# SIS Epidemic Propagation on Scale-Free Hypernetwork
## 3 Epidemic Models in Hypernetworks
### Theoretical Analysis
In fact, extending the conclusion in BA scale-free networks in simple scale-free networks, given any hyperedge, the probability that it encircles a node with hyperdegree \(s\) is proportional to \(sP(s)\), yielding  
\[\Theta(\beta,\gamma)=\sum\limits_{k}\frac{kP(k)\rho_{k}}{\sum_{s}sP(s)} \tag{4}\]  
Substituting Equation (3) into Equation (4), and \(k=\sum\limits_{s}sP(s)\), we have  
\[\Theta(\beta,\gamma)=\frac{1}{\langle k\rangle}\int_{m}^{\infty}k\cdot P(k) \cdot\frac{\beta k(m_{1}+m_{2}-1)\Theta(\beta,\gamma)}{\gamma+\beta k(m_{1}+m _{2}-1)\Theta(\beta,\gamma)}dk \tag{5}\]  
The hyperdegree distribution of the constructed hypernetwork is \(P(k)=\frac{1}{m}\left(\frac{m_{1}}{m_{2}}+1\right)\left(\frac{m}{k}\right)^{ \frac{m_{1}}{m_{2}}+2}\), yielding  
\[\langle k\rangle=\int_{m}^{\infty}\mathrm{k}P(k)\mathrm{d}k=\frac{1}{m}\left( \frac{m_{1}}{m_{2}}+1\right)\mathrm{m}^{\frac{m_{1}}{m_{2}}+2}\int_{m}^{ \infty}\frac{1}{k^{\frac{m_{1}}{m_{2}}+1}}\mathrm{d}k=\frac{(m_{1}+m_{2})m}{ m_{1}} \tag{6}\]  
Substituting the hyperdegree distribution of \(P(k)\) and Equation (6) into Equation (5), leads to  
\[\Theta(\beta,\gamma)=\frac{m_{1}}{m(m_{1}+m_{2})}\int_{m}^{\infty}k\frac{1}{m }\left(\frac{m_{1}}{m_{2}}+1\right)\left(\frac{m}{k}\right)^{\frac{m_{1}}{m_{2 }}+2}\frac{\beta k(m_{1}+m_{2}-1)\Theta(\beta,\gamma)}{\gamma+\beta k(m_{1}+m _{2}-1)\Theta(\beta,\gamma)}dk=\frac{m_{1}m^{\frac{m_{1}}{m_{2}}+2}}{m^{2}m_{ 2}}\int_{m}^{\infty}\frac{1}{k^{\frac{m_{1}}{m_{2}}+1}}\frac{\beta k(m_{1}+m_ {2}-1)\Theta(\beta,\gamma)}{\gamma+\beta k(m_{1}+m_{2}-1)\Theta(\beta,\gamma )}dk \tag{7}\]  
Reducing Equation (7), we have  
\[\frac{m_{2}}{m_{1}m^{\frac{m_{1}}{m_{2}}}\beta(m_{1}+m_{2}-1)}=\int_{m}^{ \infty}\frac{1}{k^{\frac{m_{1}}{m_{2}}}}\frac{1}{\gamma+\beta k(m_{1}+m_{2}-1 )\Theta(\beta,\gamma)}dk \tag{8}\]  
Substituting Equation (3) into \(\rho(t)=\sum\limits_{k}P(k)\rho_{k}(t)\), leads to  
\[\begin{array}{c}\rho=\int_{m}^{\infty}\frac{1}{m}\left(\frac{m_{1}}{m_{2}}+1 \right)\left(\frac{m}{k}\right)^{\frac{m_{1}}{m_{2}}+2}\frac{\beta k(m_{1}+m _{2}-1)\Theta(\beta,\gamma)}{\gamma+\beta k(m_{1}+m_{2}-1)\Theta(\beta, \gamma)}dk\\ =\frac{1}{m}\left(\frac{m_{1}}{m_{2}}+1\right)\left(m\right)^{\frac{m_{1}}{m_ {2}}+2}\beta(m_{1}+m_{2}-1)\Theta(\beta,\gamma)\int_{m}^{\infty}\frac{1}{k^Equation (11) has a trivial solution \(\Theta=0\). For the equation to have a non-zero solution, the condition must be satisfied:  
\[\frac{d}{d\Theta}\left(\frac{1}{\langle k\rangle}\sum\limits_{k}{kP(k)\frac{ \lambda k((m_{1}+m_{2}-1))\Theta}{1+\lambda k((m_{1}+m_{2}-1))\Theta}}\right) \Bigg{|}_{\Theta=0}\geq 1 \tag{12}\]  
\[\frac{1}{\langle k\rangle}\sum\limits_{k}{kP(k)\lambda k((,m_{1}+m_{2}-1,))}\geq 1 \tag{13}\]  
Propagation critical value can be solved as  
\[\rho_{k}=\frac{\beta k(m_{1}+m_{2}-1)\Theta(\beta,\gamma)}{\gamma+\beta k(m_{ 1}+m_{2}-1)\Theta(\beta,\gamma)} \tag{3}\]0  
where \(\langle k\rangle=\sum\limits_{k}{kP(k)}\) is the average hyperdegree of the nodes in the hypernetwork, \(\langle k^{2}\rangle=\sum\limits_{k}{k^{2}P(k)}\) is the second-order moment of the node hyperdegree. The above formula shows that the propagation threshold is inversely proportional to \((m_{1}+m_{2}-1)\). Namely, when the total number of neighbor nodes of the selected susceptible node in each hyperedge is greater, the more favorable the propagation is, the smaller the threshold will be. When \(m_{1}=m_{2}=1\), the model degenerates into a BA scale-free network on a complex network, at this time \(\lambda_{c}=\langle k\rangle/\langle k^{2}\rangle\). When the scale of the scale-free hypernetwork tends to infinity (\(N\rightarrow\infty\)), the second-order moment \(\langle k^{2}\rangle\rightarrow\infty\), \(\lambda_{c}\to 0\), thus there is no propagation threshold for the global propagation based on the reaction process policy on the scale-free hypernetwork, which indicates that even a small source of infection can spread across a large network.