, when the initial fraction of cooperators in the system exceeds a critical threshold, there arises a giant spanning cooperator cluster, resulting from the merging of many small cooperator clusters. The phase transition behavior is validated by various scaling laws at the critical point, such as the normalized size of the largest cooperator cluster and the susceptibility scale with the system size. Strikingly, the phase transition belongs to different universality classes, depending on the temptation of defect. The results on square lattices demonstrate that the phase transition is subject to the class of regular site percolation when 1 < b < 4/3 or invasion percolation with trapping when 3/2 < b < 2. Whereas in the parameter region 4/3 < b < 3/2 and b > 2, the percolation threshold is 1, indicating that even one defector can prevent the formation of large cooperator clusters. Interestingly, we found that the partition of the parameter region in terms of percolation is exactly the same as previous findings based on the chaotic spatial patterns in literature [4,72]. The agreement offers an underlying connection between the phase transition and the spatial chaos in evolutionary games.
Our findings presented here raise a number of questions, answers to which could further deepen our understanding of the persistence and dominance of cooperation in terms of a large cooperator cluster. For example, for strategy updating rules rather than the currently used best-take-over rule, such as the Fermi rule [63], if the percolation transition remains? If the answer is positive, will the universality class change? Another significant question pertaining to the network structure is how does the structural property affect the percolation phenomena, e.g., small-world and scale-free topology. Taken together, our results indicate that cooperation in many evolutionary games can be explored from the perspective of cooperator clusters in the combination with the tools for quantifying phase transition and percolation, opening new avenues to deepening our understanding of cooperative behaviors widely observed in many aspects in nature and society.