## 3 Materials and Methods
### Developing a Graph Attention Network across Multiple Perspectives and Scales
#### 3.3.1 Node Feature Update Mechanism  
In graph convolutional neural networks, node feature updates are achieved by aggregating features from neighboring nodes, enabling the network to discern local connectivity patterns [40]. This neighborhood-focused feature aggregation renders GCNs suitable for graph data analysis, facilitating direct engagement with graph structures and the acquisition of complex and advanced node feature representations, which offers deeper insights into various graph analysis tasks. In our research, we harnessed a multi-head graph convolutional neural network to independently extract features from two distinct graphs, \(G_{co-exp}\) and \(G_{ppi}\). Employing the graph attention network (GAT) with a multi-head attention mechanism, we updated the node features [41]. This framework allowed each node's feature representation to be influenced not only by its inherent features but also by the weighted features of its adjacent nodes. This multi-head attention approach enabled each head to focus independently on varying information facets, allowing for a more compre  
Figure 2: Visualization of the MVMSGAT Network Architecture, highlighting the integration of two distinct biological graphs: the gene co-expression graph \(G_{co-exp}\) and the protein–protein interaction graph \(G_{ppi}\). For each graph, \(G\) is represented as \(G(V,E,X)\) where \(V\) denotes the node set, \(E\) denotes the edge set, and \(X\) denotes the node feature matrix. This representation captures the structure and attributes of the graph, with each layer beneath the network diagram indicating the number of nodes \(|V|\), the number of edges \(|E|\), and the feature-length \(|X|\) at that layer. Such detail illustrates the evolving complexity and informational depth as the data progresses through the network. hensive capture of the intricate inter-node interactions within the graph's structure. This method is advantageous for generating distinctive feature representations, providing a potent means for decoding the complex information embedded within the graph. Attention coefficients \(\alpha_{ij}^{k}\) for each node pair \(i\) and \(j\) were individually calculated for each head using the following formula:  
\[\alpha_{ij}^{k}=\text{softmax}(\epsilon_{ij}^{k})=\frac{\exp(\epsilon_{ij}^{k} )}{\sum_{n\in\mathcal{N}_{i}}\exp(\epsilon_{in}^{k})} \tag{2}\]  
where \(\epsilon_{ij}^{k}\) denotes the attention score between nodes \(i\) and \(j\) for the \(k\)-th head, computed as follows:  
\[\epsilon_{ij}^{k}=\text{LeakyReLU}(\vec{a}^{k}[\vec{W}^{k}h_{i}\|\vec{W}^{k}h_ {j}]). \tag{3}\]  
Here, \(\vec{W}^{k}\) signifies the linear transformation weight matrix for the \(k\)-th head, \(\vec{a}^{k}\) is the corresponding attention vector, \(h_{i}\) and \(h_{j}\) are the feature vectors for nodes \(i\) and \(j\), respectively, and \(\|\) represents vector concatenation. Building upon the multi-head attention mechanism, we refined the node feature update process by layering multiple graph attention layers. Each node's new feature \(h_{i}^{k}\) was updated using the equation:  
\[h_{i}^{\prime k}=\sigma\left(\sum_{j\in\mathcal{N}_{i}}\alpha_{ij}^{k}\vec{W}^ {k}h_{j}\right). \tag{4}\]  
In this expression, \(\sigma\) denotes an activation function, and \(\sum_{j\in\mathcal{N}_{i}}\) represents the weighted summation of the neighbors \(j\) of node \(i\), incorporating neighbor information. The weighting \(\alpha_{ij}^{k}\), as derived from the described attention mechanism, signifies the degree of contribution from neighbor node \(j\) to node \(i\), with \(\vec{W}^{k}\) representing the linear transformation weights for the head, and \(h_{j}\) as the initial feature vector for node \(j\). Additionally, our approach integrates a Graph Embedding Module, as depicted in Figure 3. This module employs a two-layered structure, each layer comprising a GAT convolutional layer followed by batch normalization and an activation layer (ReLU).