# Cooperation in Social Dilemmas: A Group Game Model with Double-Layer Networks
## 4 Experiment and Analysis
### PGG-SP on the Static Network  
#### 4.1.1 Comparison of the Single-layer and Double-layer Networks  
A set of questions was designed to compare single-layer and double-layer networks. Figure 4 shows \(f_{c}\) obtained by different approaches in terms of the single- and double-layer NW networks. We observe that \(f_{c}\) increased with the increase in \(r\). When \(r\) is small, the double-layer NW network had a higher \(f_{c}\) value than the single-layer NW network. Moreover, the \(f_{c}\) of the double-layer NW network increased with the increase in \(\omega\) as the reward level of the group for cooperation is low if the \(r\) value is small. The information transmission ability of the double-layer network was higher than that of the single-layer network, such that the higher the probability of different layer learning, the stronger the influence between the layers of the double-layer network. Accordingly, cooperation in the network was more likely to emerge and be maintained, which resulted in the improvement of the level of cooperation. Thus, it can be seen that the double-layer NW network promotes cooperation; however, as \(r\) increases, when \(r>r_{x}\) (\(r_{x}\) is a constant), the \(f_{c}\) of the single-layer NW network gradually becomes higher than that of the double-layer NW network. In addition, the critical value \(r_{x}\) increases with the increase in \(\omega\). It means that the larger \(\omega\) is, the larger \(r\) is needed when the \(f_{c}\) of the single-layer network exceeds the double-layer network. When \(r>r_{y}\) (\(r_{y}\) is a constant), under the same \(r\), the \(f_{c}\) of the double-layer NW network decreases with the increase in \(\omega\). Through analysis, it is found that there are constants (\(r_{x}\), \(r_{y}\)) that exist as critical values, and they change with different network parameters. Furthermore, with the aforementioned conditions, the cooperation growth rate becomes slower. Understandably, when a group's reward level for cooperation is high enough, cooperative clusters are created in the network via the increase in \(r\). Currently, the \(\omega\) of the double-layer NW network becomes an interference factor. Moreover, it increases with the increase in \(\omega\), such that the \"free rider\" phenomenon appears and the growth rate of cooperation decreases. Figure 5 shows \(f_{c}\) for the single- and double-layer BA networks. It can be observed that the BA network's ability to promote cooperation was higher than that of the NW network. Likewise, the various changes are similar to the NW network. When \(r\) was small (such as \(r~{}<~{}2.5\)), the double-layer BA network promoted greater cooperation than the single-layer BA network. Additionally, the \(f_{c}\) of a network with a large \(\omega\) is slightly higher under the same \(r\) value; however, the growth rate of \(f_{c}\) decreases with the increase in \(\omega\). Even when \(\omega\) is too large, the \(f_{c}\) of the single-layer network exceeds the double-layer network with a larger \(r\) (such as \(\omega\) = 0.9, \(r\) = 3). This is an interesting result. In a network with a relatively large \(r\), a single-layer network can produce a sufficient cooperative effect. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Meaning** & **Default Value** \\ \hline \(\Delta\) & Influence factor & 0.01 \\ \(k\) & Influence factor & 0.1 \\ \(T\) & The probability of reconnection edge & 0.5 \\ \(N\) & Network size & 900 \\ \(p\) & The edge addition probability in the NW network & \\ \(M_{0}\) & The number of edges added per time step in the BA network & \\ \(\omega\) & Learning probability for different layers & \\ \(r\) & Average enhancement factor & \\ \(r_{1}\) & Enhancement factor of the first layer network & \\ \(r_{2}\) & Enhancement factor of the second layer network & \\ \(f_{c}\) & Fraction of cooperators & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The list of parameters. NW: [PERSON]–Watts. BA: [PERSON]–Albert.