## 3 Method
### Graph Convolutional Neural Network  
Convolutional neural networks (CNNs) have achieved remarkable success in extracting local features from Euclidean data possessing regular spatial structures, such as in image analysis. The utilization of translation invariance, local connectivity, and image structure enables CNNs to extract shared local features from entire datasets. However, when applied to complex domain data without regular spatial structures, such as social networks in social sciences, distributed interconnected sensing networks, knowledge graphs, and networks of interacting protein molecules, CNNs are not as effective. This has led to an increasing focus on graph-structured convolutional neural networks (GCNNs) as a deep learning method suited for graph-structured data. GCNNs involve the construction of convolutional and pooling operators. Convolutional operators aim to capture local node structures, while pooling operators aim to learn hierarchical representations and reduce parameters. For node-level tasks, the primary focus is on enhancing the expression of each node. In practical applications, pooling operators may not be required, and early work has concentrated on constructing convolution operators on the graph. Pooling operators are typically employed for graph-level tasks. The key challenge in GCNNs is developing convolution operators on the graph that capture the local connectivity of nodes. The graph structure determines the adjacency matrix, which represents node connectivity. Various convolution operators have been proposed, including spectral-based and spatial-based methods. In summary, GCNNs provide a powerful tool for analyzing complex domain data represented by graphs. The construction of convolution and pooling operators is critical to develop effective GCNN models. GCNNs have a broad range of applications, including social network analysis, protein interaction analysis, and knowledge graph analysis. A common CNN algorithm pseudocode example is presented in the following Algorithm 2. ```
Input: Graph \(\mathcal{G}(\mathcal{V};\mathcal{E})\), input node features \(\{x_{v},\forall v\in\mathcal{V}\}\) ; Number of layers \(K\) trainable functions \(\text{MsG}_{1}^{(k)}(\cdot)\) for nodes with identity coloring, \(\text{MSG}_{0}^{(k)}(\cdot)\) for the rest of nodes; EGO \((v,k)\) extracts the \(K\)-hop ego network centered at node \(v\), indicator function \(\mathcal{H}[s=v]=1\) if \(s=v\) else \(0\) Output: Node embeddings \(\mathbf{h}_{v}\) for all \(v\in\mathcal{V}\)
1:\(v\in\mathcal{V}\)do
2:\(\mathcal{G}_{v}^{(K)}\leftarrow\) EGO \((v,K)\), \(\mathbf{h}_{u}^{(0)}\)\(\leftarrow\)\(\mathbf{x}_{u}\)\(\forall u\in\mathcal{G}_{v}^{(K)}\)
3: for \(k=1,\ldots,K\)do
4: for \(u\in\mathcal{G}_{v}^{(K)}\)do
5:\(\mathbf{h}_{u}^{(k)}\leftarrow\)\(\text{AGG}^{(k)}\)
6:\(\mathbf{h}_{v}\leftarrow\mathbf{h}_{v}^{(K)}\)
```  
**Algorithm 2**ID-GNN embedding computation algorithm  
Existing graph convolutional neural networks are divided into two categories: spectral methods, which use the convolution theorem on the graph to define the graph convolution from the spectral domain, and spatial methods, which start from the node domain and aggregate each central node and its neighboring nodes by defining an aggregation function. The absence of translation invariance on the graph poses difficulties in defining convolutional neural networks in the nodal domain, and spectral methods use the convolution theorem to define the graph convolution from the spectral domain, and we first give the background knowledge of the convolution theorem convolution theorem: the Fourier transform of the signal convolution is equivalent to the product of the Fourier transforms of the signal:  
\[F(f*g)=F(f)*F(g) \tag{6}\]  
where \(f\), \(g\) denotes the two original signals, \(F(f)\) denotes the Fourier transform off, - denotes the product operator, and * denotes the convolution operator.