## 4 Methodology
### Neighborhood Information Aggregate
#### 4.3.2 Hop-Attention  
Neighbors of nodes of each order in the previous aggregation level indicate that the outputs are connected via skips, and the computation of weight allocation is carried out at this level, and the attention mechanism takes into account all the inputs through clever manipulation, and after the attention scores, the most important part of the information is judged. The outputs \(s_{1},s_{2},s_{3}\ldots s_{n}\) of each aggregation level are able to fully consider the importance of different orders of neighbor information during the multihop neighbor aggregation process through the self-attention mechanism at the hopping level, which reduces the influence of the noisy information aggregated during the aggregation of the neighbor information on the final model determination. On the other hand, the hop-attention connection has a lower risk of overfitting than a simple summation or mean fusion, which gives the same weight to each hop neighborhood and will over-learn useless features in the training data, achieving an accuracy on the training dataset that is difficult to achieve on the test set. The specific approach of the attention mechanism is to take multiple inputs with correlations, map them into _Query_, _Key_, and _Value_, and then compute the respective attention scores in order to reassign weights to each input [34]. The weight matrix \(W\) is first used to map \(s\) into \(Q\), \(K\), and \(V\).  
\[Q^{i}=W^{q}s_{i} \tag{8}\]  
\[K^{i}=W^{k}s_{i} \tag{9}\]  
\[V^{i}=W^{\infty}s_{i}, \tag{10}\]  
where \(W^{q},W^{k},W^{v}\) are weight matrices that map the outputs of each aggregation layer to \(Q,K,V\).  
\[\alpha_{i,j}=\frac{Q^{i}\cdot K^{j}}{\sqrt{d}}, \tag{11}\]  
where \(\alpha\) is the attention score, representing the correlation or similarity between \(q\) and \(k\). It is a probability distribution used to determine which \(v\) is weighted when calculating the output. A higher attention weight means that elements more related to \(q\) will receive more weight while a lower weight means that less-related elements will receive less weight. Here, we used scaled dot product attention, where the denominator \(d\) represents the dimension of \(q\), which can limit the size of the dot product of \(q\) and \(k\) in the numerator, ensuring that the distribution of the attention scores does not become too scattered due to the input dimension.  
\[\alpha_{i,j}^{i}=\frac{\exp(\alpha_{i,j})}{\sum_{j}\exp(\alpha_{i,j})} \tag{12}\]  
\(\hat{\alpha}\) is a softmax operation on \(\alpha\), which normalizes the spatial correlation of the outputs of each layer.  
\[\hat{s}_{i}=\sum_{j}\alpha_{i,j}V^{j} \tag{13}\]  
The primary function of the \"\(V\)\" operation is to generate the ultimate output, denoted as \(\hat{s}\), based on the calculated attention weights, which allows the mechanism to select and combine values in a targeted manner based on the correlation between \(Q\) and \(K\); the modification allows the model to more effectively concentrate on pertinent information within the input data, and concentrating the features learned by the model on the key partsto improve the performance of the task, thus far we obtain a node vector representation of the processed nodes.