## 2 Network Architecture  
### Graph Convolutional Network  
GCN is a convolutional neural network and is based on graph theory, the main components of a graph neural network are Node (N) and Edge (E) [1]; the nodes will be connected to each other through directed edges or undirected edges in order to represent the relationship and influence of the nodes to each other in the graph. Figure 1 is an example of a graph; this graph depicts the influence of nodes on each other. For example, all three nodes, D, C and A, act on each other, E and D are two nodes that have no direct relationship between them. The relationship between nodes is represented by an adjacency matrix \(A_{N\times N}\), where N is the number of nodes in the graph and the elements of the adjacency matrix A are set according to the principle: \(A_{ij}=1\) when the \(i^{\text{th}}\) node and \(j^{\text{th}}\) node are connected and vice versa if there is no edge between the \(i^{\text{th}}\) and \(j^{\text{th}}\) node, \(A_{ij}=0\) (with \(i,j\in[1,N]\)). According to the above principle, the adjacency matrix of the graph of Figure 1 is represented as the Equation (1). \[A_{7\times 7}= \begin{array}{ccccc}A&B&C&D&E&F&G\\ A&B&\left[\begin{array}{ccccc}\mathbf{0}&\mathbf{0}&\mathbf{1}&\mathbf{1}& \mathbf{0}&\mathbf{0}&\mathbf{0}\\ \mathbf{0}&\mathbf{0}&\mathbf{0}&\mathbf{0}&\mathbf{0}&\mathbf{1}\\ C&\mathbf{1}&\mathbf{0}&\mathbf{0}&\mathbf{1}&\mathbf{0}&\mathbf{0}& \mathbf{1}\\ D&\mathbf{1}&\mathbf{0}&\mathbf{1}&\mathbf{0}&\mathbf{1}&\mathbf{0}& \mathbf{0}\\ E&\mathbf{0}&\mathbf{0}&\mathbf{0}&\mathbf{1}&\mathbf{0}&\mathbf{1}&\mathbf{0} \\ F&\mathbf{0}&\mathbf{0}&\mathbf{0}&\mathbf{0}&\mathbf{1}&\mathbf{0}&\mathbf{0} \\ G&\mathbf{0}&\mathbf{1}&\mathbf{1}&\mathbf{0}&\mathbf{0}&\mathbf{0}&\mathbf{0} \\ \end{array}\right]\end{array}  \tag{1}\]  
After obtaining the matrix depicting graph A, the matrix A is normalized to the form of a Laplacian matrix \(\Delta_{norm}=I_{N}-D^{-1/2}\)\(AD^{-1/2}\)[39] for training with the neural network using the gradient-based method. Where \(I_{N}\) is an identity matrix of size \(N\times N\), and \(D\) is a diagonal matrix with the elements on the principal diagonal equal to the degrees of the nodes \(D_{ii}=\sum_{j}A_{ij}(1\leq i,j\leq N)\). Since \(\Delta_{norm}\) is a symmetric positive definite matrix, \(\Delta_{norm}\) can be decomposed into \(\Delta_{norm}=I_{N}-D^{-1/2}AD^{-1/2}=U\Lambda U^{T}\). \(U\) and \(\Lambda\) are matrices representing eigenvectors and eigenvalues of \(\Delta_{norm}\), respectively. Figure 1: Nodes and edges in a graph structure. An input signal \(x\in\mathbb{R}^{N}\) is filtered by \(g_{\theta}=diag(\theta)\) (with \(\theta\in\mathbb{R}^{N}\)) which is defined in (2):  
\[g_{\theta}\star x=Ug_{\theta}(\Lambda)U^{T}x \tag{2}\]  
In this Equation (2), the function \(g_{\theta}(\Lambda)\) (3) is approximated by Chebyshev polynomial according to Equation (4) [1; 39]. \[g_{\theta}(\Lambda)\approx\sum_{k=0}^{K}\theta_{k}T_{k}\Big{(}\widetilde{ \Lambda}_{norm}\Big{)} \tag{3}\]  
\[with\left\{ \begin{array}{l}T_{0}(x)=\mathbf{1}\\ T_{1}(x)=x\\ T_{k}(x)=\mathbf{2}xT_{k-\mathbf{1}}(x)-T_{k-\mathbf{2}}(x)\end{array} \right. \tag{4}\]  
In (3), \(\widetilde{\Lambda}_{norm}=\frac{2}{\lambda_{max}}\Lambda-I_{N}\) is the matrix \(\lambda\) rescaled to the maximum eigenvalue \(\lambda_{max}\) and \(\theta_{k}\in\mathbb{R}\) is the Chebyshev coefficient of degree \(k\). From (2), (3) and \(\widetilde{\Delta}_{norm}=\frac{2}{\lambda_{max}^{2}}\Delta_{norm}-I_{N}\), the convolution of signal \(x\) is rewritten as (5). \[g_{\theta}\star x\approx\sum_{k=0}^{K}\theta_{k}T_{k}\Big{(}\widetilde{ \Delta}_{norm}\Big{)}x \tag{5}\]  
To facilitate the training process of GCN and reduce the number of learning parameters, \(k\) is chosen to be \(1\), \(\widetilde{\Delta}_{norm}\approx\Delta_{norm}-I_{N}\) and \(\theta=\theta_{0}=-\theta_{1}\), so the Equation (5) is rewritten as (6):  
\[