Roles of mixing patterns in cooperation on a scale-free networked game

Effects of tag mediation and structural balance on the evolution of cooperation on signed networks

Tag-based ethnocentrism is the basic mechanism for explaining social identity in human behaviors. Combining game theory with structural balance, the present study proposed a novel evolutionary game model in signed networks considering the tag-mediated effect, which provides a new perspective for explaining collective dynamics in social systems. Experiments show that negative relations promote the unconditional cooperation, and the conditional strategy is less likely to appear in the evolution in signed networks. Network adaption is helpful in reducing the proportion of unconditional defectors, but it can be mediated by the tag-based effect. The unconditional cooperation prevails when the speed of relation updating is faster than that of strategy updating increases to a certain extent. The evolution of structural balance can be capable of reducing the proportion of ethnocentric players. From a global point of view, the tag-mediated effect stimulates the formation of attractors or repeller structures, but the dynamic structural balance prevents the formation.

Promoting cooperation in the voluntary prisoner's dilemma game via reinforcement learning

Reinforcement learning technology has been empirically demonstrated to facilitate cooperation in game models. However, traditional research has primarily focused on two-strategy frameworks (cooperation and defection), which inadequately captures the complexity of real-world scenarios. To address this limitation, we integrated Q-learning into the prisoner's dilemma game, incorporating three strategies: cooperation, defection, and going it alone. We defined each agent's state based on the number of neighboring agents opting for cooperation and included social payoff in the Q-table update process. Numerical simulations indicate that this framework significantly enhances cooperation and average payoff as the degree of social-attention increases. This phenomenon occurs because social payoff enables individuals to move beyond narrow self-interest and consider broader social benefits. Additionally, we conducted a thorough analysis of the mechanisms underlying this enhancement of cooperation.

Promotion of cooperation in a structured population with environmental feedbacks

Cooperation is a representative altruistic behavior in which individuals contribute public goods to benefit their neighborhoods and even larger communities in social networks. The defective behavior is more likely to bring higher payoffs than the cooperative behavior, which makes the cooperative behavior hard to maintain and sustain. Many mechanisms were proposed to promote cooperation within a social dilemma, in which some recent studies introduced the impact of dynamically changing environments on players' payoffs and strategies in social-ecological systems, and evolutionary-ecological systems. However, degree heterogeneity, an important structural property of many real-world complex networks such as social networks, academic collaboration networks, and communication networks, is rarely explored and studied in such eco-evolutionary games. In this research, we propose a Public Goods Game model on social networks with environmental feedback and analyze how the environmental factor and network structure affect the evolution of cooperation. It is found that as the initial environmental factors and the cooperation-enhancement defection-degradation ratio increase, the steady cooperation level of the social network significantly increases, and the dynamic environment will eventually evolve into a high-return environment; On the other hand, even if the initial environmental benefit coefficient is high, when the cooperation-enhancement defection-degradation ratio is less than a threshold, the dynamic environment will gradually degrade into a low-return environment. The steady cooperation level of the social network first gradually increases as the network structure becomes more heterogeneous, but it will decrease once the heterogeneity of the social network exceeds a certain threshold.

Evolution of cooperation with early social influence for explaining collective action

The logic of collective action has laid a foundation for the research of public choice, and the success of collective action has been a long-term discussion when free-riding mechanism is considered in the dynamics. This study proposes a , which provides a novel dimension for explaining the logic of collective action. Under the framework, the accumulation of early social influence, conformity, and the pressure of relationship updating in small groups is discussed. The experiment results show that the accumulation of early social influence indirectly promotes the participants of collective action; conformity is conducive to stimulating collective action, but relies on the accumulation of early social influence; the pressure of relationship updating plays the small-group role, which promotes the participation of collective actions; all these effects are helpful in forming the cascade of cooperators, and prevent the coexistence of participants and non-participants of collective action.

Memory and target payoff enhance cooperation in evolutionary social dilemmas

We proposed a neighbor selection mechanism based on memory and target payoff, where the target payoff is the maximum value of the group's average expected payoff. According to this mechanism, individuals prioritize selecting neighbors whose average payoffs in the last M rounds are close to the target payoff for strategy learning, aiming to maximize the group's expected payoff. Simulation results on the grid-based Prisoner's Dilemma and Snowdrift games demonstrate that this mechanism can significantly improve the group's payoff and cooperation level. Furthermore, the longer the memory length, the higher the group's payoff and cooperation level. Overall, the combination of memory and target payoff can lead to the emergence and persistence of cooperation in social dilemmas as individuals are motivated to cooperate based on both their past experiences and future goals. This interplay highlights the significance of taking into account numerous variables in comprehending and promoting cooperation within evolutionary frameworks.

The impact of dynamic linking on cooperation on complex networks

In complex social systems, individual relationships and the surrounding environment are constantly changing, allowing individuals to interact on dynamic networks. This study aims to investigate how individuals in a dynamic network engaged in a prisoner's dilemma game adapt their competitive environment through random edge breaks and reconnections when faced with incomplete information and adverse local conditions, thereby influencing the evolution of cooperative behavior. We find that random edge breaks and reconnections in dynamic networks can disrupt cooperative clusters, significantly hindering the development of cooperation. This negative impact becomes more pronounced over larger time scales. However, we also observe that nodes with higher degrees of connectivity exhibit greater resilience to this cooperation disruption. Our research reveals the profound impact of dynamic network structures on the evolution of cooperation and provides new insights into the mechanisms of cooperation in complex systems.

Conformity effect on the evolution of cooperation in signed networks

Human behaviors are often subject to conformity, but little research attention has been paid to social dilemmas in which players are assumed to only pursue the maximization of their payoffs. The present study proposed a generalized prisoner dilemma model in a signed network considering conformity. Simulation shows that conformity helps promote the imitation of cooperative behavior when positive edges dominate the network, while negative edges may impede conformity from fostering cooperation. The logic of homophily and xenophobia allows for the coexistence of cooperators and defectors and guides the evolution toward the equality of the two strategies. We also find that cooperation prevails when individuals have a higher probability of adjusting their relation signs, but conformity may mediate the effect of network adaptation. From a population-wide view, network adaptation and conformity are capable of forming the structures of attractors or repellers.

Impact of social reward on the evolution of cooperation in voluntary prisoner's dilemma

The existence and sustainability of cooperation is a critical issue in nature and social systems. Reward is an essential mechanism to enhance cooperation. Meanwhile, some individuals loathe competition and then choose to escape and become a loner in competition. In this scenario, we propose a four-strategy networked evolutionary game model involving rewarders, loners, cooperators, and defectors. The classical square lattice and the Erdös-Rényi random network are adopted to describe the interaction between individuals. The four-strategy model is an extension of the classic prisoner's dilemma game model. The simulation results show that the introduction of new strategic choices can significantly improve cooperation in the population. The promotion level of cooperation is directly correlated with reward intensity and negatively correlated with reward cost. With regard to the evolution of altruistic behaviors, the fixed income from interactions with loners has an impact that is connected to the temptation to defect. Furthermore, by analyzing characteristic snapshots of four strategies, we further dissect the essence of the evolution of cooperation. As the temptation value increases, cooperators and rewarders first form compact clusters, then more and more loners join to resist the intrusion of defectors. Eventually, the three strategies coexist stably in a spatially structured population. Our research may shed some light on exploring the nature of cooperation and solving social dilemmas in the future.

Two-layer network model of public goods games with intervention and corruption

Public goods games are widely used to model social dilemmas involving multiple agents. Though defection is the only rational choice for an individual in a public goods game, cooperative behavior is observed in a variety of social dilemmas, which is the subject of our research. Punishing defectors has been shown to be an effective mechanism for promoting cooperation, but it relies on the third-party umpire being fair. In this article, an umpire intervention model with corruption is proposed to explore the impact of corruption on punishment mechanisms. In our model, players and umpires operate in a multilayer network. The players play public goods games, which are overseen by umpires. Fair umpires punish defectors, whereas corrupt umpires take bribes from defectors rather than meting out a punishment. We separately explore the effects of the fraction of fair umpires _ρ, the spatial distribution, and the fine cost α and bribe cost β. Our Monte Carlo simulation shows that the above factors have a significant impact on cooperation. Intervention by an umpire always improves social efficiency, even for an entirely corrupt system. Moreover, relatively developed systems can resist corruption. Staggered and centralized distributions always have opposite effects on cooperative behavior, and these effects depend on _ρ and r. We also find that whether cooperators fully occupy the player layer depends only on whether β reaches a certain threshold.

Hypergraph assortativity: A dynamical systems perspective

The largest eigenvalue of the matrix describing a network's contact structure is often important in predicting the behavior of dynamical processes. We extend this notion to hypergraphs and motivate the importance of an analogous eigenvalue, the expansion eigenvalue, for hypergraph dynamical processes. Using a mean-field approach, we derive an approximation to the expansion eigenvalue in terms of the degree sequence for uncorrelated hypergraphs. We introduce a generative model for hypergraphs that includes degree assortativity, and use a perturbation approach to derive an approximation to the expansion eigenvalue for assortative hypergraphs. We define the dynamical assortativity, a dynamically sensible definition of assortativity for uniform hypergraphs, and describe how reducing the dynamical assortativity of hypergraphs through preferential rewiring can extinguish epidemics. We validate our results with both synthetic and empirical datasets.

Enhancement of Cooperation and Reentrant Phase of Prisoner’s Dilemma Game on Signed Networks

We studied the prisoner’s dilemma game as applied to signed networks. In signed networks, there are two types of links: positive and negative. To establish a payoff matrix between players connected with a negative link, we multiplied the payoff matrix between players connected with a positive link by −1. To investigate the effect of negative links on cooperating behavior, we performed simulations for different negative link densities. When the negative link density is low, the density of the cooperator becomes zero because there is an increasing temptation payoff, b. Here, parameter b is the payoff received by the defector from playing the game with a cooperator. Conversely, when the negative link density is high, the cooperator density becomes almost 1 as b increases. This is because players with a negative link will suffer more payoff damage if they do not cooperate with each other. The negative link forces players to cooperate, so cooperating behavior is enhanced.

Evolutionary dynamics of cooperation with the celebrity effect in complex networks

How long-term cooperation is maintained in a society is an important and interesting question. The evolutionary game theory is often used as the basic framework to study this topic. The social status of game participants has an important influence on individual decision-making. Enlightened by this thought, we present a classification imitation model where the mechanisms of the celebrity effect and incomplete egoism are presented. The celebrity effect is reflected in each strategy update process to probe how individual decision-making is dynamically adjusted by comparing the social status of both parties in the game. The incomplete egoism refers to the irrational imitation of celebrities while self-interest is ignored. With this model, the group cooperation decision-making mechanism led by celebrities is revealed. Large-scale Monte Carlo simulations show that the incomplete egoism of individuals cannot stimulate cooperation but guarantee the stable existence of cooperation. Furthermore, the scale-free and community structure of the network enables cooperation to spread widely and maintains long-term survival. Our conclusion might provide practically new insight into the understanding and controlling of cooperation in the complex social systems.

Mixing protocols in the public goods game

If interaction partners in social dilemma games are not selected randomly from the population but are instead determined by a network of contacts, it has far reaching consequences for the evolutionary dynamics. Selecting partners randomly leads to a well-mixed population, where pattern formation is essentially impossible. This rules out important mechanisms that can facilitate cooperation, most notably network reciprocity. In contrast, if interactions are determined by a lattice or a network, then the population is said to be structured, where cooperators can form compact clusters that protect them from invading defectors. Between these two extremes, however, there is ample middle ground that can be brought about by the consideration of temporal networks, mobility, or other coevolutionary processes. The question that we here seek to answer is, when does mixing on a lattice actually lead to well-mixed conditions? To that effect, we use the public goods game on a square lattice, and we consider nearest-neighbor and random mixing with different frequencies, as well as a mix of both mixing protocols. Not surprisingly, we find that nearest-neighbor mixing requires a higher frequency than random mixing to arrive at the well-mixed limit. The differences between the two mixing protocols are most expressed at intermediate mixing frequencies, whilst at very low and very high mixing frequencies the two almost converge. We also find a near universal exponential growth of the average size of cooperator clusters as their fraction increases from zero to one, regardless of whether this increase is due to increasing the multiplication factor of the public goods, decreasing the frequency of mixing, or gradually shifting the mixing from random to nearest neighbors.

Cooperation in a generalized age-structured spatial game

The emergence and prevalence of cooperative behavior within a group of selfish individuals remains a puzzle for evolutionary game theory precisely because it conflicts directly with the central idea of natural selection. Accordingly, in recent years, the search for an understanding of how cooperation can be stimulated, even when it conflicts with individual interest, has intensified. We investigate the emergence of cooperation in an age-structured evolutionary spatial game. In it, players age with time and the payoff that they receive after each round depends on their age. We find that the outcome of the game is strongly influenced by the type of distribution used to modify the payoffs according to the age of each player. The results show that, under certain circumstances, cooperators may not only survive but dominate the population.

Heterogeneous cooperative leadership structure emerging from random regular graphs

This paper investigates the evolution of cooperation and the emergence of hierarchical leadership structure in random regular graphs. It is found that there exist different learning patterns between cooperators and defectors, and cooperators are able to attract more followers and hence more likely to become leaders. Hence, the heterogeneous distributions of reputation and leadership can emerge from homogeneous random graphs. The important directed game-learning skeleton is then studied, revealing some important structural properties, such as the heavy-tailed degree distribution and the positive in-in degree correlation.

Aspiration induced interdependence leads to optimal cooperation level

How to couple different networks is a key issue in interdependent networks, where information sharing and payoff coupling are two frequently used methods. Unlike previous studies, in this paper, we propose a new coupling mode and test its performance in interdependent networks. Specifically, a player tends to seek additional support on another network only if his environment (defined as the proportion of holding different strategies in the neighborhood) is worse enough and exceeds an aspiration level. Conversely, it turns to the traditional version on single network if his environment is pleasing enough (the value of environment is small). Whether to establish additional support will directly influence the range of selecting fittest learning objects. As we can see from numerical results, moderate aspiration introduces diversity into the system and cooperation evolves with the support of network coupling. Besides, we also demonstrate that players with external links on the boundary of cooperative clusters protect internal cooperators and attract more players to cooperate under preferential selection rule.

Robustness on interdependent networks with a multiple-to-multiple dependent relationship

Interdependent networks as an important structure of the real system not only include one-to-one dependency relationship but also include more realistic undirected multiple interdependent relationship. The study on interdependent networks plays an important role in designing more resilient real systems. Here, we mainly focus on the case of interdependent networks with a multiple-to-multiple correspondence of interdependent nodes by generalizing feedback and nonfeedback conditions. We develop a new mathematical framework and study numerically and analytically the percolation of interdependent networks with partial multiple-to-multiple dependency links by using percolation theory. By analyzing the process of cascading failure, the size of the giant component and the critical threshold are analytically obtained and testified by simulation results for couple Erdös-Re˙nyi and scale-free networks. The results imply that the system shows a discontinuous phase transition for different coupling strengths. We find that the system becomes more resilient and easy to defend by increasing the coupling strength and the connectivity density. Our model has the potential to shed light on designing more resilient real-world dependent systems.

Identification of influential invaders in evolutionary populations

The identification of the most influential nodes has been a vibrant subject of research across the whole of network science. Here we map this problem to structured evolutionary populations, where strategies and the interaction network are both subject to change over time based on social inheritance. We study cooperative communities, which cheaters can invade because they avoid the cost of contributions that are associated with cooperation. The question that we seek to answer is at which nodes cheaters invade most successfully. We propose the weighted degree decomposition to identify and rank the most influential invaders. More specifically, we distinguish two kinds of ranking based on the weighted degree decomposition. We show that a ranking strategy based on negative-weighted degree allows to successfully identify the most influential invaders in the case of weak selection, while a ranking strategy based on positive-weighted degree performs better when the selection is strong. Our research thus reveals how to identify the most influential invaders based on statistical measures in dynamically evolving cooperative communities.

General formulation of long-range degree correlations in complex networks

We provide a general framework for analyzing degree correlations between nodes separated by more than one step (i.e., beyond nearest neighbors) in complex networks. One joint and four conditional probability distributions are introduced to fully describe long-range degree correlations with respect to degrees k and k^{'} of two nodes and shortest path length l between them. We present general relations among these probability distributions and clarify the relevance to nearest-neighbor degree correlations. Unlike nearest-neighbor correlations, some of these probability distributions are meaningful only in finite-size networks. Furthermore, as a baseline to determine the existence of intrinsic long-range degree correlations in a network other than inevitable correlations caused by the finite-size effect, the functional forms of these probability distributions for random networks are analytically evaluated within a mean-field approximation. The utility of our argument is demonstrated by applying it to real-world networks.

Impact of interaction style and degree on the evolution of cooperation on Barabási-Albert scale-free network

In this work, we study an evolutionary prisoner's dilemma game (PDG) on Barabási-Albert scale-free networks with limited player interactions, and explore the effect of interaction style and degree on cooperation. The results show that high-degree preference interaction, namely the most applicable interaction in the real world, is less beneficial for emergence of cooperation on scale-free networks than random interaction. Besides, cooperation on scale-free networks is enhanced with the increase of interaction degree regardless whether the interaction is high-degree preference or random. If the interaction degree is very low, the cooperation level on scale-free networks is much lower than that on regular ring networks, which is against the common belief that scale-free networks must be more beneficial for cooperation. Our analysis indicates that the interaction relations, the strategy and the game payoff of high-connectivity players play important roles in the evolution of cooperation on scale-free networks. A certain number of interactions are necessary for scale-free networks to exhibit strong capability of facilitating cooperation. Our work provides important insight for members on how to interact with others in a social organization.

Promoting cooperation by preventing exploitation: The role of network structure

A growing body of empirical evidence indicates that social and cooperative behavior can be affected by cognitive and neurological factors, suggesting the existence of state-based decision-making mechanisms that may have emerged by evolution. Motivated by these observations, we propose a simple mechanism of anonymous network interactions identified as a form of generalized reciprocity-a concept organized around the premise "help anyone if helped by someone'-and study its dynamics on random graphs. In the presence of such a mechanism, the evolution of cooperation is related to the dynamics of the levels of investments (i.e., probabilities of cooperation) of the individual nodes engaging in interactions. We demonstrate that the propensity for cooperation is determined by a network centrality measure here referred to as neighborhood importance index and discuss relevant implications to natural and artificial systems. To address the robustness of the state-based strategies to an invasion of defectors, we additionally provide an analysis which redefines the results for the case when a fraction of the nodes behave as unconditional defectors.

Extortion provides alternative routes to the evolution of cooperation in structured populations

In this paper, we study the evolution of cooperation in structured populations (individuals are located on either a regular lattice or a scale-free network) in the context of repeated games by involving three types of strategies, namely, unconditional cooperation, unconditional defection, and extortion. The strategy updating of the players is ruled by the replicator-like dynamics. We find that extortion strategies can act as catalysts to promote the emergence of cooperation in structured populations via different mechanisms. Specifically, on regular lattice, extortioners behave as both a shield, which can enwrap cooperators inside and keep them away from defectors, and a spear, which can defeat those surrounding defectors with the help of the neighboring cooperators. Particularly, the enhancement of cooperation displays a resonance-like behavior, suggesting the existence of optimal extortion strength mostly favoring the evolution of cooperation, which is in good agreement with the predictions from the generalized mean-field approximation theory. On scale-free network, the hubs, who are likely occupied by extortioners or defectors at the very beginning, are then prone to be conquered by cooperators on small-degree nodes as time elapses, thus establishing a bottom-up mechanism for the emergence and maintenance of cooperation.

Win-stay-lose-learn promotes cooperation in the prisoner's dilemma game with voluntary participation

Voluntary participation, demonstrated to be a simple yet effective mechanism to promote persistent cooperative behavior, has been extensively studied. It has also been verified that the aspiration-based win-stay-lose-learn strategy updating rule promotes the evolution of cooperation. Inspired by this well-known fact, we combine the Win-Stay-Lose-Learn updating rule with voluntary participation: Players maintain their strategies when they are satisfied, or players attempt to imitate the strategy of one randomly chosen neighbor. We find that this mechanism maintains persistent cooperative behavior, even further promotes the evolution of cooperation under certain conditions.

Evolution of cooperative imitators in social networks

Many evolutionary game models for network reciprocity are based on an imitation dynamics, yet how semirational imitators prevail has seldom been explained. Here we use a model to investigate the coevolutionary dynamics of cooperation and partnership adjustment in a polygenic population of semirational imitators and rational payoff maximizers. A rational individual chooses a strategy best responding to its neighbors when updating strategy and switches to a new partner who can bring it the maximal payoff from all candidates when adjusting the partnership. In contrast, a semirational individual imitates its neighbor's strategy directly and adjusts its partnership based upon a simple reputation rule. Individual-based simulations show that cooperation cannot evolve in a population of all best responders even if they can switch their partners to somebody who can reward them best in game playing. However, when imitators exist, a stable community that consists of cooperative imitators emerges. Further, we show that a birth-death selection mechanism can eliminate all best responders, cultivating a social regime of all cooperative imitators. Compared with parallel simulations that assume fixed networks, cooperative imitators are evolutionarily favored, provided they are able to adjust their partners.

Coevolutionary dynamics of phenotypic diversity and contingent cooperation

Phenotypic diversity is considered beneficial to the evolution of contingent cooperation, in which cooperators channel their help preferentially towards others of similar phenotypes. However, it remains largely unclear how phenotypic variation arises in the first place and thus leads to the construction of phenotypic complexity. Here we propose a mathematical model to study the coevolutionary dynamics of phenotypic diversity and contingent cooperation. Unlike previous models, our model does not assume any prescribed level of phenotypic diversity, but rather lets it be an evolvable trait. Each individual expresses one phenotype at a time and only the phenotypes expressed are visible to others. Moreover, individuals can differ in their potential of phenotypic variation, which is characterized by the number of distinct phenotypes they can randomly switch to. Each individual incurs a cost proportional to the number of potentially expressible phenotypes so as to retain phenotypic variation and expression. Our results show that phenotypic diversity coevolves with contingent cooperation under a wide range of conditions and that there exists an optimal level of phenotypic diversity best promoting contingent cooperation. It pays for contingent cooperators to elevate their potential of phenotypic variation, thereby increasing their opportunities of establishing cooperation via novel phenotypes, as these new phenotypes serve as secret tags that are difficult for defector to discover and chase after. We also find that evolved high levels of phenotypic diversity can occasionally collapse due to the invasion of defector mutants, suggesting that cooperation and phenotypic diversity can mutually reinforce each other. Thus, our results provide new insights into better understanding the coevolution of cooperation and phenotypic diversity.

Phenotypic variation is commonly observed from human cells to the intestinal pathogen Salmonella enterica serovar Typhimurium to the wrinkly-spreader morphs. Such phenotypic diversity proves effective in promoting cooperation, or confers survival advantage against unfavorable environmental changes. Prior studies show that interactions based on phenotypic similarity can promote cooperation. Yet in these models, the level of phenotypic diversity is prescribed such that individuals each possess the same number of available phenotypes, and thereby no evolution of phenotypic diversity per se. We here take into consideration important aspects of the diversity of phenotype and contingent cooperation and show that phenotypic diversity coevolves with cooperation under a variety of conditions. Our work provides a potential mechanism for the evolution of cooperation, and individuals, especially cooperators, endowed with diverse phenotypes constitute the backbone in inducing the coevolution.

Constructing Robust Cooperative Networks using a Multi-Objective Evolutionary Algorithm

The design and construction of network structures oriented towards different applications has attracted much attention recently. The existing studies indicated that structural heterogeneity plays different roles in promoting cooperation and robustness. Compared with rewiring a predefined network, it is more flexible and practical to construct new networks that satisfy the desired properties. Therefore, in this paper, we study a method for constructing robust cooperative networks where the only constraint is that the number of nodes and links is predefined. We model this network construction problem as a multi-objective optimization problem and propose a multi-objective evolutionary algorithm, named MOEA-Net_rc, to generate the desired networks from arbitrary initializations. The performance of MOEA-Net_rc is validated on several synthetic and real-world networks. The results show that MOEA-Net_rc can construct balanced candidates and is insensitive to the initializations. MOEA-Net_rc can find the Pareto fronts for networks with different levels of cooperation and robustness. In addition, further investigation of the robustness of the constructed networks revealed the impact on other aspects of robustness during the construction process.

Roles of mixing patterns in the network reconstruction

Compressive sensing is an effective way to reconstruct the network structure. In this paper, we investigate the effect of the mixing patterns, measured by the assortative coefficient, on the performance of network reconstruction. First, we present a model to generate networks with different assortativity coefficients, then we reconstruct the network structure by using the compressive sensing method. The experimental results show that when the assortativity coefficient r=0.2, the accuracy of the network reconstruction reaches the maximum value, which suggests that the compressive sensing is more effective for uncovering the links of social networks. Moreover, the accuracy of the network reconstruction will be higher as the network size increases.

Reciprocity in spatial evolutionary public goods game on double-layered network

Spatial evolutionary games have mainly been studied on a single, isolated network. However, in real world systems, many interaction topologies are not isolated but many different types of networks are inter-connected to each other. In this study, we investigate the spatial evolutionary public goods game (SEPGG) on double-layered random networks (DRN). Based on the mean-field type arguments and numerical simulations, we find that SEPGG on DRN shows very rich interesting phenomena, especially, depending on the size of each layer, intra-connectivity, and inter-connected couplings, the network reciprocity of SEPGG on DRN can be drastically enhanced through the inter-connected coupling. Furthermore, SEPGG on DRN can provide a more general framework which includes the evolutionary dynamics on multiplex networks and inter-connected networks at the same time.

Speed of evolution on graphs

The likelihood that a mutant fixates in the wild population, i.e., fixation probability, has been intensively studied in evolutionary game theory, where individuals' fitness is frequency dependent. However, it is of limited interest when it takes long to take over. Thus the speed of evolution becomes an important issue. In general, it is still unclear how fixation times are affected by the population structure, although the fixation times have already been addressed in the well-mixed populations. Here we theoretically address this issue by pair approximation and diffusion approximation on regular graphs. It is shown (i) that under neutral selection, both unconditional and conditional fixation time are shortened by increasing the number of neighbors; (ii) that under weak selection, for the simplified prisoner's dilemma game, if benefit-to-cost ratio exceeds the degree of the graph, then the unconditional fixation time of a single cooperator is slower than that in the neutral case; and (iii) that under weak selection, for the conditional fixation time, limited neighbor size dilutes the counterintuitive stochastic slowdown which was found in well-mixed populations. Interestingly, we find that all of our results can be interpreted as that in the well-mixed population with a transformed payoff matrix. This interpretation is also valid for both death-birth and birth-death processes on graphs. This interpretation bridges the fixation time in the structured population and that in the well-mixed population. Thus it opens the avenue to investigate the challenging fixation time in structured populations by the known results in well-mixed populations.

Effects of random rewiring on the degree correlation of scale-free networks

Random rewiring is used to generate null networks for the purpose of analyzing the topological properties of scale-free networks, yet the effects of random rewiring on the degree correlation are subject to contradicting interpretations in the literature. We comprehensively analyze the degree correlation of randomly rewired scale-free networks and show that random rewiring increases disassortativity by reducing the average degree of the nearest neighbors of high-degree nodes. The effect can be captured by the measures of the degree correlation that consider all links in the network, but not by analogous measures that consider only links between degree peers, hence the potential for contradicting interpretations. We furthermore find that random and directional rewiring affect the topology of a scale-free network differently, even if the degree correlation of the rewired networks is the same. Consequently, the network dynamics is changed, which is proven here by means of the biased random walk.

Emergence of scale-free characteristics in socio-ecological systems with bounded rationality

Socio-ecological systems are increasingly modelled by games played on complex networks. While the concept of Nash equilibrium assumes perfect rationality, in reality players display heterogeneous bounded rationality. Here we present a topological model of bounded rationality in socio-ecological systems, using the rationality parameter of the Quantal Response Equilibrium. We argue that system rationality could be measured by the average Kullback--Leibler divergence between Nash and Quantal Response Equilibria, and that the convergence towards Nash equilibria on average corresponds to increased system rationality. Using this model, we show that when a randomly connected socio-ecological system is topologically optimised to converge towards Nash equilibria, scale-free and small world features emerge. Therefore, optimising system rationality is an evolutionary reason for the emergence of scale-free and small-world features in socio-ecological systems. Further, we show that in games where multiple equilibria are possible, the correlation between the scale-freeness of the system and the fraction of links with multiple equilibria goes through a rapid transition when the average system rationality increases. Our results explain the influence of the topological structure of socio-ecological systems in shaping their collective cognitive behaviour, and provide an explanation for the prevalence of scale-free and small-world characteristics in such systems.

Universal scaling for the dilemma strength in evolutionary games

Why would natural selection favor the prevalence of cooperation within the groups of selfish individuals? A fruitful framework to address this question is evolutionary game theory, the essence of which is captured in the so-called social dilemmas. Such dilemmas have sparked the development of a variety of mathematical approaches to assess the conditions under which cooperation evolves. Furthermore, borrowing from statistical physics and network science, the research of the evolutionary game dynamics has been enriched with phenomena such as pattern formation, equilibrium selection, and self-organization. Numerous advances in understanding the evolution of cooperative behavior over the last few decades have recently been distilled into five reciprocity mechanisms: direct reciprocity, indirect reciprocity, kin selection, group selection, and network reciprocity. However, when social viscosity is introduced into a population via any of the reciprocity mechanisms, the existing scaling parameters for the dilemma strength do not yield a unique answer as to how the evolutionary dynamics should unfold. Motivated by this problem, we review the developments that led to the present state of affairs, highlight the accompanying pitfalls, and propose new universal scaling parameters for the dilemma strength. We prove universality by showing that the conditions for an ESS and the expressions for the internal equilibriums in an infinite, well-mixed population subjected to any of the five reciprocity mechanisms depend only on the new scaling parameters. A similar result is shown to hold for the fixation probability of the different strategies in a finite, well-mixed population. Furthermore, by means of numerical simulations, the same scaling parameters are shown to be effective even if the evolution of cooperation is considered on the spatial networks (with the exception of highly heterogeneous setups). We close the discussion by suggesting promising directions for future research including (i) how to handle the dilemma strength in the context of co-evolution and (ii) where to seek opportunities for applying the game theoretical approach with meaningful impact.

Dependency links can hinder the evolution of cooperation in the prisoner's dilemma game on lattices and networks

Networks with dependency links are more vulnerable when facing the attacks. Recent research also has demonstrated that the interdependent groups support the spreading of cooperation. We study the prisoner's dilemma games on spatial networks with dependency links, in which a fraction of individual pairs is selected to depend on each other. The dependency individuals can gain an extra payoff whose value is between the payoff of mutual cooperation and the value of temptation to defect. Thus, this mechanism reflects that the dependency relation is stronger than the relation of ordinary mutual cooperation, but it is not large enough to cause the defection of the dependency pair. We show that the dependence of individuals hinders, promotes and never affects the cooperation on regular ring networks, square lattice, random and scale-free networks, respectively. The results for the square lattice and regular ring networks are demonstrated by the pair approximation.

Impact of heterogeneous activity and community structure on the evolutionary success of cooperators in social networks

Recent empirical studies suggest that heavy-tailed distributions of human activities are universal in real social dynamics [L. Muchnik, S. Pei, L. C. Parra, S. D. S. Reis, J. S. Andrade Jr., S. Havlin, and H. A. Makse, Sci. Rep. 3, 1783 (2013)]. On the other hand, community structure is ubiquitous in biological and social networks [M. E. J. Newman, Nat. Phys. 8, 25 (2012)]. Motivated by these facts, we here consider the evolutionary prisoner's dilemma game taking place on top of a real social network to investigate how the community structure and the heterogeneity in activity of individuals affect the evolution of cooperation. In particular, we account for a variation of the birth-death process (which can also be regarded as a proportional imitation rule from a social point of view) for the strategy updating under both weak and strong selection (meaning the payoffs harvested from games contribute either slightly or heavily to the individuals' performance). By implementing comparative studies, where the players are selected either randomly or in terms of their actual activities to play games with their immediate neighbors, we figure out that heterogeneous activity benefits the emergence of collective cooperation in a harsh environment (the action for cooperation is costly) under strong selection, whereas it impairs the formation of altruism under weak selection. Moreover, we find that the abundance of communities in the social network can evidently foster the formation of cooperation under strong selection, in contrast to the games evolving on randomized counterparts. Our results are therefore helpful for us to better understand the evolution of cooperation in real social systems.

Spatial epidemiology of networked metapopulation: an overview

An emerging disease is one infectious epidemic caused by a newly transmissible pathogen, which has either appeared for the first time or already existed in human populations, having the capacity to increase rapidly in incidence as well as geographic range. Adapting to human immune system, emerging diseases may trigger large-scale pandemic spreading, such as the transnational spreading of SARS, the global outbreak of A(H1N1), and the recent potential invasion of avian influenza A(H7N9). To study the dynamics mediating the transmission of emerging diseases, spatial epidemiology of networked metapopulation provides a valuable modeling framework, which takes spatially distributed factors into consideration. This review elaborates the latest progresses on the spatial metapopulation dynamics, discusses empirical and theoretical findings that verify the validity of networked metapopulations, and the sketches application in evaluating the effectiveness of disease intervention strategies as well.

Degree mixing in multilayer networks impedes the evolution of cooperation

Traditionally, the evolution of cooperation has been studied on single, isolated networks. Yet a player, especially in human societies, will typically be a member of many different networks, and those networks will play different roles in the evolutionary process. Multilayer networks are therefore rapidly gaining on popularity as the more apt description of a networked society. With this motivation, we here consider two-layer scale-free networks with all possible combinations of degree mixing, wherein one network layer is used for the accumulation of payoffs and the other is used for strategy updating. We find that breaking the symmetry through assortative mixing in one layer and/or disassortative mixing in the other layer, as well as preserving the symmetry by means of assortative mixing in both layers, impedes the evolution of cooperation. We use degree-dependent distributions of strategies and cluster-size analysis to explain these results, which highlight the importance of hubs and the preservation of symmetry between multilayer networks for the successful resolution of social dilemmas.

Stochastic heterogeneous interaction promotes cooperation in spatial prisoner's dilemma game

Previous studies mostly investigate player's cooperative behavior as affected by game time-scale or individual diversity. In this paper, by involving both time-scale and diversity simultaneously, we explore the effect of stochastic heterogeneous interaction. In our model, the occurrence of game interaction between each pair of linked player obeys a random probability, which is further described by certain distributions. Simulations on a 4-neighbor square lattice show that the cooperation level is remarkably promoted when stochastic heterogeneous interaction is considered. The results are then explained by investigating the mean payoffs, the mean boundary payoffs and the transition probabilities between cooperators and defectors. We also show some typical snapshots and evolution time series of the system. Finally, the 8-neighbor square lattice and BA scale-free network results indicate that the stochastic heterogeneous interaction can be robust against different network topologies. Our work may sharpen the understanding of the joint effect of game time-scale and individual diversity on spatial games.

Social dilemma alleviated by sharing the gains with immediate neighbors

We study the evolution of cooperation in the evolutionary spatial prisoner's dilemma game (PDG) and snowdrift game (SG), within which a fraction α of the payoffs of each player gained from direct game interactions is shared equally by the immediate neighbors. The magnitude of the parameter α therefore characterizes the degree of the relatedness among the neighboring players. By means of extensive Monte Carlo simulations as well as an extended mean-field approximation method, we trace the frequency of cooperation in the stationary state. We find that plugging into relatedness can significantly promote the evolution of cooperation in the context of both studied games. Unexpectedly, cooperation can be more readily established in the spatial PDG than that in the spatial SG, given that the degree of relatedness and the cost-to-benefit ratio of mutual cooperation are properly formulated. The relevance of our model with the stakeholder theory is also briefly discussed.

Effect of initial fraction of cooperators on cooperative behavior in evolutionary prisoner's dilemma game

We investigate the influence of initial fraction of cooperators on the evolution of cooperation in spatial prisoner's dilemma games. Compared with the results of heterogeneous networks, we find that there is a relatively low initial fraction of cooperators to guarantee higher equilibrium cooperative level. While this interesting phenomenon is contrary to the commonly shared knowledge that higher initial fraction of cooperators can provide better environment for the evolution of cooperation. To support our outcome, we explore the time courses of cooperation and find that the whole course can be divided into two sequent stages: enduring (END) and expanding (EXP) periods. At the end of END period, thought there is a limited number of cooperator clusters left for the case of low initial setup, these clusters can smoothly expand to hold the whole system in the EXP period. However, for high initial fraction of cooperators, superfluous cooperator clusters hinder their effective expansion, which induces many remaining defectors surrounding the cooperator clusters. Moreover, through intensive analysis, we also demonstrate that when the tendency of three cooperation cluster characteristics (cluster size, cluster number and cluster shape) are consistent within END and EXP periods, the state that maximizes cooperation can be favored.

Robustness of cooperation on scale-free networks under continuous topological change

In this paper, we numerically investigate the robustness of cooperation clusters in prisoner's dilemma played on scale-free networks, where the network topologies change by continuous removal and addition of nodes. Each removal and addition can be either random or intentional. We therefore have four different strategies in changing network topology: random removal and random addition (RR), random removal and preferential addition (RP), targeted removal and random addition (TR), and targeted removal and preferential addition (TP). We find that cooperation clusters are most fragile against TR, while they are most robust against RP, even for large values of the temptation coefficient for defection. The effect of the degree mixing pattern of the network is not the primary factor for the robustness of cooperation under continuous change in network topology, which is quite different from the cases observed in static networks. Cooperation clusters become more robust as the number of links of hubs occupied by cooperators increase. Our results might infer the fact that a huge variety of individuals is needed for maintaining global cooperation in social networks in the real world where each node representing an individual is constantly removed and added.

Impact of social punishment on cooperative behavior in complex networks

Social punishment is a mechanism by which cooperative individuals spend part of their resources to penalize defectors. In this paper, we study the evolution of cooperation in 2-person evolutionary games on networks when a mechanism for social punishment is introduced. Specifically, we introduce a new kind of role, punisher, which is aimed at reducing the earnings of defectors by applying to them a social fee. Results from numerical simulations show that different equilibria allowing the three strategies to coexist are possible as well as that social punishment further enhance the robustness of cooperation. Our results are confirmed for different network topologies and two evolutionary games. In addition, we analyze the microscopic mechanisms that give rise to the observed macroscopic behaviors in both homogeneous and heterogeneous networks. Our conclusions might provide additional insights for understanding the roots of cooperation in social systems.

Observability transitions in correlated networks

Yang, Wang, and Motter [Phys. Rev. Lett. 109, 258701 (2012)] analyzed a model for network observability transitions in which a sensor placed on a node makes the node and the adjacent nodes observable. The size of the connected components comprising the observable nodes is a major concern of the model. We analyze this model in random heterogeneous networks with degree correlation. With numerical simulations and analytical arguments based on generating functions, we find that negative degree correlation makes networks more observable. This result holds true both when the sensors are placed on nodes one by one in a random order and when hubs preferentially receive the sensors. Finally, we numerically optimize networks with a fixed degree sequence with respect to the size of the largest observable component. Optimized networks have negative degree correlation induced by the resulting hub-repulsive structure; the largest hubs are rarely connected to each other, in contrast to the rich-club phenomenon of networks.

Difference of reciprocity effect in two coevolutionary models of presumed two-player and multiplayer games

Unlike other natural network systems, assortativity can be observed in most human social networks; however, it has been reported that a social dilemma situation represented by a 2×2 prisoner's dilemma game favors dissortativity to enhance cooperation. Our simulations successfully reveal that a public goods game with coevolution for both agents' strategy and network topology encourages assortativity, although it only slightly enhances cooperation as compared to a 2×2 donor and recipient game with a strong dilemma to be solved. This outcome occurs because the network dynamics in a multiplayer game discourages emerging cooperation unlike its beneficial result in a 2×2 prisoner's dilemma game.

Spreading of cooperative behaviour across interdependent groups

Recent empirical research has shown that links between groups reinforce individuals within groups to adopt cooperative behaviour. Moreover, links between networks may induce cascading failures, competitive percolation, or contribute to efficient transportation. Here we show that there in fact exists an intermediate fraction of links between groups that is optimal for the evolution of cooperation in the prisoner's dilemma game. We consider individual groups with regular, random, and scale-free topology, and study their different combinations to reveal that an intermediate interdependence optimally facilitates the spreading of cooperative behaviour between groups. Excessive between-group links simply unify the two groups and make them act as one, while too rare between-group links preclude a useful information flow between the two groups. Interestingly, we find that between-group links are more likely to connect two cooperators than in-group links, thus supporting the conclusion that they are of paramount importance.

Optimal interdependence between networks for the evolution of cooperation

Recent research has identified interactions between networks as crucial for the outcome of evolutionary games taking place on them. While the consensus is that interdependence does promote cooperation by means of organizational complexity and enhanced reciprocity that is out of reach on isolated networks, we here address the question just how much interdependence there should be. Intuitively, one might assume the more the better. However, we show that in fact only an intermediate density of sufficiently strong interactions between networks warrants an optimal resolution of social dilemmas. This is due to an intricate interplay between the heterogeneity that causes an asymmetric strategy flow because of the additional links between the networks, and the independent formation of cooperative patterns on each individual network. Presented results are robust to variations of the strategy updating rule, the topology of interdependent networks, and the governing social dilemma, thus suggesting a high degree of universality.