## 3 The Proposed SSP-AA Model
### Jump Knowledge Module
\(x_{t}\) represents the \(t\)-th node feature, and \(h_{t}\) is obtained through the calculations of the forget gate, memory gate, and output gate, with the specific formula shown below:  
\[f_{t}=\sigma(W_{f}\cdot[h_{t-1},x_{t}]+b_{f}) \tag{9}\]  
\[i_{t}=\sigma(W_{i}\cdot[h_{t-1},x_{t}]+b_{i}) \tag{10}\]  
\[\widetilde{C}=\tanh(W_{c}\cdot[h_{t-1},x_{t}]+b_{c}) \tag{11}\]  
\[C_{t}=f_{t}\cdot C_{t-1}+i_{t}\cdot\widetilde{C}_{t} \tag{12}\]  
\[o_{t}=\sigma(W_{o}\cdot[h_{t-1},x_{t}]+b_{o}) \tag{13}\]  
\[h_{t}=o_{t}\cdot\tanh(C_{t}) \tag{14}\]  
(2) Attention-based aggregation method  
LSTM uses forward and backward LSTMs to capture hidden features \(\overset{\leftarrow}{h}_{v}\) and \(\overset{\rightarrow}{h}_{v}\). Additionally, to address the aggregation problem caused by the different importance of nodes in different layers, an attention mechanism is applied to the LSTM module to enhance the attention weighting of node representations, making the aggregation process more focused on important nodes and features. The LSTM attention working mechanism in each layer \(l\) is shown in \[H^{final}=Combine(H^{1},H^{2},\dots,H^{n}) \tag{8}\]0  
\[H^{final}=Combine(H^{1},H^{2},\dots,H^{n}) \tag{8}\]1 LSTM structure diagram. the latent distribution. LSTM attention sends the representations of each layer into the bidirectional LSTM, generating the forward LSTM and backward LSTM hidden features \(\overset{\leftarrow l}{h}_{v}\) and \(\overset{\rightarrow l}{h}_{v}\) in each layer. These features \(\overset{\leftarrow l}{h}_{v}\) and \(\overset{\rightarrow l}{h}_{v}\) are then concatenated and linearly mapped to a score in the linear layer. After normalizing the scores of each layer \(l\), we obtain the attention score \(Z_{v}^{l}\) for each layer \(l\). With the attention score \(Z_{v}^{l}\), the most useful neighborhood range of each node \(v\) can be recognized (\(\sum_{l}Z_{v}^{l}=1\)). This allows the model to better balance the importance of different nodes when aggregating neighbor node information. \(Z_{v}^{l}\) indicates the importance of node \(v\) in layer \(l\), and \(\left\{Z_{v}^{l}\right\}_{l=1}^{n}\) shows the attention node \(v\) pays to its neighborhood at different ranges. Finally, take the sum of \(\left\{\overset{\leftarrow l}{h}_{v}^{l}\parallel\overset{\rightarrow l}{h} _{v}^{l}\right\}\) through weighting to obtain the final layer representation:  
\[H^{final}=Combine(H^{1},H^{2},\dots,H^{n}) \tag{8}\]2  
For each node, the representation in different layers is modeled in sequence through LSTM. The LSTM module can learn node representations and capture long-term dependencies of node features. Attention mechanisms are applied to the output of LSTM for each node. By calculating attention weights, the node representations in different layers can be weighted to emphasize the more important layers and nodes for the target task. Finally, the weighted node representations are aggregated to obtain the final representation of the node, which is used for subsequent tasks, like link prediction. In summary, LSTM attention in the jump knowledge module is a powerful mechanism that enhances the expressiveness and task relevance of node representations by combining the sequence modeling of LSTM and the attention mechanism. It provides an effective way to learn the interaction between node representations and tasks, and it can better capture the influence of nodes in tasks by adaptively allocating the importance weights of different nodes. Compared with traditional aggregation methods, LSTM attention has more flexibility and expressiveness in cross-layer information transmission, especially when dealing with dependencies and graph structures. At the same time, due to the sequential nature of LSTM, it can better handle information within nodes, rather than only focusing  
\[H^{final}=Combine(H^{1},H^{2},\dots,H^{n}) \tag{8}\]3 LSTM attention aggregation. on local connections between nodes. Therefore, LSTM attention is a key mechanism with broad application value in jump knowledge networks.