# Study of Information Dissemination in Hypernetworks with Adjustable Clustering Coefficient
## 3 Information Dissemination Model Based on RP Strategy
### Theoretical Analysis
Thus, the time \(t_{i}\) follows a gamma distribution with parameters \((i,\lambda)\), which leads to the following expression  
\[P(k_{ij}^{h}(t)\geq k)=P(m((\frac{t}{t_{i}})^{\frac{m_{2}}{m_{1}+m_{2}}})\geq k )=P(t_{i}\leq(\frac{m}{k})^{\frac{m_{1}}{m_{2}}+1}t) \tag{9}\]  
By using Equations (8) and (9), we have  
\[P(t_{i}\leq(\frac{m}{k})^{\frac{m_{1}}{m_{2}}+1}t)=1-e^{-(\frac{m}{k})^{\frac{ m_{1}}{m_{2}}+1}\lambda t}\sum_{l=0}^{By substituting Equation (10) into the expression, the transient hyperdegree distribution of this hypernetwork can be derived as follows  
\[ \begin{split} P\{k_{ij}^{h},t\}\approx\frac{1}{m_{1}E[N(t)]}\sum_{ ij}P(k_{ij}^{h}(t)=k)\\ =(\frac{m_{1}}{m_{2}}+1)(\frac{m}{k})^{\frac{m_{1}}{m_{2}}+2}\frac {\lambda t}{m}\frac{(\lambda t(m/k)^{\frac{m_{1}}{m_{2}}+1})^{i-1}}{(i-1)!}e^ {-\lambda t(\frac{m}{k})\frac{m_{1}}{m_{2}}+1}\end{split}  \tag{11}\]  
Using Equation (11), we can express the average hyperdegree distribution of the clustered hypernetwork as  
\[ \begin{split} P(k)\approx\lim_{n\to\infty}m_{1}E[N(t)]\sum_{ ij}P(k_{ij}^{h}(t)=k)=\frac{1}{m}(\frac{m_{1}}{m_{2}}+1)(\frac{m}{k})^{\frac{m_{1}}{ m_{2}}+2}\end{split}  \tag{12}\]  
We then proceed by assuming that the dynamical mean-field reaction rate equation based on the mean-field theory model [19] can be expressed as  
\[ \begin{split}\partial_{t}\rho_{k}(t)=-\gamma\rho_{k}(t)+\beta k (m_{1}+m_{2}-1)[1-\rho_{k}(t)]\Theta(\rho_{k}(t))\end{split}  \tag{13}\]  
where \(\rho_{k}\) is the relative density of I-state nodes with hyperdegree \(k\). The left side of the equation is the rate of change in the density of the node in the I-state with hyperdegree \(k\) at time step \(t\). The first term on the right is the annihilation term, which indicates that the I-state node with hyperdegree \(k\) reverts to the S-state with probability \(\gamma\). The second term is the generation term, which indicates that the S-state(\(1-\rho_{k}(t)\)) node with hyperdegree \(k\) acquires information to become an I-state node with probability \(\beta\). In steady state, \(\partial_{t}\rho_{k}(t)=0\), and Equation (13) can be written as  
\[ \begin{split}\rho_{k}=\frac{\beta k(m_{1}+m_{2}-1)\Theta(\beta, \gamma)}{\gamma+\beta k(m_{1}+m_{2}-1)\Theta(\beta,\gamma)}\end{split}  \tag{14}\]  
where the probability of a hyperedge linking to an informed node is  
\[ \begin{split}\Theta(\beta,\gamma)=\sum_{k}\frac{kP(k)\rho_{k}}{ \sum_{s}sP(s)}\end{split}  \tag{15}\]  
Bringing Equation (14) into Equation (15), we obtain  
\[\begin{split}\Theta(\beta,\gamma)=\sum_{k}\frac{kP(k)}{\sum_{s} sP(s)}\frac{\beta k(m_{1}+m_{2}-1)\Theta(\beta,\gamma)}{\gamma+\beta k(m_{1}+  
Substituting Equations (12) and (14) into \(\rho(t)=\sum_{k}P(k)\rho_{k}(t)\), we have  
\[ \begin{split}&\rho=\int_{m}^{\infty}\frac{1}{m}(\frac{m_{1}}{m_{2}}+1)( \frac{m}{k})^{\frac{m_{1}}{m_{2}}+2}\frac{\beta k(m_{1}+m_{2}-1)\Theta(\beta, \gamma)}{\gamma+\beta k(m_{1}+m_{2}-1)\Theta(\beta,\gamma)}dk\\ &\quad=\frac{1}{m}(\frac{m_{1}}{m_{2}}+1)m^{\frac{m_{1}}{m_{2}}+2 }\beta(m_{1}+m_{2}-1)\Theta(\beta,\gamma)(\int_{m}^{\infty}\frac{1}{\gamma} \frac{1}{k^{\frac{m_{1}}{m_{2}}}+1}dk\\ &\quad-\frac{\beta(m_{1}+m_{2}-1)\Theta(\beta,\gamma)}{\gamma} \int_{m}^{\infty}\frac{1}{k^{\frac{m_{1}}{m_{2}}}}\frac{1}{\gamma+\beta k(m_{ 1}+m_{2}-1)\Theta(\beta,\gamma)}dk)\end{split}  \tag{19}\]  
From Equation (18), the solution of Equation (19) is  
\[\rho=\frac{m(m_{1}+m_{2})(m_{1}+m_{2}-1)\beta\Theta(\beta,\gamma)(1-\Theta( \beta,\gamma))}{\gamma m_{1}} \tag{20}\]  
Equation (20) indicates that the informed node density \(\rho\) at the steady state is a function that is not influenced by the clustering coefficient \(p\) or the time step \(t\). Since there is \(m_{1}\) in the denominator of Equation (20), the effect of \(m_{2}\) is greater than the effect of \(m_{1}\). We set the effective propagation rate \(\lambda=\beta/\gamma\), and \(\lambda\) is proportional to \(\rho\). \(\rho\) can be calculated from the structural parameters of the hypernetwork as well as the spreading rate and the recovering rate.