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Evolutionary dynamics of the cooperation clusters on interdependent networks
Physica A 517 (2019) 132–140
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Physica A journal homepage: www.elsevier.com/locate/physa
Evolutionary dynamics of the cooperation clusters on interdependent networks ∗
Zhao Jinqiu a , Luo Chao a,b , , Zheng Yuanjie a a b
School of Information Science and Engineering, Shandong Normal University, Jinan 250014, China Shandong Provincial Key Laboratory for Novel Distributed Computer Software Technology, Jinan 250014, China
highlights • The effect of interdependence on dynamics of strategy clusters. • Self-organized interdependence investigated from the perspective of resource allocation. • Co-evolution of strategies and interdependence by strategy-independent rule.
article
info
Article history: Received 21 July 2018 Received in revised form 28 September 2018 Available online 9 November 2018 Keywords: Interdependent networks Prisoner’s dilemma game Cooperation clusters Resource allocation
a b s t r a c t In this article, the evolutionary dynamics of cooperation (defection) clusters on interdependent networks is studied. The emergence of spatial structures and distributions of strategy clusters have important influence on the cooperative behaviors. Different from the existing works, we focus on the effect of the structure of interdependent networks on the dynamics of strategy clusters. In order to reveal the co-evolution of game strategies and interdependent structure, the interdependent strength is taken as a kind of limited resources, which can be reallocated among players along with the game process but keep the total quantity unchanged. Through quantitative and qualitative analysis, we find that cooperation clusters are considerably affected by the overall interdependent strength of dynamical system as well as the strategy distributions on two layers. Furthermore, the size of the largest cooperation cluster is related to the kind of strategy pairs. By studying the interdependent strength among clusters, we find that larger clusters have stronger resource concentration ability. At the same time, players with more resources tend to be more cooperative. We also discuss the related microscopic system properties and demonstrate the related discussion by various analysis tools. © 2018 Elsevier B.V. All rights reserved.
1. Introduction Cooperation is a kind of common social behaviors in community. However, it is still an open problem that how to understand the emergence and persistence of cooperative behaviors in groups when defection can bring short-term benefits to rational individuals. In the last decades, different social dilemmas have been proposed, such as the prisoner’s dilemma game [1], the snow-drift game [2] and the public goods game [3] etc., which provide various models with different characteristics to study the behaviors of individuals in society. Meanwhile, complex networks as the abstract models of structured populations achieve rapid development, which provides a new perspective to gain insight into the complexity ∗ Corresponding author at: School of Information Science and Engineering, Shandong Normal University, Jinan 250014, China. E-mail address: [email protected] (C. Luo). https://doi.org/10.1016/j.physa.2018.11.018 0378-4371/© 2018 Elsevier B.V. All rights reserved.
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of dynamics. Based on structured models, works on transmission dynamics [4–6], evolutionary computation [7–10] and control [11–13], etc., have been developed rapidly. Since the network reciprocity was first proposed by Nowak et al. [14], i.e. cooperators in structured populations spontaneously aggregate into compact clusters to defense the invasion of defectors, the evolutionary games based on structured populations are attracting the attentions of researchers in this community [15]. The resources in reality are often limited. Meanwhile, the allocation of limited resources is usually related to the competition, where cooperation and defection as two common strategies would be chosen. How to make use of limited resources to maximize the cooperators is a valuable problem. In fact, it has been widely used, such as the applications of the scheduling problems [16], power systems [17,18], and the optimization problems [19]. During the evolution of game dynamics, cooperators spontaneously gather into compact clusters to resist the invasion of the defectors in structural groups, which have been considered to be the key role of cooperative evolution in multi-layer network [20–24]. But, it is few studies on the effect of resource allocation on the cooperation clusters. In recent years, multi-layer networks have gradually become a hot research topic. Different from single layer network, the multi-layer network consists of two or more layers. Apart from the connection between nodes in the same layer, there exists external links connecting the nodes residing on different layers. Nodes in the system represent individuals in the society, and edges represent social relationships. In this sense, multi-layer network can describe the systems in society more vividly in comparison to the single one. In the past few years, many achievements have been made in the study of evolutionary game on multi-layer networks. In [15], Wang et al. devoted attention to evolutionary games on multi-layer networks, and illustrated that the pattern formation and collective behavior were of great significance for the promotion of cooperation. In [25], Xiang proposed an evolutionary game model of dual-dynamic complex networks, and revealed the impact of defective leaders on average levels of cooperation. In [20], Wang et al. proposed that the evolution of public cooperation on two interdependent networks connected by a utility function. In [24], different ways of resource allocation were proposed and the effects of selforganization interdependence on the cooperation were further studied. Gómez-Gardeñes et al. analyzed the implementation of cooperative behavior interaction in mixed interactive networks in [26]. Cooperation clusters play an important role in helping partners to resist the invasion of defectors and improve the fraction of cooperation in the system [14]. A cooperation (defection) cluster is defined as a connected component that is completely occupied by cooperators (defectors) [27]. Nord et al. used mathematical methods to fill an infinite and uniform lattice, based on which the properties of clusters under certain rules were studied [28]. Yang et al. studied the transformation of cooperation penetration in different divisions of varying temptation values b based on single-layer network [27]. Based on small-world network, Grilo et al. founded that strategy exchanges contributed to the destruction of compact clusters would be favorable to cooperator agents [29]. Generally, the existing works are mainly based on single-layer network and dynamics of strategy cluster are rarely discussed. In this article, a two-layer interdependent network is implemented, where interdependent strength of external links is considered as a kind of limited resource that can be reallocated but the total amount will not be changed. Our research focuses on the formation and evolution of cooperative behavior with the change of interdependence of external links. Through quantitative and qualitative analysis, we will reveal the influence of independence resources on the dynamics of cooperation clusters. As we mentioned, the self-organization interdependence and the penetration of defectors are not all factors that determine the size of the largest cooperation cluster, which is also related to the strategy pairs residing two-layers. Some strategies will promote the formation of the largest cooperation cluster, while others will hinder the process. By exploring the interdependent strength among clusters, we find that larger clusters have stronger ability for resource concentration. In order to prove the correlation between the distribution of cooperation clusters and interdependent strength, we have compared the distribution of the first and the second largest cooperation clusters with the evolution of interdependent strength. Meanwhile, we find that the individuals are more inclined to cooperate with others when they possess more resources. The above results have also been illustrated in snow-drift game. This article is as follows: in the Section model, we mainly introduce the models and formulas used in the study. In the main results, we analyze the effects of interdependence resources on the size of the cluster, and reveal the factors that affect the size as well as the internal properties of the cooperation clusters. We summarized the results in the discussion. 2. Model In this article, the model is composed of two square lattices of size L × L with periodic boundary. In the proposed system, player x is connected to its four nearest neighbors located on the same layer, and the corresponding player x′ on another layer by means of an external link. The interdependent strength between x and x′ is represented by the parameter α (α ∈ [0, 1]), where α = 0 represents that there is no connection between two players and α = 1 represents the strongest interdependent strength. In our model, interdependent strength can represent some certain resources, e.g. the energy, the wealth or the natural resources, which can be relocated among players by repeated game but keep a fixed total amount. Obviously, the greater the interdependence strength the players have, the more chance to win in the game. The total interdependence resource in the system is as follows: 2
Γ =
L ∑ i=1
α i.
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Fig. 1. As the increase of the temptation to defect b, the fraction of cooperators and the proportion of largest cooperation clusters vary under the different initial interdependent strength. The size of the network is L = 200.
We define δ as the basic unit of resource transfer. At the beginning of game, the interdependent strength of all individuals is assumed to be α = γ and the total resources in the whole system are Γ = L2 γ . As disused above, during the process of evolution, interdependent resources can be reallocated among players by means of a concise rule based on the results of game, but the total amount of resources will be fixed, which can reasonably describe some dynamics in reality by illustrating the reallocation of resources in social activities. In the previous works [30], the introduction of ‘‘teaching rule’’ in evolutionary games shows rich dynamical features compared with ‘‘copying rule’’. Therefore, the concept of ‘teaching activity’ is adopted in following discussion. Game strategy, i.e. cooperation or defection, is assigned to each player on networks with the same probability. At each time, players implement the PDG with their four nearest neighbors, and the payoff matrix is as follows:
[ M=
RAMZ @PS TAMZ @PP
]
where the reward for mutual cooperation R = 1, the temptation to defect T = b (1 < b ≤ 2), the punishment for mutual defection P = 0 and sucker’s payoff S = 0. Initially, a player is randomly selected to play the PDG with its nearest four neighbors to gain payoff pxt . At the same ′ time, we calculate the corresponding payoff on the other layer pt x . The Fitness of player x is calculated as ft (x) = pt x + x′ αxx′ pt [23,31]. Similarly, a player y from four nearest neighbors of player x is randomly selected and calculate its payoff ′ y ft (y) = pt + αyy′ pt y . Subsequently, the strategy of player x is try to transfer to its neighbor y with a certain probability as W ( x → y) =
1 1 + exp {[ft (y) − ft (x)] /k0 }
where k0 = 0.1 represents noise factor. When the player x succeeds in teaching its strategy to y, it will obtain the basic unit δ of resources at a certain probability under the condition of αxx′ + δ ≤ 1 and αyy′ − δ ≥ 0. Meanwhile, player y will lose the corresponding resources to keep the total number of interdependence resources unchanged. Here, we focus on the influence of interdependent strength on the size of cooperation clusters. Sˆ1 is defined as the number of players in the largest cooperation cluster and Sˆ2 as the number of players in the second largest cooperation clusters. s1 and s2 represent the proportions of the largest cooperation cluster and the second largest cooperation cluster, respectively. The formula is as follows: s1 =
Sˆ1 L2
, s2 =
Sˆ2
L2 Simulations results are obtained from sufficiently large system sizes (from L = 200 to L = 400) to avoid finite size effects. The equilibration is required up to 105 MCS, and final results were averaged over up to 20 independent realizations. 3. Main results In order to study the impact of the interdependence resources on the size of the cooperation clusters, we study the varying of s1 and fc with the increase of b at γ = 0.2, 0.5, 0.8. The trend of s1 and fc is generally identical, which gradually decreases with the increase of b until the cooperators disappear. In Fig. 1(a)∼(c), the cooperators and cooperation clusters disappear when b = 1.21,1.24,1.196, respectively. Due to the penetration of the defectors, the cooperation clusters are divided by the defectors, which lead to the decline of s1 faster than fc . In previous studies [24], too much or too little interdependence resources will cause the system to lose interaction of self-organization, which hampers the survival of cooperators. The cooperation cluster itself plays an important role in improving the cooperation level, which also reflects the size of cooperation cluster to some extent. We can conclude that γ = 0.5 is more beneficial for the cooperators to cluster together. When it comes to the analysis of Fig. 1(b) and (c), their fc are both around 0.5 when b = 1.15, but s1 in Fig. 1(b) is 0.15 and 0.072 in Fig. 1(c). It can be further explained that in addition to the penetration of the defectors, the number of interdependence resources itself plays a signification role in the size of the cooperation clusters.
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Fig. 2. The fraction of the cooperators and the proportion of the largest cooperation cluster as a function of the initial interdependent strength γ are shown under different temptation to defect b.
Fig. 3. In panel (a), in dependence of the initial interdependent strength, the numbers of different strategy couples (C–C, C–D, and D–D couples) in whole system are presented. In panel (b), the ratio of the numbers of C–C and C–D strategy couples in the largest cooperation cluster is studied in the form of Ncc /Ncd. The temptation to defect is b = 1.09 and the system size is L = 200.
In order to further investigate the impact of interdependence resources, the fraction of the cooperation and the proportion of the largest cooperation cluster as a function of γ are shown in Fig. 2. In Fig. 2(a), When b = 1.06, fc and s1 reach the highest values at γ = 0.5. As the value of γ continues to increase, the values of fc and s1 are decreasing. The similar trend can be found in Fig. 2(b) and (c). As temptation to defect b takes different values, the proportion of cooperation clusters will not increase significantly with the increase of interdependence resources, just like the cooperation level, but at γ = 0.4 it turns out to be different. In order to further explore the internal characteristics of the cooperation clusters which affect the cooperation clusters, we will gain insight into the structure of cooperation clusters. There are three kinds of strategy couples residing in two layers: C–C (Cooperation–Cooperation), C–D (Cooperation–Defection) and D–D (Defection–Defection). Fig. 3(a) shows the number of different strategy couples between the two layers in the whole system in dependence of the initial interdependent strength γ with b = 1.09. It can be observed that the variance of the number of C–C is the same as the one of s1 in Fig. 2(b), where the highest value occurs at γ = 0.4, and the number of C–C is greater than the ones of C–D and D–D. In order to study the quantitative relationship between different strategy couples, we further explore the ratio of the numbers of C–C and C–D strategy couples in the largest cooperation cluster shown in Fig. 3(b), where Ncc , Ncd represent the numbers of C–C and C–D couples in the largest cooperation cluster, respectively. The value of Ncc /Ncd increases with the increase of γ , and achieves its highest value until γ = 0.4. Therefore, different strategy couples can promote or inhibit the formation of cooperation clusters. So, it can be speculated that for C–C, the relevant cooperation groups can be easily constructed, and the size of cooperation clusters can be significantly increased through the emergence of interaction reciprocity [32]. To further prove that C–C does have a significant effect on the size of the cooperation clusters, we calculate the average number of cooperators around the different strategies with varying the initial interdependent strength γ shown in Fig. 4. We find that, under different interdependence resources, the average number of cooperators around C–C couples is always the highest, followed by the one of C–D couples. And, the average number of cooperators around D–D couples is the lowest. Because of the influence of feedback, it will enhance interaction reciprocity or information sharing [33] to promote cooperative behavior. In previous studies [34,35], the authors explain the relationship between the circular C-cluster growth and network reciprocity, which can explain the END-period and EXP-period of dynamics which directly attribute to emerging network reciprocity in evolution game. It is difficult for C–D to promote the formation of cooperative groups, and D–D will
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Fig. 4. ⟨Nc⟩ as a function of the initial interdependent strength are shown, where ⟨Nc⟩ are the average numbers of cooperators around C–C, C–D, and D–D couples in the system. Dark blue represents the average number of cooperators around C–C couples. Light blue represents the average number of cooperators around C–D couples, and yellow represents the average number of cooperators around D–D couples. The temptation to defect is b = 1.09. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
hinder the survival of cooperators. This further proves the size of the largest cooperation cluster is related to the composition of strategy couples between the two layers. It should be noted that when the temptation to defect b is changed, the fraction of cooperation and the size of the largest cooperation cluster would be varied, but the overall dynamical behaviors can be maintained. In Fig. 5(a), the ratio of the number of external links with the interdependent strength α to the total amount of external links in the top five cooperation clusters are described, where a ‘‘seesaw effect’’ is shown. The ratio in all the five cooperation clusters have a large value at α = 0, α = 1, and a small value when α ∈ [0.1, 0.9]. Moreover, the larger the cooperation clusters is, the more uniform the middle part is distributed. As to the largest cooperation cluster, the sum of the ratios at α = 0 and α = 1 reaches 70%. The ratio shows an increasing trend around α = 0.8 and has a large increase in the range of α = [0.9, 1]. As to the second largest cooperation cluster, the sum of the ratios at α = 0 and α = 1 reaches 90%, which shows a large increase in the range of α ∈ [0.9, 1]. As to the third largest cooperation clusters, the sum of the ratios at α = 0 and α = 1 reaches 70%. And it is equal to the number of individuals between 0.2 and 0.5, which changes little but is not zero. When α ∈ [0.6, 0.9], the ratio grows slightly but has a significant increase in the range of α ∈ [0.9, 1]. Since the fourth and fifth largest cooperation clusters are smaller than the first three large cooperation clusters, the proportion of the players numbers is unevenly distributed at α = [0.1, 0.9]. Since each player has the same interdependent strength α = γ = 0.5, and the total resource in the system is Γ = 2000. When the system is stable, the proportion of the largest cooperation cluster is 60%. Even if all the resources are concentrated together, the proportion of the individual whose interdependence resource reaches 1 only accounts for 50%, which means that there are many players in the largest cooperation group whose interdependent strength is less than 1. By accounting the proportion of players whose interdependent strength lie in α ∈ [0.8, 1], we find that the result of the largest cooperation cluster is 58.65%, and the second largest cooperation cluster is 57.38%. It is obvious that the limited resources in the largest cooperation cluster are more rich compared with the ones in the second largest cooperation cluster. In order to further study the resource concentration of the cooperation clusters, we use the formula
P(Rα,γ ) =
L2 P(α )α L2
γ
=
α P(α ) γ
where L represents the size of the network. In Fig. 5(b), we see the distribution of α P(α )/γ is obviously different from the normal distribution, exponential distribution and the binomial distribution. The cooperation clusters have the lowest concentration of resources at α = 0. The concentration of resources in the range of α = [0.1, 0.8] is not obvious, but, when α reaches 0.9, there is a big jump and the concentration reaches its highest level at α = 1. Compared with the first three largest cooperation clusters, the fourth and fifth largest cooperation clusters have the lower concentration of resources at α = 1. To verify the above discussion, α P(α )/γ calculated in the first tenth largest cooperation clusters at α = 1 is shown in Fig. 6. In Fig. 6(a), at γ = 0.1, 0.2, the results show irregular fluctuations. In Fig. 6(b), the results of the top six largest cooperation clusters are almost identical, for instance, the results shown for γ = 0.5. In Fig. 6(c), the resource concentration abilities of the top five largest cooperation clusters are generally identical when γ = 0.8, 0.9. With different quantities of interdependence resources, the resource concentration ability of the first and second largest cooperation clusters are similar compared with other cooperation clusters. These different results can attribute to the self-organizations of interdependence. Generally speaking, no matter how the overall interdependence resources existing in the system changes, larger cooperation clusters have similar resource concentration ability.
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Fig. 5. The distribution of interdependent strength of the top five cooperation clusters as a function of self-organized interdependence. In panel (a), with the increase of α , the variance of P(α ) in the top five cooperation clusters. P(α ) represents the proportion of individuals whose interdependent strength is α in its cooperation clusters. Panel (b) shows the variance of α P(α )/γ as a function of α , where α P(α )/γ represents the ratio of the accumulated amount of interdependent strength existing in external links of strength α to the total amount of interdependent strength in the whole system.
Fig. 6. α P(α )/γ calculated in the first ten largest cooperation clusters, where α = 1.
Next, we study the distribution of strategies and the evolution process of interdependent strength on the two-layer network denoted by layer A and layer B. Here, the first and second largest cooperation clusters are taken as examples. In Fig. 7, at 1 MCS, the interdependent strength of the whole system is 0.5. Due to the randomness of the initial strategy choices, we can see the different distribution of cooperators and defectors on the networks A and B. As the evolution time goes by, it comes to 10 MCS, and the snapshot shows a variety of colors. The defectors gather together to become a great defection cluster, while the cooperators also cluster to against the invasion of the defectors. The size and distribution of the cooperation clusters vary. At 50 MCS, with the process of the accumulation of interdependent resources, a larger number of cooperators and defectors gather together to form clusters, respectively. The above process would be identical on two layers. In Fig. 7(c), we can see that the yellow regions where the interdependent strength approaches 1 are also the places where the first and second largest cooperation clusters appear (shown in Fig. 7(g) and (k)). The first largest cooperation clusters on the two-layer network locates at the same regions. Meanwhile, it can be observed that the blue regions in Fig. 7(c), where the interdependent strength is close to 0, are mainly occupied by defectors. At 10 000 MCS, it can be found that color distinction is more obvious in Fig. 7(d). The interdependent resources owned by players are approaching 0 or 1, and there are few middle values. Furthermore, the regions with the interdependent strength 1 are almost completely connected into a whole. The distribution of cooperators and defectors in Fig. 7(h) and (l) is generally similar. The first largest cooperation clusters lies in the yellow area shown in Fig. 7(d). However, the distribution of the first and second largest cooperation clusters on A and B is distinct, where the size of the first largest cooperation clusters on layer B is significantly larger than that on layer A, which can attribute to the randomness brought by the Monte Carlo method. Finally, in order to verify the robustness of the above results, snow-drift game is carried out. At first, every player has the same interdependent strength α = γ = 0.5. Only after 10 MCS, the distribution of interdependent strength has obviously changed. The cooperation level of snow-drift game is higher than that of prisoner’s game. In the two-layer network, the defectors are difficult to gather together, while cooperators begin to form a large cooperation cluster. At 50 MCS, a large connected component region in yellow appears in Fig. 8(c), where the largest cooperation clusters are emerging shown in Fig. 8(g) and (k). The defectors are surrounded by the cooperators. At 100000 MCS, only two kinds of interdependent strength, i.e. zero or one, can be observed. It shows that interdependence resources allocation has reached the serious polarization, where the yellow areas occupy almost all the places and only a small number of defectors survive. In Fig. 9, we studied the distribution of interdependent strength of cooperation clusters in snow-drift game. Since the cooperation level of snow-drift game is high, almost all cooperators have been connected into a cluster. In Fig. 9(a), the
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Fig. 7. (a)∼(d) is snapshots of the distributions of the interdependent strength of all the external links between two networks. (e)∼(h) is snapshots of the distributions of the cooperation clusters and the defectors cluster on network A. (i) ∼(l) is snapshots of the distributions of the cooperators cluster and the defectors cluster on network A. We provided color bars for reference to the interdependent strength. Yellow denotes the largest cooperation cluster, orange denotes the second largest cooperation cluster, light blue denotes the other cooperation clusters and dark blue denotes the defection clusters. At first, every individual has the same interdependent strength α = γ = 0.5. The temptation to defect is b = 1.09 and the system size is L = 200. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
proportion of individuals with different interdependent strength in this cluster is shown. In Fig. 9(b), a large amount of interdependence resources are concentrated until α = 1. At α = 0, the value of α P(α )/γ is zero. And the ability of resource concentration is not obvious when α ∈ [0.1, 0.8]. At α = 0.9, the ability for resource concentration has a big jump, and the value of α P(α )/γ reaches its highest value at α = 1. This is consistent with the results we obtained in the prisoner’s game, which indicates the finding has robustness. 4. Discussion In this article, based on interdependent networks, we have studied the dynamics of strategy clusters. In the proposed model, interdependent strength as limited resources can be reallocated among players during the evolutionary process and co-evolve with evolutionary game. Different from the existing works relate to strategy clusters, besides of investigation on the formation and distribution of clusters, the impact of the co-evolution of interdependent strength and game strategies on the dynamics of strategy clusters have been detailedly discussed. We illustrate that the size of the largest cooperation cluster is related to the kind of strategy pairs residing on the two layers, where the formation of C–C can establish a local structure similar to network reciprocity, which would greatly promote the formation of cooperation clusters. Besides, during the evolution process, the larger clusters have stronger ability for resource concentration, which is similar to the Matthew effect. And, the players with more resources are inclined to be cooperative. By a series of experiments, the relationship between the cooperation clusters and the interdependent strength has been illustrated. The results show the formation of
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Fig. 8. (Snow-drift Game) In the evolution process, the interdependent strength of external links between two networks and the distribution snapshots of the first and second largest cooperation clusters. Panels (a)∼(d) are the evolutionary processes of interdependent strength, and panels (e)∼(h) are the distribution and evolution process of the first and second largest cooperation clusters on network A. Panels (i)∼(l) are the distribution and evolution process of the first and second largest cooperation clusters on network B. We provide color bars for reference to the interdependent strength. Yellow denotes the largest cooperation cluster, orange denotes the second largest cooperation cluster, light blue denotes the other cooperation clusters and dark blue denotes the defector clusters. At first, every individual has same interdependent strength α = γ = 0.5. b = 1.4, c = 0.8, and the system size is L = 200. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 9. The distribution of interdependent strength of cooperation clusters in snow-drift game. Panel (a) shows P(α ) as a function α , where P(α ) represents the proportion of individuals whose interdependent strength is α in the cooperation cluster. Panel (b) shows the variance of α P(α )/γ in dependence on α , where α P(α )/γ shows the distributions of accumulated amount of interdependent strength as a function of α .
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cooperation clusters, especially for larger clusters, is considerably affected by the distribution of interdependent strength. Furthermore, the experiments utilizing snow-drift Game further verify the robustness of the results. Acknowledgments This research is supported by the National Natural Science Foundation of China (Nos: 61402267; 61572300; 81871508; 61773246); Shandong Provincial Natural Science Foundation (ZR2014FQ004); Major Program of Shandong Province Natural Science Foundation (No. ZR2018ZB0419). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35]
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Physica A journal homepage: www.elsevier.com/locate/physa
Evolutionary dynamics of the cooperation clusters on interdependent networks ∗
Zhao Jinqiu a , Luo Chao a,b , , Zheng Yuanjie a a b
School of Information Science and Engineering, Shandong Normal University, Jinan 250014, China Shandong Provincial Key Laboratory for Novel Distributed Computer Software Technology, Jinan 250014, China
highlights • The effect of interdependence on dynamics of strategy clusters. • Self-organized interdependence investigated from the perspective of resource allocation. • Co-evolution of strategies and interdependence by strategy-independent rule.
article
info
Article history: Received 21 July 2018 Received in revised form 28 September 2018 Available online 9 November 2018 Keywords: Interdependent networks Prisoner’s dilemma game Cooperation clusters Resource allocation
a b s t r a c t In this article, the evolutionary dynamics of cooperation (defection) clusters on interdependent networks is studied. The emergence of spatial structures and distributions of strategy clusters have important influence on the cooperative behaviors. Different from the existing works, we focus on the effect of the structure of interdependent networks on the dynamics of strategy clusters. In order to reveal the co-evolution of game strategies and interdependent structure, the interdependent strength is taken as a kind of limited resources, which can be reallocated among players along with the game process but keep the total quantity unchanged. Through quantitative and qualitative analysis, we find that cooperation clusters are considerably affected by the overall interdependent strength of dynamical system as well as the strategy distributions on two layers. Furthermore, the size of the largest cooperation cluster is related to the kind of strategy pairs. By studying the interdependent strength among clusters, we find that larger clusters have stronger resource concentration ability. At the same time, players with more resources tend to be more cooperative. We also discuss the related microscopic system properties and demonstrate the related discussion by various analysis tools. © 2018 Elsevier B.V. All rights reserved.
1. Introduction Cooperation is a kind of common social behaviors in community. However, it is still an open problem that how to understand the emergence and persistence of cooperative behaviors in groups when defection can bring short-term benefits to rational individuals. In the last decades, different social dilemmas have been proposed, such as the prisoner’s dilemma game [1], the snow-drift game [2] and the public goods game [3] etc., which provide various models with different characteristics to study the behaviors of individuals in society. Meanwhile, complex networks as the abstract models of structured populations achieve rapid development, which provides a new perspective to gain insight into the complexity ∗ Corresponding author at: School of Information Science and Engineering, Shandong Normal University, Jinan 250014, China. E-mail address: [email protected] (C. Luo). https://doi.org/10.1016/j.physa.2018.11.018 0378-4371/© 2018 Elsevier B.V. All rights reserved.
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of dynamics. Based on structured models, works on transmission dynamics [4–6], evolutionary computation [7–10] and control [11–13], etc., have been developed rapidly. Since the network reciprocity was first proposed by Nowak et al. [14], i.e. cooperators in structured populations spontaneously aggregate into compact clusters to defense the invasion of defectors, the evolutionary games based on structured populations are attracting the attentions of researchers in this community [15]. The resources in reality are often limited. Meanwhile, the allocation of limited resources is usually related to the competition, where cooperation and defection as two common strategies would be chosen. How to make use of limited resources to maximize the cooperators is a valuable problem. In fact, it has been widely used, such as the applications of the scheduling problems [16], power systems [17,18], and the optimization problems [19]. During the evolution of game dynamics, cooperators spontaneously gather into compact clusters to resist the invasion of the defectors in structural groups, which have been considered to be the key role of cooperative evolution in multi-layer network [20–24]. But, it is few studies on the effect of resource allocation on the cooperation clusters. In recent years, multi-layer networks have gradually become a hot research topic. Different from single layer network, the multi-layer network consists of two or more layers. Apart from the connection between nodes in the same layer, there exists external links connecting the nodes residing on different layers. Nodes in the system represent individuals in the society, and edges represent social relationships. In this sense, multi-layer network can describe the systems in society more vividly in comparison to the single one. In the past few years, many achievements have been made in the study of evolutionary game on multi-layer networks. In [15], Wang et al. devoted attention to evolutionary games on multi-layer networks, and illustrated that the pattern formation and collective behavior were of great significance for the promotion of cooperation. In [25], Xiang proposed an evolutionary game model of dual-dynamic complex networks, and revealed the impact of defective leaders on average levels of cooperation. In [20], Wang et al. proposed that the evolution of public cooperation on two interdependent networks connected by a utility function. In [24], different ways of resource allocation were proposed and the effects of selforganization interdependence on the cooperation were further studied. Gómez-Gardeñes et al. analyzed the implementation of cooperative behavior interaction in mixed interactive networks in [26]. Cooperation clusters play an important role in helping partners to resist the invasion of defectors and improve the fraction of cooperation in the system [14]. A cooperation (defection) cluster is defined as a connected component that is completely occupied by cooperators (defectors) [27]. Nord et al. used mathematical methods to fill an infinite and uniform lattice, based on which the properties of clusters under certain rules were studied [28]. Yang et al. studied the transformation of cooperation penetration in different divisions of varying temptation values b based on single-layer network [27]. Based on small-world network, Grilo et al. founded that strategy exchanges contributed to the destruction of compact clusters would be favorable to cooperator agents [29]. Generally, the existing works are mainly based on single-layer network and dynamics of strategy cluster are rarely discussed. In this article, a two-layer interdependent network is implemented, where interdependent strength of external links is considered as a kind of limited resource that can be reallocated but the total amount will not be changed. Our research focuses on the formation and evolution of cooperative behavior with the change of interdependence of external links. Through quantitative and qualitative analysis, we will reveal the influence of independence resources on the dynamics of cooperation clusters. As we mentioned, the self-organization interdependence and the penetration of defectors are not all factors that determine the size of the largest cooperation cluster, which is also related to the strategy pairs residing two-layers. Some strategies will promote the formation of the largest cooperation cluster, while others will hinder the process. By exploring the interdependent strength among clusters, we find that larger clusters have stronger ability for resource concentration. In order to prove the correlation between the distribution of cooperation clusters and interdependent strength, we have compared the distribution of the first and the second largest cooperation clusters with the evolution of interdependent strength. Meanwhile, we find that the individuals are more inclined to cooperate with others when they possess more resources. The above results have also been illustrated in snow-drift game. This article is as follows: in the Section model, we mainly introduce the models and formulas used in the study. In the main results, we analyze the effects of interdependence resources on the size of the cluster, and reveal the factors that affect the size as well as the internal properties of the cooperation clusters. We summarized the results in the discussion. 2. Model In this article, the model is composed of two square lattices of size L × L with periodic boundary. In the proposed system, player x is connected to its four nearest neighbors located on the same layer, and the corresponding player x′ on another layer by means of an external link. The interdependent strength between x and x′ is represented by the parameter α (α ∈ [0, 1]), where α = 0 represents that there is no connection between two players and α = 1 represents the strongest interdependent strength. In our model, interdependent strength can represent some certain resources, e.g. the energy, the wealth or the natural resources, which can be relocated among players by repeated game but keep a fixed total amount. Obviously, the greater the interdependence strength the players have, the more chance to win in the game. The total interdependence resource in the system is as follows: 2
Γ =
L ∑ i=1
α i.
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Fig. 1. As the increase of the temptation to defect b, the fraction of cooperators and the proportion of largest cooperation clusters vary under the different initial interdependent strength. The size of the network is L = 200.
We define δ as the basic unit of resource transfer. At the beginning of game, the interdependent strength of all individuals is assumed to be α = γ and the total resources in the whole system are Γ = L2 γ . As disused above, during the process of evolution, interdependent resources can be reallocated among players by means of a concise rule based on the results of game, but the total amount of resources will be fixed, which can reasonably describe some dynamics in reality by illustrating the reallocation of resources in social activities. In the previous works [30], the introduction of ‘‘teaching rule’’ in evolutionary games shows rich dynamical features compared with ‘‘copying rule’’. Therefore, the concept of ‘teaching activity’ is adopted in following discussion. Game strategy, i.e. cooperation or defection, is assigned to each player on networks with the same probability. At each time, players implement the PDG with their four nearest neighbors, and the payoff matrix is as follows:
[ M=
RAMZ @PS TAMZ @PP
]
where the reward for mutual cooperation R = 1, the temptation to defect T = b (1 < b ≤ 2), the punishment for mutual defection P = 0 and sucker’s payoff S = 0. Initially, a player is randomly selected to play the PDG with its nearest four neighbors to gain payoff pxt . At the same ′ time, we calculate the corresponding payoff on the other layer pt x . The Fitness of player x is calculated as ft (x) = pt x + x′ αxx′ pt [23,31]. Similarly, a player y from four nearest neighbors of player x is randomly selected and calculate its payoff ′ y ft (y) = pt + αyy′ pt y . Subsequently, the strategy of player x is try to transfer to its neighbor y with a certain probability as W ( x → y) =
1 1 + exp {[ft (y) − ft (x)] /k0 }
where k0 = 0.1 represents noise factor. When the player x succeeds in teaching its strategy to y, it will obtain the basic unit δ of resources at a certain probability under the condition of αxx′ + δ ≤ 1 and αyy′ − δ ≥ 0. Meanwhile, player y will lose the corresponding resources to keep the total number of interdependence resources unchanged. Here, we focus on the influence of interdependent strength on the size of cooperation clusters. Sˆ1 is defined as the number of players in the largest cooperation cluster and Sˆ2 as the number of players in the second largest cooperation clusters. s1 and s2 represent the proportions of the largest cooperation cluster and the second largest cooperation cluster, respectively. The formula is as follows: s1 =
Sˆ1 L2
, s2 =
Sˆ2
L2 Simulations results are obtained from sufficiently large system sizes (from L = 200 to L = 400) to avoid finite size effects. The equilibration is required up to 105 MCS, and final results were averaged over up to 20 independent realizations. 3. Main results In order to study the impact of the interdependence resources on the size of the cooperation clusters, we study the varying of s1 and fc with the increase of b at γ = 0.2, 0.5, 0.8. The trend of s1 and fc is generally identical, which gradually decreases with the increase of b until the cooperators disappear. In Fig. 1(a)∼(c), the cooperators and cooperation clusters disappear when b = 1.21,1.24,1.196, respectively. Due to the penetration of the defectors, the cooperation clusters are divided by the defectors, which lead to the decline of s1 faster than fc . In previous studies [24], too much or too little interdependence resources will cause the system to lose interaction of self-organization, which hampers the survival of cooperators. The cooperation cluster itself plays an important role in improving the cooperation level, which also reflects the size of cooperation cluster to some extent. We can conclude that γ = 0.5 is more beneficial for the cooperators to cluster together. When it comes to the analysis of Fig. 1(b) and (c), their fc are both around 0.5 when b = 1.15, but s1 in Fig. 1(b) is 0.15 and 0.072 in Fig. 1(c). It can be further explained that in addition to the penetration of the defectors, the number of interdependence resources itself plays a signification role in the size of the cooperation clusters.
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Fig. 2. The fraction of the cooperators and the proportion of the largest cooperation cluster as a function of the initial interdependent strength γ are shown under different temptation to defect b.
Fig. 3. In panel (a), in dependence of the initial interdependent strength, the numbers of different strategy couples (C–C, C–D, and D–D couples) in whole system are presented. In panel (b), the ratio of the numbers of C–C and C–D strategy couples in the largest cooperation cluster is studied in the form of Ncc /Ncd. The temptation to defect is b = 1.09 and the system size is L = 200.
In order to further investigate the impact of interdependence resources, the fraction of the cooperation and the proportion of the largest cooperation cluster as a function of γ are shown in Fig. 2. In Fig. 2(a), When b = 1.06, fc and s1 reach the highest values at γ = 0.5. As the value of γ continues to increase, the values of fc and s1 are decreasing. The similar trend can be found in Fig. 2(b) and (c). As temptation to defect b takes different values, the proportion of cooperation clusters will not increase significantly with the increase of interdependence resources, just like the cooperation level, but at γ = 0.4 it turns out to be different. In order to further explore the internal characteristics of the cooperation clusters which affect the cooperation clusters, we will gain insight into the structure of cooperation clusters. There are three kinds of strategy couples residing in two layers: C–C (Cooperation–Cooperation), C–D (Cooperation–Defection) and D–D (Defection–Defection). Fig. 3(a) shows the number of different strategy couples between the two layers in the whole system in dependence of the initial interdependent strength γ with b = 1.09. It can be observed that the variance of the number of C–C is the same as the one of s1 in Fig. 2(b), where the highest value occurs at γ = 0.4, and the number of C–C is greater than the ones of C–D and D–D. In order to study the quantitative relationship between different strategy couples, we further explore the ratio of the numbers of C–C and C–D strategy couples in the largest cooperation cluster shown in Fig. 3(b), where Ncc , Ncd represent the numbers of C–C and C–D couples in the largest cooperation cluster, respectively. The value of Ncc /Ncd increases with the increase of γ , and achieves its highest value until γ = 0.4. Therefore, different strategy couples can promote or inhibit the formation of cooperation clusters. So, it can be speculated that for C–C, the relevant cooperation groups can be easily constructed, and the size of cooperation clusters can be significantly increased through the emergence of interaction reciprocity [32]. To further prove that C–C does have a significant effect on the size of the cooperation clusters, we calculate the average number of cooperators around the different strategies with varying the initial interdependent strength γ shown in Fig. 4. We find that, under different interdependence resources, the average number of cooperators around C–C couples is always the highest, followed by the one of C–D couples. And, the average number of cooperators around D–D couples is the lowest. Because of the influence of feedback, it will enhance interaction reciprocity or information sharing [33] to promote cooperative behavior. In previous studies [34,35], the authors explain the relationship between the circular C-cluster growth and network reciprocity, which can explain the END-period and EXP-period of dynamics which directly attribute to emerging network reciprocity in evolution game. It is difficult for C–D to promote the formation of cooperative groups, and D–D will
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Fig. 4. ⟨Nc⟩ as a function of the initial interdependent strength are shown, where ⟨Nc⟩ are the average numbers of cooperators around C–C, C–D, and D–D couples in the system. Dark blue represents the average number of cooperators around C–C couples. Light blue represents the average number of cooperators around C–D couples, and yellow represents the average number of cooperators around D–D couples. The temptation to defect is b = 1.09. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
hinder the survival of cooperators. This further proves the size of the largest cooperation cluster is related to the composition of strategy couples between the two layers. It should be noted that when the temptation to defect b is changed, the fraction of cooperation and the size of the largest cooperation cluster would be varied, but the overall dynamical behaviors can be maintained. In Fig. 5(a), the ratio of the number of external links with the interdependent strength α to the total amount of external links in the top five cooperation clusters are described, where a ‘‘seesaw effect’’ is shown. The ratio in all the five cooperation clusters have a large value at α = 0, α = 1, and a small value when α ∈ [0.1, 0.9]. Moreover, the larger the cooperation clusters is, the more uniform the middle part is distributed. As to the largest cooperation cluster, the sum of the ratios at α = 0 and α = 1 reaches 70%. The ratio shows an increasing trend around α = 0.8 and has a large increase in the range of α = [0.9, 1]. As to the second largest cooperation cluster, the sum of the ratios at α = 0 and α = 1 reaches 90%, which shows a large increase in the range of α ∈ [0.9, 1]. As to the third largest cooperation clusters, the sum of the ratios at α = 0 and α = 1 reaches 70%. And it is equal to the number of individuals between 0.2 and 0.5, which changes little but is not zero. When α ∈ [0.6, 0.9], the ratio grows slightly but has a significant increase in the range of α ∈ [0.9, 1]. Since the fourth and fifth largest cooperation clusters are smaller than the first three large cooperation clusters, the proportion of the players numbers is unevenly distributed at α = [0.1, 0.9]. Since each player has the same interdependent strength α = γ = 0.5, and the total resource in the system is Γ = 2000. When the system is stable, the proportion of the largest cooperation cluster is 60%. Even if all the resources are concentrated together, the proportion of the individual whose interdependence resource reaches 1 only accounts for 50%, which means that there are many players in the largest cooperation group whose interdependent strength is less than 1. By accounting the proportion of players whose interdependent strength lie in α ∈ [0.8, 1], we find that the result of the largest cooperation cluster is 58.65%, and the second largest cooperation cluster is 57.38%. It is obvious that the limited resources in the largest cooperation cluster are more rich compared with the ones in the second largest cooperation cluster. In order to further study the resource concentration of the cooperation clusters, we use the formula
P(Rα,γ ) =
L2 P(α )α L2
γ
=
α P(α ) γ
where L represents the size of the network. In Fig. 5(b), we see the distribution of α P(α )/γ is obviously different from the normal distribution, exponential distribution and the binomial distribution. The cooperation clusters have the lowest concentration of resources at α = 0. The concentration of resources in the range of α = [0.1, 0.8] is not obvious, but, when α reaches 0.9, there is a big jump and the concentration reaches its highest level at α = 1. Compared with the first three largest cooperation clusters, the fourth and fifth largest cooperation clusters have the lower concentration of resources at α = 1. To verify the above discussion, α P(α )/γ calculated in the first tenth largest cooperation clusters at α = 1 is shown in Fig. 6. In Fig. 6(a), at γ = 0.1, 0.2, the results show irregular fluctuations. In Fig. 6(b), the results of the top six largest cooperation clusters are almost identical, for instance, the results shown for γ = 0.5. In Fig. 6(c), the resource concentration abilities of the top five largest cooperation clusters are generally identical when γ = 0.8, 0.9. With different quantities of interdependence resources, the resource concentration ability of the first and second largest cooperation clusters are similar compared with other cooperation clusters. These different results can attribute to the self-organizations of interdependence. Generally speaking, no matter how the overall interdependence resources existing in the system changes, larger cooperation clusters have similar resource concentration ability.
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Fig. 5. The distribution of interdependent strength of the top five cooperation clusters as a function of self-organized interdependence. In panel (a), with the increase of α , the variance of P(α ) in the top five cooperation clusters. P(α ) represents the proportion of individuals whose interdependent strength is α in its cooperation clusters. Panel (b) shows the variance of α P(α )/γ as a function of α , where α P(α )/γ represents the ratio of the accumulated amount of interdependent strength existing in external links of strength α to the total amount of interdependent strength in the whole system.
Fig. 6. α P(α )/γ calculated in the first ten largest cooperation clusters, where α = 1.
Next, we study the distribution of strategies and the evolution process of interdependent strength on the two-layer network denoted by layer A and layer B. Here, the first and second largest cooperation clusters are taken as examples. In Fig. 7, at 1 MCS, the interdependent strength of the whole system is 0.5. Due to the randomness of the initial strategy choices, we can see the different distribution of cooperators and defectors on the networks A and B. As the evolution time goes by, it comes to 10 MCS, and the snapshot shows a variety of colors. The defectors gather together to become a great defection cluster, while the cooperators also cluster to against the invasion of the defectors. The size and distribution of the cooperation clusters vary. At 50 MCS, with the process of the accumulation of interdependent resources, a larger number of cooperators and defectors gather together to form clusters, respectively. The above process would be identical on two layers. In Fig. 7(c), we can see that the yellow regions where the interdependent strength approaches 1 are also the places where the first and second largest cooperation clusters appear (shown in Fig. 7(g) and (k)). The first largest cooperation clusters on the two-layer network locates at the same regions. Meanwhile, it can be observed that the blue regions in Fig. 7(c), where the interdependent strength is close to 0, are mainly occupied by defectors. At 10 000 MCS, it can be found that color distinction is more obvious in Fig. 7(d). The interdependent resources owned by players are approaching 0 or 1, and there are few middle values. Furthermore, the regions with the interdependent strength 1 are almost completely connected into a whole. The distribution of cooperators and defectors in Fig. 7(h) and (l) is generally similar. The first largest cooperation clusters lies in the yellow area shown in Fig. 7(d). However, the distribution of the first and second largest cooperation clusters on A and B is distinct, where the size of the first largest cooperation clusters on layer B is significantly larger than that on layer A, which can attribute to the randomness brought by the Monte Carlo method. Finally, in order to verify the robustness of the above results, snow-drift game is carried out. At first, every player has the same interdependent strength α = γ = 0.5. Only after 10 MCS, the distribution of interdependent strength has obviously changed. The cooperation level of snow-drift game is higher than that of prisoner’s game. In the two-layer network, the defectors are difficult to gather together, while cooperators begin to form a large cooperation cluster. At 50 MCS, a large connected component region in yellow appears in Fig. 8(c), where the largest cooperation clusters are emerging shown in Fig. 8(g) and (k). The defectors are surrounded by the cooperators. At 100000 MCS, only two kinds of interdependent strength, i.e. zero or one, can be observed. It shows that interdependence resources allocation has reached the serious polarization, where the yellow areas occupy almost all the places and only a small number of defectors survive. In Fig. 9, we studied the distribution of interdependent strength of cooperation clusters in snow-drift game. Since the cooperation level of snow-drift game is high, almost all cooperators have been connected into a cluster. In Fig. 9(a), the
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Fig. 7. (a)∼(d) is snapshots of the distributions of the interdependent strength of all the external links between two networks. (e)∼(h) is snapshots of the distributions of the cooperation clusters and the defectors cluster on network A. (i) ∼(l) is snapshots of the distributions of the cooperators cluster and the defectors cluster on network A. We provided color bars for reference to the interdependent strength. Yellow denotes the largest cooperation cluster, orange denotes the second largest cooperation cluster, light blue denotes the other cooperation clusters and dark blue denotes the defection clusters. At first, every individual has the same interdependent strength α = γ = 0.5. The temptation to defect is b = 1.09 and the system size is L = 200. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
proportion of individuals with different interdependent strength in this cluster is shown. In Fig. 9(b), a large amount of interdependence resources are concentrated until α = 1. At α = 0, the value of α P(α )/γ is zero. And the ability of resource concentration is not obvious when α ∈ [0.1, 0.8]. At α = 0.9, the ability for resource concentration has a big jump, and the value of α P(α )/γ reaches its highest value at α = 1. This is consistent with the results we obtained in the prisoner’s game, which indicates the finding has robustness. 4. Discussion In this article, based on interdependent networks, we have studied the dynamics of strategy clusters. In the proposed model, interdependent strength as limited resources can be reallocated among players during the evolutionary process and co-evolve with evolutionary game. Different from the existing works relate to strategy clusters, besides of investigation on the formation and distribution of clusters, the impact of the co-evolution of interdependent strength and game strategies on the dynamics of strategy clusters have been detailedly discussed. We illustrate that the size of the largest cooperation cluster is related to the kind of strategy pairs residing on the two layers, where the formation of C–C can establish a local structure similar to network reciprocity, which would greatly promote the formation of cooperation clusters. Besides, during the evolution process, the larger clusters have stronger ability for resource concentration, which is similar to the Matthew effect. And, the players with more resources are inclined to be cooperative. By a series of experiments, the relationship between the cooperation clusters and the interdependent strength has been illustrated. The results show the formation of
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Fig. 8. (Snow-drift Game) In the evolution process, the interdependent strength of external links between two networks and the distribution snapshots of the first and second largest cooperation clusters. Panels (a)∼(d) are the evolutionary processes of interdependent strength, and panels (e)∼(h) are the distribution and evolution process of the first and second largest cooperation clusters on network A. Panels (i)∼(l) are the distribution and evolution process of the first and second largest cooperation clusters on network B. We provide color bars for reference to the interdependent strength. Yellow denotes the largest cooperation cluster, orange denotes the second largest cooperation cluster, light blue denotes the other cooperation clusters and dark blue denotes the defector clusters. At first, every individual has same interdependent strength α = γ = 0.5. b = 1.4, c = 0.8, and the system size is L = 200. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 9. The distribution of interdependent strength of cooperation clusters in snow-drift game. Panel (a) shows P(α ) as a function α , where P(α ) represents the proportion of individuals whose interdependent strength is α in the cooperation cluster. Panel (b) shows the variance of α P(α )/γ in dependence on α , where α P(α )/γ shows the distributions of accumulated amount of interdependent strength as a function of α .
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cooperation clusters, especially for larger clusters, is considerably affected by the distribution of interdependent strength. Furthermore, the experiments utilizing snow-drift Game further verify the robustness of the results. Acknowledgments This research is supported by the National Natural Science Foundation of China (Nos: 61402267; 61572300; 81871508; 61773246); Shandong Provincial Natural Science Foundation (ZR2014FQ004); Major Program of Shandong Province Natural Science Foundation (No. ZR2018ZB0419). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35]
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