# GRNN: Graph-Retraining Neural Network for Semi-Supervised Node Classification
## 1 Introduction  
Convolutional neural networks (CNNs) achieve outstanding performance in a wide range of tasks that are based on Euclidean data including computer vision [1] and recommender systems [2; 3]. However, an increasing number of application data are in the form of graph-structured non-Euclidean data, such as literature-citation networks and knowledge graphs. In this case, graph convolutional networks (GCNs) commonly outperform other models and are successfully applied in social analysis [4; 5], citation networks [6; 7; 8], transport forecasting [9; 10], and other promising fields. For example, in a literature-citation network, articles are usually presented as nodes and citation relationships as edges between nodes. When faced with the challenge of semi-supervised node classification on such non-Euclidean graph data, classical GCNs first extract and aggregate features of articles using the graph Fourier transform and the convolution theorem and then classify unlabeled articles based on graph convolution output features. Graph neural networks (GNNs) are capable of reading prospective data from the network architecture. Graph convolution networks (GCNs), in which nodes learn their potential representations by aggregating features from neighboring nodes, have been shown to be effective and practical. For example, GCN [11] calculates the weight of aggregated neighbor nodes based on the degree of its nodes and its first-order neighbor nodes; GAT [12] injects the graph structure into the mechanism by performing masked attention, which means that the weight of aggregated neighbor nodes is a learnable parameter; GraphSAGE [13] uniformly samples a fixed number of neighbor nodes to aggregate information rather than using all neighbor nodes. MPNN [14] uses message-passing functions to unify almost all variants of the spatial graph neural network. However, a critical limitation is that the labels of nodes are independently predicted based on their representations and their neighbor-node representations. Each layer of the graph convolution neural network is a special kind of Laplace smoothing [6]. In other words, they attach importanceto the smoothness of the nodes of the training set but neglect the smoothness of the whole network. Although C&S [15] and UniMP [16] perform smoothing of the predictions over the graph via label propagation, the representations are neglected in the smoothing process. In this paper, we propose a novel approach called the graph-retraining neural network (GRNN), which conducts smoothing over the graph using a graph convolution neural network. The key idea behind the approach is similar to the expectation-maximum algorithm [17], alternating between a learning procedure and an inference procedure. It is different from GMNN [18], which contains a conditional random field [19] to model the joint distribution of the labels conditioned on representations. In the learning procedure, instead of maximizing the likelihood function, GRNN uses graph neural networks to train all node representations to minimize loss function or maximum accuracy, which can adjust the graph neural network's weight. In the inference procedure, instead of calculating the expectations of the likelihood function, GRNN predicts all node labels using the final outputs of graph neural networks. Inspired by the idea of keeping the known representations at training nodes fixed [20; 21], GRNN updates the predicted node labels by correcting the training labels in the inference procedure. To alleviate the problem that the expectation-maximum algorithm has a significant influence on the initially predicted labels, GRNN uses graph neural networks to train the training set and adjust the graph neural network's weight as the initial weight. The architecture of GRNN alternating between a learning procedure and an inference procedure is illustrated in Figure 1.  
In general, there are three main contributions of the proposed GRNN to semi-supervised node classification.