^{1}

^{2}

^{1}

^{1}

^{1}

^{1}

^{2}

In order to study the influence of network’s structure on cooperation level of repeated snowdrift game, in the frame of two kinds of topologically alterable network models, the relation between the cooperation density and the topological parameters was researched. The results show that the network’s cooperation density is correlated reciprocally with power-law exponent and positively with average clustering coefficient; in other words, the more homogenous and less clustered a network, the lower the network’s cooperation level; and the relation between average degree and cooperation density is nonmonotonic; when the average degree deviates from the optimal value, the cooperation density drops.

Recently,

During a long period of time, all complex networks were regarded as random graphs. However, the American scientists Barabási and Albert overturned this presumption by their research on a

Firstly, we distribute alterable normalized weights to isolated nodes; then we link the nodes with corresponding probability. We can control the power-law coefficient by tuning the weights and the total number of edges. The specific algorithm is as follows.

The average degree is

Clearly, parameter

Barabási and Albert proposed the BA scale-free model; the degree distribution complies with power-law distribution; however, the network’s average clustering coefficient is relatively low which does not agree with realistic conditions. The clustering coefficient is the average connecting probability of two nodes which belong to the neighborhood of a common node, and it describes the network’s local structural property. We adopt the clustering coefficient alterable model in the literature [

The game theory is about the rational behavior in interdependent relations [

There are some classical models, such as the prisoner’s dilemma game, the hawk-dove game, the snowdrift game, and the public goods game [

In a pairwise game, each player can choose either to cooperate or to defect; he gets different payoffs according to the strategies that he and his rival adopt. The payoff matrix is shown in Table

Payoff matrix.

Cooperation | Defection | |
---|---|---|

Cooperation | ( |
( |

Defection | ( |
( |

In every round of game, each individual interacts with all their neighbors and then obtains payoff according to the payoff matrix. After each round, individual

Cooperation density

When pursuing the analytical solutions, theoretical tools such as the mean field theory have been very successful on regular networks. However, they fail when dealing with complicated network structures while numerical simulations are still effective. In this section, we simulate the repeated snowdrift game on power-law alterable and clustering coefficient alterable networks and analyze the influence of power-law coefficient, average degree, and clustering coefficient on the emergence of cooperation.

In this section, we simulate and analyze the influence of power-law coefficient on the emergence of cooperation in a power-law coefficient alterable model where the total nodes number is

Figure

Figure

By observing the three curves in Figure

The average degree portrays the denseness of the network from a microscopic perspective; in other words, the network gets denser as we increase the average degree. In this section, we study the effect of different average degrees on the cooperation density. Every data is averaged over 100 model simulations. In every model, we conducted 10000 time steps of which the former 8000 time steps were to guarantee that the system has entered a steady state, and the latter 2000 time steps were to generate the average cooperation density

Figure

The relation between the cooperation density

The average clustering coefficient describes the clustering characteristic of a complex network. Most realistic networks are clustered; for instance, one’s friends know each other in social network.

BA scale-free networks have relatively low average clustering coefficient which disagree with realistic conditions. In that, we studied the influence of average clustering coefficient on the emergence of cooperation on a clustering coefficient alterable scale-free network. Let

As shown in Figure

Figure

The emergence of cooperation on complex networks is of great importance, and it provides a new explanation of the population behavior evolution in both nature and society. Based on a power-law alterable model and a clustering coefficient alterable model, we studied the prisoners dilemma game on scale-free networks with synchronous update of Schlag strategy.

The simulation results show that the cooperation level is correlated reciprocally with the power-law coefficient of the network and the relation between the cooperation density and the network average degree because there exists an optimal average degree corresponding to the highest cooperation level. The cooperation level has a positive relation with the average clustering coefficient when the latter is limited to a certain range.

The authors declare that there is no conflict of interests regarding the publication of this paper.