### Theoretical Framework
#### 3.3.3 The Impact of Time Decay on Information Diffusion  
The dissemination of information in social networks is often subject to diminishing influence over time, much like the cooling of an object as described by [PERSON]'s law of cooling. In the context of information propagation, as time passes, the relevance and impact of a given piece of information gradually decreases until it is eventually ignored or forgotten by the recipients. To model this phenomenon, this study adapts [PERSON]'s law of cooling to capture the decay in information influence over time. The law of cooling states that the rate at which an object's temperature decreases is proportional to the difference between the object's temperature and the ambient temperature of its surroundings. The mathematical expression for this law is given by Equation (16). \[T^{\prime}(t)=-\alpha(T(t)-H) \tag{16}\]  
where \(T(t)\) represents the temperature of the object at time \(t\), \(T^{\prime}(t)\) is the rate of temperature decrease, \(H\) is the ambient temperature, and \(\alpha\) is a constant that regulates the rate of cooling. The difference (\(T(t)-H\)) reflects how much hotter the object is compared to its environment. This differential equation can be solved to model the temperature (\(T(t)\)) at any given time as follows:  
\[\frac{T^{\prime}(t)}{T(t)-H}=-\alpha \tag{17}\] \[\int\frac{T^{\prime}(t)}{T(t)-H}dt=\int(-\alpha)dt\] (18) \[\ln(T(t)-H)=-\alpha t+c\] (19) \[T(t)-H=e^{-\alpha t+c}\] (20) \[T(t)=H+Ce^{-\alpha t} \tag{21}\]  
Assuming that at time \(t_{0}\), the object's temperature is \(T(t_{0})\) (abbreviated as \(T_{0}\)), substitution into the above equation yields the following:  
\[T(t_{0})=H+Ce^{-\alpha t_{0}} \tag{22}\] \[C=(T(t_{0})-H)e^{\alpha t_{0}} \tag{23}\]  
Substituting Equation (23) into Equation (21) yields  
\[T(t)=H+(T(t_{0})-H)e^{-\alpha(t-t_{0})} \tag{24}\]  
Assuming that all objects cool to the ambient temperature of zero (i.e., \(H=0\)), this formula is simplified as follows:  
\[T(t)=T(t_{0})e^{-\alpha(t-t_{0})} \tag{25}\]  
This paper hypothesizes that the influence of information on users decays over time in a similar manner to the cooling of an object. A \"heat\" value for each piece of information (e.g., a tweet) is defined, which decreases over time for each recipient. The specific formula is shown in Equation (26). \[\frac{T^{\prime}(t)}{T(t)-H}=-\alpha \tag{17}\]0  
where \(P(u^{\prime},v^{\prime})\) represents the reduced influence of information on user \(v^{\prime}\) from user \(u^{\prime}\) at time \(t\) and \(t_{u}\) is the time at which user \(u\) first received the information. The heat of the information decays exponentially over time, as described by the adapted cooling law shown in \[\frac{T^{\prime}(t)}{T(t)-H}=-\alpha \tag{17}\]1  
3.3.4. The Role of Time Decay in Social Media Information Dissemination  
On social media platforms, users are constantly exposed to an overwhelming amount of information, which is continually updated. Over time, older information is quickly replaced by new content. During the information dissemination process, users typically react most strongly to a piece of information when they first receive it. For example, assume that user \(A\) follows three other users, namely \(B\), \(C\), and \(D\). Over a period, users \(B\), \(C\), and \(D\) each share the same message (\(I\)) but at different times. Their actions have varying impacts on user \(A\)'s decision making. Assume that user \(B\) shares message \(I\) first. User \(A\) sees the message from \(B\) but decides not to share it. Next, user \(C\) also shares the same message (\(I\)), and again, user \(A\) sees it but chooses not to forward it. Finally, user \(D\) shares the message (\(I\)), and at this moment, user \(A\) decides to share it. According to the linear threshold model, the order in which user \(A\) receives the information does not matter; only the cumulative influence of \(B\), \(C\), and \(D\) is considered.