## 3 Structure of the GCN-Based Residential Classification Model  
After gaining the feature vectors for the training set, the validation set, and the test set, the data are inserted into a graph convolutional neural network. When constructing an areal settlement into a graph structure, the skeleton lines of the areal settlement need to be mapped to the graph structure. In this research, the skeleton line characteristics of the surface settlement morphology are embedded in the nodes of the graph and the main morphology of the areal settlement is expressed by the graph structure of the skeleton line. The attributes of the nodes in the graph carry local and overall morphological features of the surface settlement, which are extracted by different methods because the residential areas have different shapes. Thus, when performing convolutional operations on the graph, features of the shape can be extracted from different aspects, and in classification prediction, the shape features are aggregated by a fully connected neural network and their category is predicted according to the obtained probability values of the shape.  
Figure 7 displays the GCN architecture used in this study. In the forward propagation process, the initial input signal is the feature matrix \(X_{0}\in R^{N\times D}\) of the skeleton line, where \(N\) is the number of skeleton line nodes and \(D\) is the number of skeleton line features. Deep information about the areal settlement is extracted through graph convolution and activation calculation of hidden layers. The number of neurons in each hidden layer is I, that is, the number of convolution kernels. Finally, the probability \(Y\in R^{1\times 9}\) of the nine skeleton lines is output in the \((L+1)^{th}\) layer, or the fully connected layer. The graph convolution and activation calculation formula of the GCN hidden layer is as follows:  
\[H_{j}^{[l+1]}=ReLU\left(\sum_{i=1}^{l}\left(\hat{D}^{-\frac{1}{2}}\ \hat{A}\ \hat{D}^{-\frac{1}{2}}\ H_{i}^{[l]}W^{[l+1]}\right)+b_{j}^{[l+1]}\right) \tag{6}\]  
where \(ReLU(\cdot)\) represents a rectified linear function, \(H_{i}^{[l]}\) is the output of the \(i^{th}\) neuron in the \(l^{th}\) layer, and \(H_{j}^{[l+1]}\) is the output of the \(j^{th}\) neuron in the \((L+1)^{th}\) layer. \(H^{[0]}\) is set as \(X_{0}\). \(W^{[l+1]}\in R^{l_{\times}l_{\times}l_{\times}l_{\times}l}\) and \(b_{j}^{[l+1]}\in R^{l\times l_{\times}l_{\times}l}\) are the trainable weight matrix and bias, respectively, which are the training parameters of the GCN. Furthermore, \(\hat{A}\ =I_{N}+A\) is the adjacency matrix and \(\hat{D}\) is the degree matrix of \(\hat{A}\).  
Figure 7: Architecture of the GCN.  
After the graph convolution operator is applied to \(L\) hidden layers, the output of the \(\left(L+1\right)^{th}\) layer is \(H^{[L+1]}\in R^{1\times 9}\), or Y. The \(\left(L+1\right)^{th}\) layer is the fully connected layer, which is calculated as follows:  
\[Y=Softmax\Big{(}reduce\_mean\Big{(}H^{[L]}W^{[L+1]}+b^{[L+1]}\Big{)}\Big{)} \tag{7}\]  
where \(W^{[L+1]}\in R^{I_{L}\times 9}\) and \(b^{[L+1]}\in R^{1\times 9}\) represent the learning weights and bias, respectively, of the \(\left(L+1\right)^{th}\) layer and \(\left(H^{[L]}W^{[L+1]}+b^{[L+1]}\right)\in R^{N\times 9}\) calculates the probability of a different classification at each node of the graph. The final classification is judged by a global probability value. Therefore, the \(reduce\_mean(\cdot)\) function is used to calculate the mean along the first dimension (node dimension). Then, the normalized exponential function \(Softmax(\cdot)\) is used to obtain the output probability Y of the nine classifications. In backpropagation, to maximize the efficiency of the model, the cross-entropy function is used to calculate the model loss, and the Adam algorithm is selected for the optimization.