# Study of Information Dissemination in Hypernetworks with Adjustable Clustering Coefficient
## 3 Information Dissemination Model Based on RP Strategy
### Theoretical Analysis  
By employing techniques of the mean field theory, the following dynamical equation can be derived for \(k_{ij}^{h}(t)\) at the preferential attachment step  
\[\frac{\partial_{t}k_{ij}^{h}}{\partial_{t}}=\lambda[m-(m-1)p]m_{2}\frac{k_{ ij}^{h}}{\sum_{w}k_{w}^{h}} \tag{2}\]  
where \(\lambda\) is the rate of node batches' arrival. If \((m-1)p\) is the number of hyperedges generated by the clustering attachment, then \(m-(m-1)p\) is the number of hyperedges generated by the preferential attachment, and the above equation multiplied by \(m_{2}\) is the total number of old nodes selected by the preferential attachment. When a new node connects with an old node \(u\) and the old node is also a neighbor of another node \(j\), the second node \(j\) is selected at random to form a new hyperedge with the  
Figure 2: Description of the global information transmission process. Red nodes represent I-state individuals and hollow nodes represent S-state individuals. (**a**) At the initial moment, all nodes are in the S-state. (**b**) When \(t=1\), the \(v_{1}\) node is set as the spreading seed (I-state). (**c**) When \(t=2\), the neighbor nodes \(v_{3}\), \(v_{7}\), \(v_{8}\), and \(v_{10}\) of \(v_{1}\) are notified to change to the I-state. (**d**) When \(t=3\), nodes \(v_{4}\), \(v_{5}\), and \(v_{9}\) transition to the I-state, whereas nodes \(v_{1}\) and \(v_{8}\) revert to the S-state. new node. During this process, \(k_{ij}^{h}(t)\) is governed by the following dynamical equation at the aggregation attachment step:  
\[\frac{\partial_{t}k_{ij}^{h}}{\partial_{t}}=\lambda(m-1)pm_{2}\sum_{u\in j} \Omega_{j}\frac{k_{iu}^{h}}{\sum_{w}k_{w}^{h}}\frac{1}{(m_{1}-1)k_{iu}^{h}}= \lambda(m-1)pm_{2}\frac{k_{ij}^{h}}{\sum_{w}k_{w}^{h}} \tag{3}\]  
where \(\Omega_{j}\) represents the set of neighbors of node \(j\). Similar to the idea discussed in Equation (2), \((m-1)pm_{2}\) is the total number of old nodes selected for the clustering attachment. Utilizing Equations (2) and (3), the dynamical equation for the total rate of change in the hyperdegree of node \(j\) within a single time step can be derived:  
\[\frac{\partial_{t}k_{ij}^{h}}{\partial_{t}}=\lambda[m-(m-1)p]m_{2}\frac{k_{ij }^{h}}{\sum_{w}k_{w}^{h}}+\lambda(m-1)pm_{2}\frac{k_{ij}^{h}}{\sum_{w}k_{w}^{ h}}=\lambda mm_{2}\frac{k_{ij}^{h}}{\sum_{w}k_{w}^{h}} \tag{4}\]  
If \(m_{1}\) new nodes and \(m_{2}\) existing nodes form a new hyperedge, the hyperdegree of the respective node increases by \((m_{1}+m_{2})\) at each time step. At moment \(t\), a total of \(m\) new hyperedges are formed. Using the notation \(\sum_{ij}k_{ij}^{h}(t)\approx m(m_{1}+m_{2})E[N(t)]=\lambda m(m_{1}+m_{2})t\), it follows that  
\[\frac{\partial_{t}k_{ij}^{h}}{\partial_{t}}=\lambda mm_{2}\frac{k_{ij}^{h}}{m (m_{1}+m_{2})t}=\frac{m_{2}k_{ij}^{h}}{(m_{1}+m_{2})t} \tag{5}\]  
The initial condition for Equation (5) conforms to the following equation:  
\[k_{ij}^{h}(t_{i})=m \tag{6}\]  
where \(t_{i}\) represent the time at which the \(i_{th}\) batch node enters the hypernetwork. Furthermore, the node of the \(i_{th}\) batch emerges at time \(t_{i}\). Therefore, \(k_{ij}^{h}(t_{i})\) denotes the hyperdegree of the \(j_{th}\) node in the \(i_{th}\) batch at the moment of emergence. Using Equation (6), we can obtain the solution to Equation (5) as follows  
\[k_{ij}^{h}(t_{i})=m(\frac{t}{t_{i}})^{\frac{m_{2}}{m_{1}+m_{2}}} \tag{7}\]  
Through Equation (7), we obtain  
\[P(k_{ij}^{h}(t)\geq k)=P(m((\frac{t}{t_{i}})^{\frac{m_{2}}{m_{1}+m_{2}}})\geq k )=P(t_{i}\leq(\frac{m}{k})^{\frac{m_{1}}{m_{2}}+1}t) \tag{8}\]  
It is worth observing that the process of node batch arrivals follows a Poisson process with rate \(\lambda\).