### Information Dissemination Models
#### 3.2.2 The Linear Threshold Model  
The LT model is another classical influence propagation model. The basic concept of the LT model is that the activation state of an inactive node is determined by the collective influence of its active neighbors at a certain time. A user can only be activated if enough of its neighboring active users have sufficient influence. The main difference between the IC model and the LT model lies in how neighboring nodes exert influence. Like the IC model, the LT model defines the social network as a weighted graph represented as \(G=(V,E,W)\), where each edge has a weight of \(p_{uv}\in W\). In network \(G\), each user \(v\in V\) is assigned a random threshold (\(\theta_{v}\), where \(\theta_{v}\) ranges between 0 and 1). This value represents how easily user \(v\) can be activated by information; the larger the value of \(\theta_{v}\), the more difficult it is for the node to be influenced by its neighboring nodes. Let \(u\in N(v)\) represent the set of neighboring nodes of user \(v\), and if any neighboring node (\(u\in N(v)\)) is activated, the influence of node \(u\) on \(v\) is denoted by \(l_{uv}\). The specific propagation process of the LT model is described as follows:Step 1: In the network (\(G=(V,E,W)\)), a seed set (\(S\)) is defined. At the initial time step, only the nodes in set \(S\) are in the activated state, meaning they have already been influenced by the information. * Step 2: Check whether the cumulative influence (\(l_{uv}\)) from neighbors is greater than or equal to the threshold (\(\theta_{v}\)). If this condition is met, node \(v\) is successfully activated; otherwise, the activation fails. * Step 3: This process continues until no further nodes can be activated in the network. At any time step (\(t_{c}\)), nodes activated at the previous time step (\(t_{c-1}\)) remain in the activated state. Any inactive nodes at step \(t_{c-1}\) are activated at the next time step if the influence from their neighbors meets the required conditions. The overall process of information dissemination for the LT model is illustrated in Algorithm 2. ```
Input: Social network graph \(G=(V,E)\), activation probability \(p_{uv}\), seed set \(S_{0}\), user activation threshold \(\theta_{v}\), current time step \(t_{c}\), end time step \(t_{e}\) Output: Set of activated nodes \(S_{t}\) Initialize:\(AS\gets S_{0}\), \(S_{t}\gets S_{0}\) while\(t_{c}\leq t_{e}\)do for each user \(v\in V\setminus S_{t}\)do  Compute the cumulative influence \(l_{uv}\) from each neighbor \(u\in N(v)\) if\(\sum_{u\in N(v)}l_{uv}\geq\theta_{v}\)then  Activate node \(v\) \(v\in S_{t}\setminus S_{t-1}\) \(AS\gets AS\cup\{v\}\), \(S_{t}\gets S_{t}\cup\{v\}\) else \(v\in V\setminus S_{t}\), node \(v\) remains inactive endif endfor  Increment time step \(t_{c}\gets t_{c}+1\) endwhile return\(S_{t}\)
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**Algorithm 2** Linear Threshold Model  
To better illustrate the algorithmic details of the propagation process for this model, Figure 5 depicts the complete process of propagation in the linear threshold (LT) model. In the initial state, the network topology consists of four nodes and six edges, as shown in Figure 4(a). Figure 4(b) shows the state at time \(t_{1}\), where nodes \(a\) and \(b\) (represented as black nodes) are in the activated state, meaning they have the ability to spread information. At this point, nodes \(a\) and \(b\) begin attempting to activate their neighboring nodes (\(c\) and \(d\)). At time \(t_{1}\), the cumulative influence from nodes \(a\) and \(b\) on node \(c\) (with a weighted influence of 0.3) exceeds node \(c\)'s activation threshold (0.25). As a result, node \(c\) successfully transitions from an inactive state to an active state at time \(t_{2}\), as shown in Figure 4(c). Similarly, at time \(t_{1}\), the cumulative influence from nodes \(a\) and \(b\) on node \(d\) (0.3) is less than node \(d\)'s activation threshold (0.5), so node \(d\) remains inactive. The propagation process continues at time \(t_{2}\), and now node \(c\) also gains the ability to activate other nodes.