## 4 Methodology
### Graph Convolutional Neural Network Module
The symmetric normalization of the adjacency matrix is performed as follows:

\[\hat{A}^{t}=\hat{D}^{-\frac{1}{2}}\hat{A}^{t}\hat{D}^{-\frac{1}{2}},\ \widetilde{A}^{t}=A^{t}+I_{N},\widetilde{D}_{i,i}\ =\sum_{j=1}^{n}\widetilde{A}^{t}_{i,j} \tag{5}\]

where \(\hat{A}^{t}\) is the symmetrically normalized adjacency matrix, and during the process of capturing adjacent node features, \(\widetilde{A}^{t}=A^{t}+I_{N}\) prevents the current node's feature information from being unable to propagate. \(D\) is the degree matrix, where \(D_{i,i}\) represents the degree of gantry node i, i.e., the number of edges connected to it. After generating the adjacency matrix \(A=\{A_{t_{1}},A_{t_{2}},A_{t_{3}},\cdots,A_{t_{n}},\}\) as well as the feature matrix \(X=(X_{t_{1}},X_{t_{2}},\cdots,X_{t_{n}})\), \(X_{t_{i}}\in R^{N\times P}\) of the time-slice transaction graph, we learn the representation of the graph using GCN. For each time slice, the neighbor information of the node is aggregated in the output vector. The GCN can be trained by a back propagation algorithm and the relationships between node features are learned in the network. Taking a time-slice transaction graph as an example, the GCN computes the embedding vectors of each node by performing a weighted average of the features of the neighboring nodes. This process can be viewed as a series of convolution operations. For each node i, its embedding vector can be expressed as:

\[h^{l}_{i}=\sigma(\sum_{j\in N(i)}\frac{1}{c_{ij}}h^{l-1}_{i}W^{(l-1)}) \tag{6}\]

where \(N(i)\) is the set of neighboring nodes of node i, \(c_{ij}\) is the normalization factors of the edges between nodes i and j, \(W^{(l-1)}\) is the parameter matrix from layer \(l-1\) to layer \(l\), \(\sigma(\cdot)\) is the activation function ReLU. As shown in Figure 3, each node in the figure represents the highway ETC gantry, which demonstrates that the GCN model captures the features of the current node and its neighboring nodes through a convolutional layer, and ultimately outputs the current node's state matrix \(C^{t}=GCN_{L}\left(Z^{l}\right)\) that aggregates information of the neighboring nodes, \(GCN_{L}\) represents the L-layer GCN, and the specific convolutional process of \(GCN_{L}\) is as follows:

\[Z^{l}=ReLU(\hat{A}^{t}Z^{(l-1)}W^{(l-1)}) \tag{7}\]

\[C^{t}=ReLU(\hat{A}^{t}Z^{(l-1)}W^{(l-1)}) \tag{8}\]

Figure 3: Schematic diagram for feature aggregation of node transaction graphs within time slices, where the red dots indicate the current node and the neighboring nodes to be aggregated by the current convolutional layer, and the red arrows indicate the neighboring nodes to be aggregated towards the current node. ### Attention Mechanism

After obtaining the information of the aggregated neighbor nodes of each time transaction graph, in order to capture the dynamic information of the transaction graphs of different time slices, in this paper, we propose a localized proximity time window state, construction based on a contextual attention mechanism, as shown in Figure 4. In this attention mechanism, each time step t is represented as three aspects:

1. The query vector \(q_{t}\) is used to represent the focus or information of interest at the current time step t. It captures the feature representation of the current time step and is used as a query for similarity calculation with the keys of other time steps. 2. The key vector \(k_{t}\) is used to represent the information of other time steps. It contains the target for similarity computation with the query vector \(q_{t}\) for the current time step t. The key vector can be regarded as a representation of other time steps for comparison with the query of the current time step. 3. The value vector \(v_{t}\) contains the feature representation or information for each time step. It comprises a sequence of vectors used to compute the attention weights. The value vector can contain important information related to the transaction graph.