# Cooperation in Social Dilemmas: A Group Game Model with Double-Layer Networks
## 4 Experiment and Analysis
### PGG-SP on the Dynamic Network
#### 4.2.3 Performance of the Dynamic Double-layer NW Network  
The purpose of this experiment was to explore the influence of \(\omega\) and \(r\) on the dynamic double-layer NW network. It is observed from Figure 17 that for a dynamic double-layer NW network, \(f_{c}\) increases with the increase in \(\omega\) if \(r\) is small; however, \(f_{c}\) decreases with the increase in \(\omega\) when \(r\) is large. This is a significant result. It is because when \(r\) is very small, the original single-layer network reward is very small, and the larger \(\omega\) can communicate with the two layers and transfer cooperation quickly. But when \(r\) is large enough, a cluster of cooperators can be formed in the single-layer network, so a larger \(\omega\) affects the divergence of cooperation. Accordingly, it is similar with the performance of a static double-layer NW network, which indicates that the existence of a learning probability \(\omega\) may not only promote the production of cooperation, but also limit the divergence of cooperation. The following analysis is used to express the synergy effect in the dynamic double-layer NW network. It can be seen from Figure 18 that the change of the first layer dynamic network change rate is greater than the second layer static network. For example, when the network faces the risk of full defections, it will accelerate the collapse of the network; however, when the network tends to comprehensively cooperate, the speed of network cooperation will be accelerated, which indicates that the dynamic network layer is more sensitive to numerous changes in the network. In Figure 19, when the network gradually collapses, the choices of the double-layer network tend to defect. On the contrary, when the enhancement factor is large enough, the double-layer network has an increasing tendency to choose cooperation at the same time. This reflects the synergy of the double-layer network, which is similar with the situation in reality, where individuals often make the same choices in different social or network relationships due to various attributes, personalities, and abilities. Figure 17: When the value of \(r\) is different, \(f_{c}\) of the dynamic double-layer NW network varies with \(\omega\), where \(p\) = 0.0005, \(\omega\) = 0.2. Figure 19: The strategy evolution of the dynamic double-layer NW network in the game, where a pair of nodes simultaneously choose to cooperate (orange), or to defect (purple), or one is cooperative but the other defects (green). The results were acquired with the combination of \(p\) = 0.0005 and \(\omega\) = 0.2. (a) r = 4, (b) r = 7, (c) r = 10. Figure 20: _Cont._  
Figure 18: The evolution of the dynamic double-layer NW network in the game, with the cooperation rate of the first layer network (black solid line) and the cooperation rate of the second layer network (red solid line). The results were acquired with the combination of \(p\) = 0.0005, \(\omega\) = 0.2. (a) r = 4, (b) r = 7, (c) r = 10.  
cooperators, while the network with a relatively large enhancement factor \(r\) can already show the embryonic form of a \"cooperators cluster\". Over time, the network with a small enhancement factor will be completely swallowed by the defectors; however, a network with a large enhancement factor will form a stronger and larger \"cooperators cluster\". At this time, only sporadic defectors are distributed around the network, so that it is no longer possible to form a piece of structure. The greater the enhancement factor, the greater and faster the chance of a defector being assimilated into a cooperator, and thus it is more likely to form \"comprehensive cooperation\". The purpose of Figure 21 is to show the edge changes when the double-layer network changes dynamically. As can be seen from Figure 21, when the enhancement factor is larger, the number of change edges in the dynamic layer network gradually decreases. Moreover, the greater the value of \(r\), the faster the dynamic edges decrease.