Feedback reciprocity mechanism promotes the cooperation of highly clustered scale-free networks

Human group size puzzle: why it is odd that we live in large societies

Human groups tend to be much larger than those of non-human primates. This is a puzzle. When ecological factors do not limit primate group size, the problem of coordination creates an upper threshold even when cooperation is guaranteed. This paper offers a model of group coordination towards behavioural synchrony to spell out the mechanics of group size limits, and thus shows why it is odd that humans live in large societies. The findings suggest that many of our species' evolved social behaviours and culturally maintained social technologies emerged as solutions to this problem.

Clustering drives cooperation on reputation networks, all else fixed

Reputation-based cooperation on social networks offers a causal mechanism between graph properties and social trust. Using a simple model, this paper demonstrates the underlying mechanism in a way that is accessible to scientists not specializing in networks or mathematics. The paper shows that when the size and degree of the network is fixed (i.e. all graphs have the same number of agents, who all have the same number of connections), it is the clustering coefficient that drives differences in how cooperative social networks are.

Prisoner's dilemma game model Based on historical strategy information

In many dilemmas, decisions are determined not by a single factor, but by multiple ones, including memory, reputation, reward and punishment. In recent years, how to design a mechanism to promote cooperation has become a research hot-spot. However, most of the previous studies mainly consider the historical benefits of the game, and pay less attention to the stability of the strategy (the frequency of strategy changes in the length of memory) and the proportion of memory in decision-making. The decision-making process of group evolution involves the influence of memory information on cooperative evolution in multi round games. It makes up for the lack of stability factors and weights in previous studies. Based on the above factors, a new strategy update rule is proposed to study the influence of the stability of historical strategy information on the evolution of cooperation in prisoner's dilemma game, and the influence of memory weight on cooperation is considered. The stability of the current strategy is measured by the strategy in historical memory (the number of times the strategy in memory is continuous and consistent with the current strategy), which can determine the probability of an individual learning the neighbor strategy next time. Numerical simulation shows that an appropriate increase in the length of historical memory is more conducive to the emergence of cooperation, and the greater the weight of historical strategy information is, the more conducive to promoting cooperation, which shows that historical strategy information is still the main factor in decision-making. This study will help us understand the cooperative evolution of many real systems, such as nature, biology, society and so on, and effectively design reasonable mechanisms to promote cooperation.

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.

Networks of reliable reputations and cooperation: a review

Reputation has been shown to provide an informal solution to the problem of cooperation in human societies. After reviewing models that connect reputations and cooperation, we address how reputation results from information exchange embedded in a social network that changes endogenously itself. Theoretical studies highlight that network topologies have different effects on the extent of cooperation, since they can foster or hinder the flow of reputational information. Subsequently, we review models and empirical studies that intend to grasp the coevolution of reputations, cooperation and social networks. We identify open questions in the literature concerning how networks affect the accuracy of reputations, the honesty of shared information and the spread of reputational information. Certain network topologies may facilitate biased beliefs and intergroup competition or in-group identity formation that could lead to high cooperation within but conflicts between different subgroups of a network. Our review covers theoretical, experimental and field studies across various disciplines that target these questions and could explain how the dynamics of interactions and reputations help or prevent the establishment and sustainability of cooperation in small- and large-scale societies.

This article is part of the theme issue ‘The language of cooperation: reputation and honest signalling’.

Critical phenomena and strategy ordering with hub centrality approach in the aspiration-based coordination game

We study the coordination game with an aspiration-driven update rule in regular graphs and scale-free networks. We prove that the model coincides exactly with the Ising model and shows a phase transition at the critical selection noise when the aspiration level is zero. It is found that the critical selection noise decreases with clustering in random regular graphs. With a non-zero aspiration level, the model also exhibits a phase transition as long as the aspiration level is smaller than the degree of graphs. We also show that the critical exponents are independent of clustering and aspiration level to confirm that the coordination game belongs to the Ising universality class. As for scale-free networks, the effect of aspiration level on the order parameter at a low selection noise is examined. In model networks (the Barabási-Albert network and the Holme-Kim network), the order parameter abruptly decreases when the aspiration level is the same as the average degree of the network. In contrast, in real-world networks, the order parameter decreases gradually. We explain this difference by proposing the concepts of hub centrality and local hub. The histogram of hub centrality of real-world networks separates into two parts unlike model networks, and local hubs exist only in real-world networks. We conclude that the difference of network structures in model and real-world networks induces qualitatively different behavior in the coordination game.

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.

Zero-Determinant Strategies in Iterated Public Goods Game

Recently, Press and Dyson have proposed a new class of probabilistic and conditional strategies for the two-player iterated Prisoner’s Dilemma, so-called zero-determinant strategies. A player adopting zero-determinant strategies is able to pin the expected payoff of the opponents or to enforce a linear relationship between his own payoff and the opponents’ payoff, in a unilateral way. This paper considers zero-determinant strategies in the iterated public goods game, a representative multi-player game where in each round each player will choose whether or not to put his tokens into a public pot, and the tokens in this pot are multiplied by a factor larger than one and then evenly divided among all players. The analytical and numerical results exhibit a similar yet different scenario to the case of two-player games: (i) with small number of players or a small multiplication factor, a player is able to unilaterally pin the expected total payoff of all other players; (ii) a player is able to set the ratio between his payoff and the total payoff of all other players, but this ratio is limited by an upper bound if the multiplication factor exceeds a threshold that depends on the number of players.

Evolutionary dynamics of synergistic and discounted group interactions in structured populations

The emergence of cooperation between unrelated individuals enables researchers to study how the collective cooperative behavior survives in a world where egoists could get more short-term benefits. The spatial multi-player games, which invoke interactions between individuals who are not directly linked by the interactive networks, are drawing more and more attention in exploring the evolution of cooperation. Here we address the evolutionary dynamics in infinite structured populations with discounted, linear, and synergistic group interactions. The five classical scenarios are recovered from the dynamics: (i) dominating defection, (ii) dominating cooperation, (iii) co-existence, (iv) bi-stability, and (v) neutral variants. For linear interactions, the evolutionary dynamics is equivalent to that in finite as well as the well-mixed counterparts, which can be achieved by a payoff matrix transformation, and it illustrates that the more neighbors there are, the harder the cooperators survive. Yet both cooperation and defection emerge easier in finite populations than in infinite for discounted and synergistic interactions. Counterintuitively, we find that the synergistic group interactions always raise cooperators׳ barriers to occupy the population with the increase of the number of neighbors in infinite structured populations. Our results go against the common belief that synergistic interactions are necessarily beneficial for the cooperative behavior.

Analytical description for the critical fixations of evolutionary coordination games on finite complex structured populations

Evolutionary game theory is crucial to capturing the characteristic interaction patterns among selfish individuals. In a population of coordination games of two strategies, one of the central problems is to determine the fixation probability that the system reaches a state of networkwide of only one strategy, and the corresponding expectation times. The deterministic replicator equations predict the critical value of initial density of one strategy, which separates the two absorbing states of the system. However, numerical estimations of this separatrix show large deviations from the theory in finite populations. Here we provide a stochastic treatment of this dynamic process on complex networks of finite sizes as Markov processes, showing the evolutionary time explicitly. We describe analytically the effects of network structures on the intermediate fixations as observed in numerical simulations. Our theoretical predictions are validated by various simulations on both random and scale free networks. Therefore, our stochastic framework can be helpful in dealing with other networked game dynamics.

Peer pressure: enhancement of cooperation through mutual punishment

An open problem in evolutionary game dynamics is to understand the effect of peer pressure on cooperation in a quantitative manner. Peer pressure can be modeled by punishment, which has been proved to be an effective mechanism to sustain cooperation among selfish individuals. We investigate a symmetric punishment strategy, in which an individual will punish each neighbor if their strategies are different, and vice versa. Because of the symmetry in imposing the punishment, one might intuitively expect the strategy to have little effect on cooperation. Utilizing the prisoner's dilemma game as a prototypical model of interactions at the individual level, we find, through simulation and theoretical analysis, that proper punishment, when even symmetrically imposed on individuals, can enhance cooperation. Also, we find that the initial density of cooperators plays an important role in the evolution of cooperation driven by mutual punishment.

Percolation on interacting networks with feedback-dependency links

When real networks are considered, coupled networks with connectivity and feedback-dependency links are not rare but more general. Here, we develop a mathematical framework and study numerically and analytically the percolation of interacting networks with feedback-dependency links. For the case that all degree distributions of intra- and inter- connectivity links are Poissonian, we find that for a low density of inter-connectivity links, the system undergoes from second order to first order through hybrid phase transition as coupling strength increases. It implies that the average degree k of inter-connectivity links has a little influence on robustness of the system with a weak coupling strength, which corresponds to the second order transition, but for a strong coupling strength corresponds to the first order transition. That is to say, the system becomes robust as k increases. However, as the average degree k of each network increases, the system becomes robust for any coupling strength. In addition, we find that one can take less cost to design robust system as coupling strength decreases by analyzing minimum average degree kmin of maintaining system stability. Moreover, for high density of inter-connectivity links, we find that the hybrid phase transition region disappears, the first order region becomes larger and second order region becomes smaller. For the case of two coupled scale-free networks, the system also undergoes from second order to first order through hybrid transition as the coupling strength increases. We find that for a weak coupling strength, which corresponds to the second order transitions, feedback dependency links have no effect on robustness of system relative to no-feedback condition, but for strong coupling strength which corresponds to first order or hybrid phase transition, the system is more vulnerable under feedback condition comparing with no-feedback condition. Thus, for designing resilient system, designers should try to avoid the feedback dependency links, because the existence of feedback-dependency links makes the system extremely vulnerable and difficult to defend.

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.

Impact of roles assignation on heterogeneous populations in evolutionary dictator game

The evolution of cooperation is a hot and challenging topic in the field of evolutionary game theory. Altruistic behavior, as a particular form of cooperation, has been widely studied by the ultimatum game but not by the dictator game, which provides a more elegant way to identify the altruistic component of behaviors. In this paper, the evolutionary dictator game is applied to model the real motivations of altruism. A degree-based regime is utilized to assess the impact of the assignation of roles on evolutionary outcome in populations of heterogeneous structure with two kinds of strategic updating mechanisms, which are based on Darwin's theory of evolution and punctuated equilibrium, respectively. The results show that the evolutionary outcome is affected by the role assignation and that this impact also depends on the strategic updating mechanisms, the function used to evaluate players' success, and the structure of populations.

Punishment based on public benefit fund significantly promotes cooperation

In prisoner's dilemma game (shortly, PD game), punishment is most frequently used to promote cooperation. However, outcome varies when different punishment approaches are applied. Here the PD game is studied on a square lattice when different punishment patterns are adopted. As is known to all, tax system, a common tool to adjust the temperature of the economy, is widely used in human society. Inspired by this philosophy, players in this study would pay corresponding taxes in accordance with their payoff level. In this way, public benefit fund is established consequently and it would be utilized to punish defectors. There are two main methods for punishing: slight intensity of punishment (shortly, SLP) and severe intensity of punishment (shortly, SEP). When the totaling of public benefit fund keeps relatively fixed, SLP extends further, which means more defectors would be punished; by contrast, SEP has a smaller coverage. It is of interest to verify whether these two measures can promote cooperation and which one is more efficient. Simulate results reveal that both of them can promote cooperation remarkably. Specifically speaking, SLP shows constant advantage from the point of view either of fractions of cooperation or average payoff.

Influence of reciprocal links in social networks

How does reciprocal links affect the function of real social network? Does reciprocal link and non-reciprocal link play the same role? Previous researches haven't displayed a clear picture to us until now according to the best of our knowledge. Motivated by this, in this paper, we empirically study the influence of reciprocal links in two representative real datasets, Sina Weibo and Douban. Our results demonstrate that the reciprocal links play a more important role than non-reciprocal ones in information diffusion process. In particular, not only coverage but also the speed of the information diffusion can be significantly enhanced by considering the reciprocal effect. We give some possible explanations from the perspectives of network connectivity and efficiency. This work may shed some light on the in-depth understanding and application of the reciprocal effect in directed online social networks.

Evolutionary dynamics of group interactions on structured populations: a review

Interactions among living organisms, from bacteria colonies to human societies, are inherently more complex than interactions among particles and non-living matter. Group interactions are a particularly important and widespread class, representative of which is the public goods game. In addition, methods of statistical physics have proved valuable for studying pattern formation, equilibrium selection and self-organization in evolutionary games. Here, we review recent advances in the study of evolutionary dynamics of group interactions on top of structured populations, including lattices, complex networks and coevolutionary models. We also compare these results with those obtained on well-mixed populations. The review particularly highlights that the study of the dynamics of group interactions, like several other important equilibrium and non-equilibrium dynamical processes in biological, economical and social sciences, benefits from the synergy between statistical physics, network science and evolutionary game theory.

Bipartite graphs as models of population structures in evolutionary multiplayer games

By combining evolutionary game theory and graph theory, “games on graphs” study the evolutionary dynamics of frequency-dependent selection in population structures modeled as geographical or social networks. Networks are usually represented by means of unipartite graphs, and social interactions by two-person games such as the famous prisoner’s dilemma. Unipartite graphs have also been used for modeling interactions going beyond pairwise interactions. In this paper, we argue that bipartite graphs are a better alternative to unipartite graphs for describing population structures in evolutionary multiplayer games. To illustrate this point, we make use of bipartite graphs to investigate, by means of computer simulations, the evolution of cooperation under the conventional and the distributed N-person prisoner’s dilemma. We show that several implicit assumptions arising from the standard approach based on unipartite graphs (such as the definition of replacement neighborhoods, the intertwining of individual and group diversity, and the large overlap of interaction neighborhoods) can have a large impact on the resulting evolutionary dynamics. Our work provides a clear example of the importance of construction procedures in games on graphs, of the suitability of bigraphs and hypergraphs for computational modeling, and of the importance of concepts from social network analysis such as centrality, centralization and bipartite clustering for the understanding of dynamical processes occurring on networked population structures.

Velocity-enhanced cooperation of moving agents playing public goods games

In this paper we study the evolutionary dynamics of the public goods game in a population of mobile agents embedded in a two-dimensional space. In this framework, the backbone of interactions between agents changes in time, allowing us to study the impact that mobility has on the emergence of cooperation in structured populations. Our results point out that a low degree of mobility enhances cooperation in the system. In addition, we study the impact of the size of the groups in which games are played on cooperation. Again we find a rise and fall of cooperation related to the percolation point of the instant interaction networks created by the set of mobile agents.

Adaptive and bounded investment returns promote cooperation in spatial public goods games

The public goods game is one of the most famous models for studying the evolution of cooperation in sizable groups. The multiplication factor in this game can characterize the investment return from the public good, which may be variable depending on the interactive environment in realistic situations. Instead of using the same universal value, here we consider that the multiplication factor in each group is updated based on the differences between the local and global interactive environments in the spatial public goods game, but meanwhile limited to within a certain range. We find that the adaptive and bounded investment returns can significantly promote cooperation. In particular, full cooperation can be achieved for high feedback strength when appropriate limitation is set for the investment return. Also, we show that the fraction of cooperators in the whole population can become larger if the lower and upper limits of the multiplication factor are increased. Furthermore, in comparison to the traditionally spatial public goods game where the multiplication factor in each group is identical and fixed, we find that cooperation can be better promoted if the multiplication factor is constrained to adjust between one and the group size in our model. Our results highlight the importance of the locally adaptive and bounded investment returns for the emergence and dominance of cooperative behavior in structured populations.

Evolutionary games defined at the network mesoscale: the Public Goods game

The evolutionary dynamics of the Public Goods game addresses the emergence of cooperation within groups of individuals. However, the Public Goods game on large populations of interconnected individuals has been usually modeled without any knowledge about their group structure. In this paper, by focusing on collaboration networks, we show that it is possible to include the mesoscopic information about the structure of the real groups by means of a bipartite graph. We compare the results with the projected (coauthor) and the original bipartite graphs and show that cooperation is enhanced by the mesoscopic structure contained. We conclude by analyzing the influence of the size of the groups in the evolutionary success of cooperation.