## 3 Methodology
### Construction of Information Propagation Graphs  
#### 3.1.1 Building the Information Propagation Graph  
In constructing the information dissemination graph, we treat each post as a node, while edges between posts are defined based on comment and retweet relationships. Specifically, if tweet A is a reply to tweet B, then there will be a directed edge between A and B in the graph to indicate the direction of information flow. Thus setting \(C=\left\{c_{1},c_{2},\ldots,c_{m}\right\}\) for the false information detection data sets, including \(c_{i}\) for the data set of the event \(i\), \(m\) for the cumulative sum of events. \(c_{i}=\left\{r_{i},w_{i}^{(1)},w_{i}^{(2)}\ldots,w_{i}^{(n_{i}-1)},G_{i}\right\}\), where \(n_{i}\) is the total number of posts for event \(i\), \(r_{i}\) represents the source post, \(w_{i}^{(j)}\in V_{i}(j=1,2,\ldots,n_{i}-1)\) is the \(j\)-th related response posts, and \(G_{i}=(V_{i},E_{i},A_{i})\) on behalf of the propagation graph of event \(i\). The undirected graph \(G_{i}=(V_{i},E_{i},A_{i})\) is constructed according to the comment or retweet relationship in the event, where \(V_{i}=\left\{r_{i},w_{i}^{i},w_{2}^{i}\ldots,w_{n_{i}-1}^{i}\right\}\) represents the set of nodes in the graph, \(E_{i}=\left\{e_{st}^{i}|s,t=0,\ldots,n_{i}-1\right\}\) indicates the set of response behaviors in the process of information propagation, and the adjacency matrix \(A_{i}\in\left\{0,1\right\}^{n_{i}\times n_{i}}\) represents the response relationship of event propagation in the propagation graph G, which is defined as Equation (1).  
\[A_{\text{st}}^{i}=\left\{\begin{array}{l}1,e_{st}^{i}\in E_{i}\\ 0,other\end{array}\right., \tag{1}\]  
Based on inter-node relationships, we construct three distinct propagation graph structures with differentiated characteristics. First, the Undirected Graph (UD-Graph) employs binary relationships to represent node connectivity, preserving adjacency relationships while disregarding interaction directionality. Second, the Top-Down Propagation Graph (TD-Graph) adheres to hierarchical information flow patterns, where directional edges from parent posts (source posts \(r_{i}\)) to child posts (response posts \(w_{i}^{(j)}\)) explicitly delineate the tree-like diffusion pathways of information dissemination. In contrast, the Bottom-Up Propagation Graph (BU-Graph) establishes reverse edge configurations (child-to-parent orientation) to form information convergence networks, a topological design that systematically captures how diverse information sources aggregate toward specific nodes.  
As can be seen from Figure 2, we construct the top-down propagation graph structure \(G_{i}^{TD}=<V_{i},E_{i}^{TD}>\) and a bottom-up diffusion graph structure \(G_{i}^{BU}=<V_{i},E_{i}^{BU}>\) for graph \(G_{i}\), and use adjacency matrices \(A_{i}^{TD}\) and \(A_{i}^{BU}\) to represent the connection relationship of  
Figure 1: Framework of ICP-BCN model.  
nodes on the graph, respectively. In this paper, the semantic feature expression obtained by processing the original post and the response text content by BERT pre-training model is used as the feature vector of the post, i.e., the initial feature matrix of the propagation structure and the diffusion structure are the same. \(X^{\textit{TD}}=X^{\textit{BU}}=[x_{0},x_{1},\dots,x_{n-1}]\in\mathbb{R}^{n_{i }\times d}\), where \(n_{i}\) represents the number of posts for event \(i\), and \(d\) is the dimensionality of the feature space. \(x_{0}\in\mathbb{R}^{d}\) denotes the feature vector of the original post, while \(x_{j}\in\mathbb{R}^{d}(j~{}=~{}1,2,\dots,n_{i}-1)\) represents the feature vector of the response post. The adjacency matrices are transpositions of each other, i.e., \(A_{i}^{\textit{TD}}=\left(A_{i}^{\textit{BU}}\right)^{T}\).