### Theoretical Framework
#### 3.3.5. User Propagation Influence-Based Linear Threshold Model (UPI-LT)  
In the UPI-LT model, a user's activation threshold is primarily influenced by two factors, namely the information propagation capability of neighboring nodes and the level of homophily among nodes in the surrounding environment. The detailed methods for calculating these factors are provided in previous sections. The formula for calculating a user's activation threshold in the UPI-LT model is expressed as follows:  
\[\theta_{i}=\alpha_{1}\frac{\sum_{v_{ij}\in neighbor^{in}}Inf_{e}(v_{i})}{|v_{ i}|}+(1-\alpha_{1})\frac{S}{E} \tag{28}\]  
where \(\theta_{i}\) is the activation threshold of node \(i\); \(Inf_{e}(v_{i})\) is the propagation influence of neighboring node \(v_{i}\); and \(\frac{S}{E}\) represents the homophily coefficient, where \(S\) is the number of edges between nodes of the same color and \(E\) is the total number of edges. Figure 10. The information dissemination process considers the time attenuation factor. After calculating the \(\theta_{i}\) value, it is necessary to normalize \(\theta_{i}\) as follows:  
\[\theta_{i}^{\prime}=\frac{\theta_{i}-\min(\theta_{i})}{\max(\theta_{i})-\min( \theta_{i})} \tag{29}\]  
The final activation threshold is given by  
\[\theta(i)=1-\theta_{i}^{\prime} \tag{30}\]  
The framework of the proposed UPI-LT model is illustrated in Figure 11. As shown in the figure, the improvements made to the LT in the UPI-LT model lie in the consideration of the propagation influence of neighboring nodes on the user's ability to receive information. On this basis, the model introduces the user's propagation influence and time decay factor, dynamically accumulating their effects over time. Figure 12 provides an example to explain the dynamic changes in propagation probability during information dissemination. In the figure, black nodes represent users currently in an activated state, while white nodes are inactive. The red dashed lines indicate the attempts of activated nodes to influence inactive nodes. Additionally, each node in Figure 12 is assigned an activation threshold, and the edges between nodes are assigned random weights to represent the influence probabilities between users. A time decay factor of \(\alpha=0.2\) is set. The process of information dissemination with time decay characteristics is explained as follows:  
The initial state of the network at time \(T_{0}\) and the connections between nodes are shown in Figure 12a. The network consists of 7 nodes and 9 directed edges. The black nodes represent activated nodes, and the red dashed lines represent activation attempts between nodes. Figure 11: Model block diagram of the UPI-LT.  
At time \(T_{1}\), node \(v_{3}\) is the seed node. As shown in Figure 12b, node \(v_{3}\) attempts to activate its followers, which include \(v_{1}\), \(v_{2}\), \(v_{4}\), and \(v_{6}\), as indicated by the red dashed arrows. The cumulative influence on each node is calculated as follows:  
* \(Sum_{P(v_{1})}=P(v_{3},v_{1})=0.15\), and \(\theta(v_{1})=0.1\), so \(P(v_{3},v_{1})>\theta(v_{1})\), resulting in node \(v_{1}\) being successfully activated. * \(Sum_{P(v_{2})}=P(v_{3},v_{2})=0.2\), and \(\theta(v_{2})=0.15\), so \(P(v_{3},v_{2})>\theta(v_{2})\), resulting in node \(v_{2}\) being successfully activated. * \(Sum_{P(v_{4})}=P(v_{3},v_{4})=0.15\), and \(\theta(v_{4})=0.3\), so \(P(v_{3},v_{4})<\theta(v_{4})\), meaning node \(v_{4}\) is not activated at this moment. * \(Sum_{P(v_{6})}=P(v_{3},v_{6})=0.3\), and \(\theta(v_{6})=0.4\), so \(P(v_{3},v_{6})<\theta(v_{6})\), meaning node \(v_{6}\) is not activated. Thus, at time \(T_{1}\), the set of activated nodes is \(A_{T_{1}}=\{v_{1},v_{2},v_{3}\}\). At time \(T_{2}\), node \(v_{1}\) attempts to activate its follower, \(v_{4}\), while node \(v_{2}\) attempts to activate its follower, \(v_{6}\).