### Theoretical Framework
#### 3.3.5. User Propagation Influence-Based Linear Threshold Model (UPI-LT)
As shown in Figure 12c, the influence values are expressed as follows:  
* For node \(v_{4}\), the total influence is \(Sum_{P(v_{4})}=P(v_{3},v_{4})\times e^{-0.2(v_{2}-t_{1})}+P(v_{1},v_{4}) \approx 0.322\), and \(Sum_{P(v_{4})}>0.3\), so node \(v_{4}\) is successfully activated. * For node \(v_{6}\), the total influence is \(Sum_{P(v_{6})}=P(v_{3},v_{6})\times e^{-0.2(v_{2}-t_{1})}+P(v_{2},v_{6}) \approx 0.346\), and \(Sum_{P(v_{6})}<\theta(v_{6})=0.4\), so node \(v_{6}\) is not yet activated. Thus, at time \(T_{2}\), the set of activated nodes is \(A_{T_{2}}=\{v_{1},v_{2},v_{3},v_{4}\}\). At time \(T_{3}\), node \(v_{4}\) attempts to activate its follower, \(v_{5}\). As shown in Figure 12d, \(Sum_{P(v_{5})}=P(v_{4},v_{5})=0.2\), and \(\theta(v_{5})=0.15\), so \(P(v_{4},v_{5})>\theta(v_{5})\), and node \(v_{5}\) is successfully activated. Thus, at time \(T_{3}\), the set of activated nodes is \(A_{T_{3}}=\{v_{1},v_{2},v_{3},v_{4},v_{5}\}\). At time \(T_{4}\), the newly activated node (\(v_{5}\)) attempts to activate its followers, \(v_{6}\) and \(v_{7}\). As shown in Figure 12e,  
* For node \(v_{6}\), the total influence is \(Sum_{P(v_{6})}=P(v_{3},v_{6})\times e^{-0.2(v_{3}-t_{1})}+P(v_{2},v_{6}) \times e^{-0.2(t_{3}-t_{2})}+P(v_{5},v_{6})\approx 0.483\), and since \(Sum_{P(v_{6})}>\theta(v_{6})\), node \(v_{6}\) is successfully activated. Figure 12: Diagram of the dynamic information dissemination process. For node \(v_{7}\), the total influence is \(Sum_{P(v_{7})}=P(v_{5},v_{7})=0.25\), and \(\theta(v_{7})=0.6\), so \(P(v_{5},v_{7})<\theta(v_{7})\), and node \(v_{7}\) is not activated. Thus, at time \(T_{4}\), the set of activated nodes is \(A_{T_{4}}=\{v_{1},v_{2},v_{3},v_{4},v_{5},v_{6}\}\). At time \(T_{5}\), the activated node (\(v_{5}\)) attempts, again, to activate its follower, \(v_{7}\). As shown in Figure 12f, the total influence on node \(v_{7}\) is \(Sum_{P(v_{7})}=P(v_{5},v_{7})\times e^{-0.2(t_{5}-t_{4})}=0.204\), and since \(\theta(v_{7})=0.6\), \(P(v_{5},v_{7})<\theta(v_{7})\), and node \(v_{7}\) remains inactive. Given the decay of information over time, node \(v_{7}\) is not activated. At this point, no further nodes can be activated, and the information propagation process concludes. The final set of activated nodes in the network is \(A_{node}=\{v_{1},v_{2},v_{3},v_{4},v_{5},v_{6}\}\). The UPI-LT model introduces the following three key innovations that distinguish it from existing models: the homophily ratio, time decay factor, and dynamic thresholds. These innovations specifically enhance the model's ability to accurately capture the propagation of influence in dynamic social networks. The homophily ratio measures the degree of similarity between users and directly influences user activation thresholds. Traditional models often assume uniform influence between users, neglecting the fact that in real social networks, individuals with similar characteristics are more likely to influence each other. The UPI-LT model incorporates homophily as a critical factor, allowing nodes with higher similarity to have a stronger likelihood of activating each other. This approach captures real-world social dynamics more effectively, making influence propagation predictions more accurate. The time decay factor is designed to reflect the temporal aspect of influence within a social network. In many existing models, the influence of a node is considered static, remaining constant over time. However, in reality, the relevance of information diminishes as time passes. The UPI-LT model introduces a time decay factor that gradually reduces a node's influence as time progresses, accurately reflecting the diminishing impact of outdated information. By incorporating the time decay factor, our model better captures the temporal nature of information dissemination, leading to more realistic modeling of influence spread in dynamic environments. Unlike traditional models that employ fixed activation thresholds, the UPI-LT model features dynamic thresholds that adapt to changing network conditions.