# SIS Epidemic Propagation on Scale-Free Hypernetwork
## 4 Simulation Experiment and Analysis
### Comparative Analysis of Hypernetwork and Complex Network Structures  
To compare the propagation process of infectious diseases under different network structures, two networks with a scale of \(N=2\) 000 were generated, respectively. The first network is a hypernetwork. The construction process is based on Section 3.1, and the parameters are set as \(m_{1}=2\), \(m_{2}=4\), and \(m=2\). The second network structure is a simple scale-free network. The construction process is based on the BA scale-free model, and the number of edges added at each time step is \(m=2\). After the analysis, the average hyperdegree in the obtained hypernetwork is \(\langle d_{H}\rangle\approx 6\), and the average degree in the complex network is \(\langle k\rangle\approx 4\). During the spread process, the node numbered 31 (with a degree of 6) and the node numbered 45 (with a degree of 4) were selected as the initial spreading nodes in the two networks, respectively. Let \(\beta=0.2\) and \(\gamma=0.05\). Figure 5a shows the change curve of the spread of infectious diseases under the two network structures. It can be seen that in the hypernetwork structure, the infectious disease spreads rapidly to the entire network, while the comprehensive spread of the infectious disease in the simple scale-free network takes a certain amount of time. Under the hypernetwork structure, when the system reaches a steady state, the proportion of infected individuals is also higher than in the complex network structure. In fact, in the hypernetwork structure, each individual will be influenced by all other neighbors in the social contact relationship. The contact between individuals is a higher-order interaction that is more complex than the pairwise interaction in a simple scale-free network, and the propagation will be faster, showing clear clustering characteristics of epidemic spread. Therefore, the hypernetwork structure can better reflect the reality and simulate the spread process and situation of infectious diseases in the real world.  
Figure 4: Densities of \(\rho\) obtained by theoretical analysis and simulations in the stable state.  
Furthermore, we construct a hypernetwork whose structural parameters are set as \(m_{1}=1\), \(m_{2}=1\), \(m=2\) (the mean value of the hyperdegree in the hypernetwork is \(\langle d_{H}\rangle\approx 4\)), aiming at comparing the infectious disease propagation process in the simple scale-free network generated by the BA model. In the two networks, the node numbered 39 (with a degree of 4) and the node numbered 41 (with a degree of 4) were selected as the initial spreading nodes. Taking \(\beta=0.2\), \(\gamma=0.1\), we test the change curve of density of infected individuals under two network structures. As shown in Figure 5b, under the same parameters of propagation and recovery rates, from the initial propagation trend to the steady state, the infectious disease propagation curves of the simple scale-free network and the hypernetwork structure are basically consistent. In fact, when the hypernetwork structure parameters are set to \(m_{1}=1\), \(m_{2}=1\), \(m=2\), the model degenerates into a simple scale-free network model, and the simulation results fully illustrate this feature. Therefore, the epidemic propagation model on the classical complex network BA model can be regarded as a special case of the model in this paper. In addition, the introduction of hyperedges in hypernetworks can better describe the interaction between multiple nodes in  
Figure 5: The simulated density of infected nodes \(\rho\) vs. time \(t\) for different network structures. (a) Contrast between scale-free hypernetwork and simple scale-free network; (b) Contrast between degenerated scale-free hypernetwork and simple scale-free network.  
the intricate social network. This unique advantage is beyond the reach of simple scale-free network methods.