# Hypergraph-Driven High-Order Knowledge Tracing with a Dual-Gated Dynamic Mechanism
## 4 Methodology
### The HGKT Model
#### 4.1.1 Hypergraph-Enhanced Embedding Layer  
Define a hypergraph \(\mathcal{G}=(\mathcal{V},\varepsilon)\), where \(\mathcal{V}\) represents a set of vertices, and \(\varepsilon\) represents a set of hyperedges. In the hypergraph \(\mathcal{G}\), an exercise \(e_{t}\) corresponds to a vertex, and a knowledge concept \(c\) corresponds to a hyperedge. The incidence matrix \(\mathbf{\mathcal{H}}\in\mathbb{R}^{|\mathcal{V}|\times|\varepsilon|}\) is based on the \(\mathbf{Q}\) matrix, an exercise-knowledge concept association matrix that represents node-hyperedge relationships. In a hypergraph, a hyperedge denotes a group relationship connecting multiple vertices simultaneously, as illustrated in Figure 3. Here, \(C\) represents knowledge concepts, \(e_{t}\) represents exercises, and each exercise forms a hyperedge with its associated knowledge concepts. Exercise-knowledge concepts incidence matrix \(\mathbf{\mathcal{H}}\) each entry is defined as follows:  
\[\mathcal{H}(\mathcal{V}_{i},\varepsilon_{j})= \begin{cases}1,\text{ }if\text{ } \mathcal{V}_{i}\in\varepsilon_{j}\\ 0,\text{ }if\text{ }\mathcal{V}_{i}\
otin\varepsilon_{j}\end{cases}  \tag{1}\]  
where if a vertex belongs to a hyperedge, then \(\mathcal{H}(\mathcal{V}_{i},\varepsilon_{j})=1\); otherwise, \(\mathcal{H}(\mathcal{V}_{i},\varepsilon_{j})=0\). By constructing the hypergraph using the exercise-knowledge concepts incidence matrix \(\mathbf{\mathcal{H}}\), each exercise and its associated knowledge concepts form hyperedges. A hypergraph neural network is employed to explicitly model the complex relationships between exercises and knowledge concepts, capturing high-order exercise correlations which latent dependencies formed when multiple exercises nonlinearly interact through shared knowledge concepts through a two-layer hypergraph convolution. Hypergraph convolution employs a message propagation mechanism, enabling exercise embeddings to aggregate features from related knowledge concepts and form enhanced  
Figure 3: The exercise–knowledge concept relationship diagram. Figure 2: The architecture of the HGKT model. representations that incorporate knowledge structure information. The generation of enhanced exercise embeddings involves two convolutional layers:  
First convolution layer: Uses a non-linear transformation with ReLU activation to fuse exercise features with their associated knowledge concepts features into embedding using incidence matrix \(\mathbf{\mathcal{H}}\). \[\mathbf{X^{(1)}}=ReLU(\mathbf{W}_{1}\cdot\mathbf{\mathcal{H}}\cdot\mathbf{e}_{t}+\mathbf{b}_{1}) \tag{2}\]  
where \(\mathbf{W}_{1}\) is the weight matrix, and \(\mathbf{b}_{1}\) is the bias term. ReLU is a non-linear activation function. Second Convolution Layer: Propagates features across concept-sharing exercises through incidence matrix operations, where hyperedges dynamically cluster exercises with overlapping concepts--mathematically formalizing high-order relationships as multi-hop paths exercise-concept-exercise in hypergraph topology. This further refines the feature representation, producing the final enhanced embedding \(\mathbf{E}_{t}\) with a dimensionality of \(d_{k}\). \[\mathbf{E}_{t}=\mathbf{W}_{2}\cdot\mathbf{\mathcal{H}}\cdot\mathbf{X^{(1)}}+\mathbf{b}_{2} \tag{3}\]  
where \(\mathbf{W}_{2}\) is the weight matrix, and \(\mathbf{b}_{2}\) is the bias term. The enhanced embeddings not only retain the semantic information of the exercises themselves but also encode the association patterns among knowledge concepts, allowing for a more precise characterization of an exercise's position within the knowledge space. During each forward propagation, the hypergraph structure is dynamically reconstructed, enabling the convolution operations to adapt to varying knowledge point association patterns across different exercises.