Prisoner＇s Dilemma Game with Heterogeneous Influential Effect on Regular Small-World Networks

The effect of heterogeneous influence of different individuals on the maintenance of co-operative behaviour is studied in an evolutionary Prisoner＇s Dilemma game with players located on the sites of regular small-world networks. The players interacting with their neighbours can either co-operate or defect and update their states by choosing one of the neighbours and adopting its strategy with a probability depending on the pay off difference. The selection of the neighbour obeys a preferential rule： the more influential a neighbour, the larger the probability it is picked. It is found that this simple preferential selection rule can promote continuously the co-operation of the whole population with the strengthening of the disorder of the underlying network.     

Prisoner's Dilemma Game on Two Types of Positively Correlated Networks

We study the effects of degree correlations on the evolution of cooperation in the prisoner＇s dilemma game with individuals located on two types of positively correlated networks. It is shown that the positive degree correlation can either promote or inhibit the emergence of cooperation depending on network configurations. Furthermore, we investigate the probability to cooperate as a function of connectivity degree, and find that high-degree individuals generally have a higher tendency to cooperate. Finally, it is found that small-degree individuals usually change their strategy more frequently, and such change is shown to be unfavourable to cooperation for both kinds of networks.     

Evolutionary snowdrift game with disordered environments in mobile societies

We investigate an evolutionary snowdrift game on a square $N=L\times L$ lattice with periodic boundary conditions, where a population of $n_{0}$ ($n_{0}\leq N$) players located on the sites of this lattice can either cooperate with or defect from their nearest neighbours. After each generation, every player moves with a certain probability $p$ to one of the player's nearest empty sites. It is shown that, when $p=0$, the cooperative behaviour can be enhanced in disordered structures. When $p>0$, the effect of mobility on cooperation remarkably depends on the payoff parameter $r$ and the density of individuals $\rho$ ($\rho=n_{0}/N$). Compared with the results of $p=0$, for small $r$, the persistence of cooperation is enhanced at not too small values of $\rho$; whereas for large $r$, the introduction of mobility inhibits the emergence of cooperation at any $\rho<1$; for the intermediate value of $r$, the cooperative behaviour is sometimes enhanced and sometimes inhibited, depending on the values of $p$ and $\rho$. In particular, the cooperator density can reach its maximum when the values of $p$ and $\rho$ reach their respective optimal values. In addition, two absorbing states of all cooperators and all defectors can emerge respectively for small and large $r$ in the case of $p>0$.     

Dynamic Evolution with Limited Learning Information on a Small-World Network

This paper investigates the dynamic evolution with limited learning information on a small-world network. In the system, the information among the interaction players is not very lucid, and the players are not allowed to inspect the profit collected by its neighbors, thus the focal player cannot choose randomly a neighbor or the wealthiest one and compare its payoff to copy its strategy. It is assumed that the information acquainted by the player declines in the form of the exponential with the geographical distance between the players, and a parameter V is introduced to denote the inspect-ability about the players. It is found that under the hospitable conditions, cooperation increases with the randomness and is inhibited by the large connectivity for the prisoner＇s dilemma; however, cooperation is maximal at the moderate rewiring probability and is chaos with the connectivity for the snowdrift game. For the two games, the acuminous sight is in favor of the cooperation under the hospitable conditions; whereas, the myopic eyes are advantageous to cooperation and cooperation increases with the randomness under the hostile condition.     

Adaptive co-evolution of strategies and network leading to optimal cooperation level in spatial prisoner’s dilemma game

We study evolutionary prisoner's dilemma game on adaptive networks where a population of players co-evolves with their interaction networks. During the co-evolution process, interacted players with opposite strategies either rewire the link between them with probability $p$ or update their strategies with probability $1-p$ depending on their payoffs. Numerical simulation shows that the final network is either split into some disconnected communities whose players share the same strategy within each community or forms a single connected network in which all nodes are in the same strategy. Interestingly, the density of cooperators in the final state can be maximised in an intermediate range of $p$ via the competition between time scale of the network dynamics and that of the node dynamics. Finally, the mean-field analysis helps to understand the results of numerical simulation. Our results may provide some insight into understanding the emergence of cooperation in the real situation where the individuals' behaviour and their relationship adaptively co-evolve.     

Spatial Games Based on Pursuing the Highest Average Payoff

We propose a strategy updating mechanism based on pursuing the highest average payoff to investigate the prisoner＇s dilemma game and the snowdrift game. We apply the new rule to investigate cooperative behaviours on regular, small-world, scale-free networks, and find spatial structure can maintain cooperation for the prisoner＇s dilemma game. fn the snowdrift game, spatial structure can inhibit or promote cooperative behaviour which depends on payoff parameter. We further study cooperative behaviour on scale-free network in detail. Interestingly, non-monotonous behaviours observed on scale-free network with middle-degree individuals have the lowest cooperation level. We also find that large-degree individuals change their strategies more frequently for both games.     

Integrating the environmental factor into the strategy updating rule to promote cooperation in evolutionary games

Many previous studies have shown that the environment plays an important role for social individuals.In this paper,we integrate the environmental factor,which is defined as the average payoff of all a player’s neighbours,with the standard Fermi updating rule by introducing a tunable parameter,ω.It is found that the level of cooperation increases remarkably,and that the cooperators can better resist the invasion of defection with an increase in ω.This interesting phenomenon is then explained from a microscopic view.In addition,the universality of this mechanism is also proved with the help of the small-world network and the random regular graph.This work may be helpful in understanding cooperation behaviour in species from unicellular organisms up to human beings.     

Cooperation influenced by the correlation degree of two-layered complex networks in evolutionary prisoner’s dilemma games

An evolutionary prisoner's dilemma game is investigated on two-layered complex networks respectively representing interaction and learning networks in one and two dimensions. A parameter q is introduced to denote the correlation degree between the two-layered networks. Using Monte Carlo simulations we studied the effects of the correlation degree on cooperative behaviour and found that the cooperator density nontrivially changes with q for different payoff parameter values depending on the detailed strategy updating and network dimension. An explanation for the obtained results is provided.     

Effect of Mixing Assortativity on Extreme Events in Complex Networks

We investigate the effect of mixing assortativity on the occurrence of extreme events in complex networks. The bias random walk model is adopted with a preferential transition probability tuned by a parameter We derive exact expressions for the stationary distribution probability and for the occurrence probability of extreme events. They reveal that the occurrence of extreme events strongly depends on the mixing assortativity of the network. It is shown that, for non-assortative （rk = 0）, assortative （rk = 0.15） and disassortative （rk = -0.15） scale-free networks, the minimal occurrence of extreme events will happen at α= -1.0, α = -0.6 ands = 0.2, respectively.     

Evolutionary prisoner’s dilemma game on highly clustered community networks

This paper studies the evolutionary prisoner's dilemma game on a highly clustered community network in which the clustering coefficient and the community size can be tuned. It finds that the clustering coefficient in such a degree-homogeneous network inhibits the emergence of cooperation for the entire range of the payoff parameter. Moreover, it finds that the community size can also have a marked influence on the evolution of cooperation, with a larger community size leading to not only a lower cooperation level but also a smaller threshold of the payoff parameter above which cooperators become extinct.