# Cooperation in Social Dilemmas: A Group Game Model with Double-Layer Networks
## 4 Experiment and Analysis
### PGG-SP on the Dynamic Network
#### 4.2.2 Comparison of the Dynamic Single-layer and Double-layer NW Networks  
To compare the difference between dynamic single-layer and double-layer NW networks, we run PGG-SP on two networks separately. Figure 16 shows the changes of \(f_{c}\) under different \(r\) in the dynamic single-layer network and double-layer network, respectively. It is found from Figure 16b that the game results of the dynamic double-layer NW network are approximately the same, in which \(f_{c}\) can be distributed above or near the average value during each game; however, the results show dispersion and randomness in the dynamic single-layer NW network in Figure 16a, especially under the medium-sized enhancement factor, the dispersion of the evolutionary results is stronger. Comparing the two results, the evolution of PGG-SP on a dynamic double-layer network is more evenly distributed. In addition, under the same conditions, \(f_{c}\) in a dynamic double-layer network is greater. In a single-layer NW network, the cooperators will quickly form clusters due to the existence of dynamic topology; however, there is still the possibility of defectors invading. If the cooperators do not form a stable cluster, they are still facing a crisis of erosion. On the contrary, in the double-layer NW network, the second layer network will increase the ability of the dynamic layer network to resist risks, thus it is more conducive to the formation and maintenance of stable clusters. In other words, the dynamic layer of the double-layer NW network improves the ability to generate cooperation, and the static layer improves the ability to maintain cooperation. In summary, these results provide important insights into the evolutionary stability of PGG-SP on dynamic double-layer networks.  
Figure 14: The strategy evolution of the double-layer BA network in the game, where a pair of nodes simultaneously choose to cooperate (orange), or choose to defect (purple), or one cooperative but the other defect (green). The results were acquired with the combination of \(M_{0}=1\) and \(\omega=0.2\). (**a**) \(r_{1}=r_{2}=2\), (**b**) \(r_{1}=3\), \(r_{2}=1\), (**c**) \(r_{1}=4\), \(r_{2}=2\).  
Figure 13: The evolution of the double-layer BA network in the game; the cooperation rate of the first layer network (black solid line), the cooperation rate of the second layer network (red solid line), the average cooperation rate of the network (blue solid line), the largest node \(b_{i}\) of the first layer (green solid line), the largest node \(b_{i}\) of the second layer (purple solid line). The results were acquired with the combination of \(M_{0}=1\) and \(\omega=0.2\). (**a**) \(r_{1}=r_{2}=2\), (**b**) \(r_{1}=3\), \(r_{2}=1\), (**c**) \(r_{1}=4\), \(r_{2}=2\).  
Figure 16: When the value of \(r\) is different, (**a**) the dynamic single-layer NW network, (**b**) the dynamic double-layer NW network under the same scale for 20 operations, where \(p=0.0005\) and \(\omega=0.2\).  
Figure 15: \(f_{c}\) versus \(r\) on the static and dynamic double-layer NW networks with different \(\omega\) under the same \(p=0.0005\).