# 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
Nevertheless, the difference with the NW network is that although is larger, \(f_{c}\) is higher; however, when the result of \(r_{1}~{}+~{}r_{2}\) is the same, \(f_{c}\) of the double-layer BA network with a larger value of \(|r_{1}-r_{2}|\) is higher. Similar to the double-layer NW network, when \(r_{1}~{}+~{}r_{2}\) is small, \(f_{c}\) of a network with a large \(\omega\) is slightly higher. Moreover, with the increase in \(r_{1}~{}+~{}r_{2}\), \(f_{c}\) of a network with a small \(\omega\) exceeds the network with a large \(\omega\). The reason for the different performance of \(f_{c}\) on the double-layer BA network with different \(\omega\) in Figure 8 is linked to the double-layer NW network. The greater the value of \(r_{1}~{}+~{}r_{2}\), the more collaborators will exist by high rewards. \(\omega\) improves the transmission between the two layers of networks. When the reward is small, a network with a large \(\omega\) will promote cooperation more. On the contrary, a network with a small \(\omega\) will help maintain cooperation. Next, the following will focus on analyzing the reasons for the higher \(f_{c}\) on a network with a larger value of the \(|r_{1}-r_{2}|\) when the \(r_{1}~{}+~{}r_{2}\) is the same. It can be known from previous research that the scale-free network has a higher ability to promote cooperation than the small-world network [46]. We know that there are some nodes with high connectivity in the scale-free network due to the characteristics of the network, also known as hub nodes [17]. If the hub node in the BA network chooses to cooperate, the influence on its neighbor nodes is great. Therefore, in the double-layer BA network, if \(r\) of a layer network is large, even if \(r\) of the other layer is small, the network with a larger \(r\) value will play a leading role due to the existence of the learning probability between the layers. Thus, when is the same, the double-layer BA network with a larger \(|r_{1}-r_{2}|\) has stronger promotion ability for cooperation. Figure 7: The relationship between \(r_{1}\) and \(f_{c}\) in a double-layer NW network with different \(\omega\) and \(r_{2}\) values. Solid line: \(\omega\) = 0.2. Dashed line: \(\omega\) = 0.6. To explore the impact of different network sizes on cooperation, it is observed from Figure 9 that whether it is a double-layer NW or a double-layer BA network, for a relatively small \(r\), as \(\omega\) increases, the network cooperation levels of different scales show an increasing trend; however, for a relatively large \(r\), with the increase in \(\omega\), networks of various sizes have shown a state of declining cooperation rates. It shows that the existence of \(\omega\) is of great help to the network with a small \(r\); however, for a network with a relatively large \(r\), a lower \(\omega\) should be selected. In addition, it is seen that the relationship between \(\omega\), \(r\), and \(f_{c}\) has a certain degree of robustness to the network scale. Even under different network sizes, changes in \(\omega\) and \(r\) of the same type of network have roughly the same effect on the fraction of cooperators. At the same time, in Figure 9a,b, \(f_{c}\) of the double-layer NW network shows a decreasing characteristic according to the increase in \(N\) under the same conditions; however, the double-layer BA network does not have this feature because of the existence of hub nodes and the particularity of its structure. Figure 8: Color maps where \(f_{c}\) changes with the enhancement factors (\(r_{1}\), \(r_{2}\)) in the double-layer BA network. The results were acquired with the combination of \(M_{0}=1\), where (**a**) \(\omega=0.2\) and (**b**) \(\omega=0.6\).