## 3 The Proposed SSP-AA Model
### Jump Knowledge Module  
During link prediction tasks using the SSP-AA model, as the layers of the GNNs increase, deep networks may face the issue of over-smoothing. This is particularly prevalent in cases of smaller data volumes. The jumping knowledge [41; 42] module mitigates the risk of over-smoothing by adaptively aggregating multi-layer information through cross-layer connections. The primary goal of the jumping knowledge module is to quickly and effectively aggregate the neighborhood information of high-order nodes, combining representations from different neighborhood ranges using jumping connections. If the model directly extracts information from high-order neighbors, noise may be introduced and will lead to a decrease in model performance. Therefore, this paper proposes the integration of the jumping knowledge module into the SSP-AA model as a means to enhance model performance. The \(l\)-th layer of the jumping knowledge module is defined as:  
\[H^{l}=\sigma(W^{l}\cdot Aggregate(H^{l-1})) \tag{7}\]  
where _Aggregate_() is the model's defined function, and the definition of the jumping knowledge module between different layers is:  
\[H^{final}=Combine(H^{1},H^{2},\dots,H^{n}) \tag{8}\]  
where _Combine_() is a symbol describing aggregation methods. Figure 3 shows the connections of a four-layer jumping knowledge network, and N.A. denotes the aggregation of neighboring nodes. The motivation to use the jumping knowledge module is to efficiently aggregate information from high-order neighbors and use this module to combine representations within different neighbor scopes. As shown in Figure 3, each layer of the model enhances the influence of nodes by aggregating neighborhood information from the previous layer. However, in the final aggregation process, each node selectively filters some information from the previous intermediate representations for merging. This step is independently applied to each node. Thus, the model can adjust the effective neighborhood size for each node as needed, which achieves adaptive selective aggregation. (1) Long Short-Term Memory (LSTM) of the polymerization method  
Common aggregation strategies include direct concatenation, max-pooling, and mean-pooling, among others. However, each of them has certain drawbacks. Direct concatenation can preserve original information but may lead to information redundancy and increased sensitivity to noise. Max-pooling can highlight the most representative node features and reduce the influence of noise, but it cannot flexibly handle differences in node importance, potentially limiting the model's expressive ability. Mean-pooling can capture the overall node features, but it is sensitive to noisy nodes and unable to differentiate the importance among nodes, possibly overly focusing on unimportant nodes or neglecting important ones. Therefore, these aggregation schemes exhibit structure dependence. Hence, it is questionable whether the radius size in aggregation strategies can achieve optimal representations for all nodes and tasks. A larger radius may lead to over-smoothing, while  
Figure 3: Jump knowledge connection. a smaller radius may lead to instability or insufficient information aggregation. To address this issue, this section introduces an adaptive aggregation mechanism in the jumping knowledge module, using LSTM to model the dependencies in sequential data. In the jumping knowledge module, LSTM can serve as a node-level aggregation method module, which is used for modeling representations of each node at different layers. Through its memory and forgetting mechanisms, it can capture sequential information of node representations, thereby providing richer node features. The structure of a single LSTM is shown in Figure 4.  
As shown in Figure 4, \(f_{t}\), \(i_{t}\), \(o_{t}\), and \(h_{t}\) represent the forget gate, memory gate, output gate, and hidden layer output, respectively. \(\sigma\) represents the activation function, while tanh denotes the hyperbolic tangent activation function.