# Orthogonal Neural Network: An Analytical Model for Deep Learning
## 4 ONN Model Applied on Graph Neural Networks
### Discussion  
The analysis of the orthogonal neural network model has been described in more detail in the section on convolutional neural networks, and the analysis of orthogonal neural network model in graph neural networks is explained in this section. The expansion method of the graph neural network is given above for the Cora dataset. Since the graph data have the characteristics of being large scale and sparse, it is not convenient to directly use the calculation results for the dataset in discussion; instead, here, a set of simple calculation data are constructed for discussion. The y-axis in Figure 4 represents the six nodes, and the coordinate values correspond to the respective node numbers, and the x-axis represents the six features. The coordinate values correspond to the feature numbers. The color of the blocks in the figure corresponds to the values within the blocks, which are extracted from the model calculation results and are used only to illustrate the analysis method, and they do not correspond to the calculations in the actual dataset. Here, it is assumed that the analysis is performed by drawing from the classification calculation result \(\mathbf{e}_{1}\) of node 1. Nodes 21 and 35 are the neighbor nodes of node 1, and nodes 1400, 1512, and 2700 are the second-order neighbor nodes; that is, the neighbor nodes of the neighbor nodes. From the data, it can be seen that the computational results of the graph neural network include not only the features of its own nodes but also the features of neighboring nodes, which is the characteristic that enables the graph neural network to learn and aggregate the features of neighboring nodes through the structural information of the graph data and significantly improves the performance of deep learning in processing graph data. In Figure 3(a), the larger positive values of feature 1400 of node 21 and feature 21 of node 35 indicate that these two features play a major role in this classification, and the larger and negative values of feature 10 of the second node 1512 indicate that this feature will also have a considerable influence on the classification. If \(\mathbf{e}_{1}\) corresponds to a correct classification, the two positive features play a positive role in the classification, while the negative features play a negative role. If \(\mathbf{e}_{1}\) corresponds to a wrong classification, the two types of features play the opposite role, with the negative features playing a positive role in control by ensuring that the classification calculation value cannot be greater than the value resulting from the correct classification. In contrast, the positive features play the exact opposite role. In Figure 3(b), the data represents a value in the matrix \(\mathbb{R}^{2708\times 16}\) of the output of the first graph convolutional layer, corresponding to a vector denoted as \(\mathbf{e}\in E\) after orthogonal modulation. The reduction in the orthogonal vector information contained therein enables the analysis of another property of the graph neural network, which is that the model contains several graph convolutional layers and is able to aggregate the features of several  
Figure 4: Analysis of (**a**) GNN and (**b**) first graph convolutional layer. orders of neighboring nodes. The figure shows the output of the first graph convolutional layer, thus aggregating the features of the first-order neighbor nodes. Since \(S(\mathbf{e})>0\), it enters the second graph convolutional layer through the activation function. The analysis here also shows again that positive and negative values in the results are important. The reason for the good performance of the deep learning model lies in the complex computing process involved, which ensures diversity in feature learning. The idea of graph purification has been proposed in the study of graph neural networks for improving the model robustness by eliminating those nodes or connected edges that look significantly different from their neighbors according to the statistical laws of the graph structure.