# Cooperation in Social Dilemmas: A Group Game Model with Double-Layer Networks
## 4 Experiment and Analysis
### PGG-SP on the Static Network
#### 4.1.2 Exploring Cooperation in Double-layer Network  
This section examines the impact of the double-layer network in terms of cooperation. Figure 6 shows the changes of \(f_{c}\) in the double-layer NW network. From the graph we can see that when the \(r_{1}+r_{2}\) is a fixed value, is in a fixed range. Moreover, \(f_{c}\) increases with the increase in \(r_{1}+r_{2}\). It can be seen that the growth rate of \(f_{c}\) decreases with the increase in \(\omega\). Accordingly, it can be seen that the critical value becomes larger when reaching basic full cooperation (Figure 6a,b). Figure 4: \(f_{c}\) versus \(r\) for the double-layer NW networks with different \(\omega\) values and a single-layer NW network, \(p=0.0005\). Figure 5: \(f_{c}\) versus \(r\) with the double-layer BA network with different \(\omega\) values and a single-layer BA network, \(M_{0}=1\). From Figure 7, when \(r_{1}\) is fixed, \(f_{c}\) increases with the increase in \(r_{2}\), which is consistent with a larger \(r_{1}~{}+~{}r_{2}\) and the larger \(f_{c}\) shown in Figure 6. When \(r_{2}\) is fixed, a network with a large \(\omega\) (dashed line) has a higher \(f_{c}\) than a network with a small \(\omega\) (solid line) when \(r_{1}\) is small; however, as \(r_{1}\) increases, the \(f_{c}\) of a network with a small \(\omega\) gradually overtakes a network with a large \(\omega\). In addition, the critical \(r_{1}\) value of the overtaking point decreases as \(r_{2}\) increases. Even for some networks with large \(r_{2}\) values (such as \(r_{2}>8\)), the \(f_{c}\) of a network with a small \(\omega\) is always higher than that of a network with a large \(\omega\). The following explains the situation of the group game in the double-layer NW network, combined with the change process of the cooperation levels shown in Figures 6 and 7 with different learning probabilities for layers and enhancement factors for the two layers. In the double-layer NW network, the larger the value of \(r_{1}~{}+~{}r_{2}\), the higher the reward for choosing cooperation. This leads to a higher cooperation level in the network. It can be known from the structure of a double-layer NW network that each layer NW network has the same characteristics of the small average path and the high clustering coefficient [16], so that as long as the sum of the enhancement factors of the two layers is consistent, the network cooperation level is fixed, showing a symmetrical status. In addition, the learning probability of different layers \(\omega\) increases the degree of connection between the upper and lower layers in the game, so that the transmission rate of cooperation is increased between  
Figure 6: Color maps where \(f_{c}\) changes with the enhancement factors (\(r_{1},r_{2}\)) in the double-layer NW network. The results were acquired with the combination of \(p=0.0005\), where (**a**) \(\omega=0.2\) and (**b**) \(\omega=0.6\). layers. Therefore, when the network with a large \(\omega\) is compared with a network with a small \(\omega\), even if the enhancement factor is small, it is difficult to maintain the cooperative cluster within the layer; however, the transfer of information between the layers because of \(\omega\) will help maintain cooperation. Thus, the structural characteristics of the double-layer network lead to a strong ability to promote cooperation so that the rate of cooperation increases rapidly; however, as the enhancement factor becomes larger, cooperative clusters within the network layer are easily formed, then the learning probability between the layers instead limits the improvement of the cooperation level. Therefore, under the larger enhancement factor, the network fraction of cooperators with a large \(\omega\) is gradually smaller than that with a small \(\omega\) due to the slow growth rate. If we now consider the double-layer BA network shown in Figure 8, the fraction of cooperators also exhibits a symmetrical distribution.