In this paper, we propose a method named Dimensional reweighting Graph Convolutional Networks (DrGCNs), to tackle the problem of variance between dimensional information in the node representations of GCNs. We prove that DrGCNs can reduce the variance of the node representations by connecting our problem to the theory of the mean field. However, practically, we find that the degrees DrGCNs help vary severely on different datasets. We revisit the problem and develop a new measure K to quantify the effect. This measure guides when we should use dimensional reweighting in GCNs and how much it can help. Moreover, it offers insights to explain the improvement obtained by the proposed DrGCNs. The dimensional reweighting block is light-weighted and highly flexible to be built on most of the GCN variants. Carefully designed experiments, including several fixes on duplicates, information leaks, and wrong labels of the well-known node classification benchmark datasets, demonstrate the superior performances of DrGCNs over the existing state-of-the-art approaches. Significant improvements can also be observed on a large scale industrial dataset. Deep neural networks (DNNs) have been widely applied in various fields, including computer vision (He et al., 2016; Hu et al., 2018) , natural language processing (Devlin et al., 2019) , and speech recognition (Abdel-Hamid et al., 2014) , among many others. Graph neural networks (GNNs) is proposed for learning node presentations of networked data (Scarselli et al., 2009) , and later be extended to graph convolutional network (GCN) that achieves better performance by capturing topological information of linked graphs (Kipf & Welling, 2017) . Since then, GCNs begin to attract board interests. Starting from GraphSAGE (Hamilton et al., 2017) defining the convolutional neural network based graph learning framework as sampling and aggregation, many follow-up efforts attempt to enhance the sampling or aggregation process via various techniques, such as attention mechanism , mix-hop connection and adaptive sampling . In this paper, we study the node representations in GCNs from the perspective of covariance between dimensions. Suprisingly, applying a dimensional reweighting process to the node representations may be very useful for the improvement of GCNs. As an instance, under our proposed reweighting scheme, the input covariance between dimensions can be reduced by 68% on the Reddit dataset, which is extremely useful since we also find that the number of misclassified cases reduced by 40%, compared with the previous SOTA method. We propose Dimensional reweighting Graph Convolutional Networks (DrGCNs), in which the input of each layer of the GCN is reweighted by global node representation information. Our discovery is that the experimental performance of GCNs can be greatly improved under this simple reweighting scheme. On the other hand, with the help of mean field theory (Kadanoff, 2009; Yang et al., 2019) , this reweighting scheme is also proved to improve the stability of fully connected networks, provding insight to GCNs. To deepen the understanding to which extent the proposed reweighting scheme can help GCNs, we develop a new measure to quantify its effectiveness under different contexts (GCN variants and datasets). Experimental results verify our theoretical findings ideally that we can achieve predictable improvements on public datasets adopted in the literature over the state-of-the-art GCNs. While studying on these well-known benchmarks, we notice that two of them (Cora, Citeseer) suffer from duplicates and feature-label information leaks. We fix these problems and offer refined datasets for fair comparisons. To further validate the effectiveness, we deploy the proposed DrGCNs on A* 1 company's recommendation system and clearly demonstrate performance improvements via offline evaluations. 2 DRGCNS: DIMENSIONAL REWEIGHTING GRAPH CONVOLUTIONAL NETWORKS We We investigate the originality of the Cora and CiteSeer dataset. The two datasets are widely used for being light-weighted and easy to handle. The most popular version is provided by Planetoid (Yang et al., 2016) . The two datasets are both citation networks where each node represents a research paper, and each edge represents a citation relationship between two papers. Edges are directed but are usually handled undirectedly by GCN methods. Each paper belongs to a sub-field in computer science and is marked as its label. Papers have features of bag-of-word(BOW) vectors that each dimension represents whether the document of the paper contains a particular word in the dictionary or not. Cora has 2,708 papers with a dictionary size of 1,433, while Citeseer has 3,327 papers with a dictionary size of 3,703. A.1 CORA Cora originates in (McCallum et al., 2000) 7 with extracted information(including titles, authors, abstracts, references, download links etc.) in plain-text form. Those download links are mostly unavailable now. Before they become unavailable, (Lu & Getoor, 2003) 8 extracts a subset of 2,708 papers and assigns labels and BOW feature vectors to the papers. The dictionary is chosen from words(after stemming) 9 that occur 10 or more times in all papers and result in a dictionary size of 1,433. Planetoid (Yang et al., 2016) reordered each node to form the benchmark Cora dataset ). There exist a lot of duplicated papers (one paper appears as multiple identical papers in the dataset) in the original Cora of (McCallum et al., 2000) , and (Lu & Getoor, 2003) inherits the problem of duplicated papers. In Cora, we find 32 duplicated papers among the 2,708 papers. Another problem is the information leak. The generation process of the dictionary chooses words that occur more than 10 times, and does not exclude the label contexts of papers. Therefore, some papers may be classified easily only by looking at their labels. For instance, 61.8% of papers labeled "reinforcement learning" contain exactly the word "reinforcement" "learning" in their title and abstract(after stemming). Altogether 1,145(42.3%) of these papers contain their label as one or some of the dimensions of their features.
We propose a simple yet effective reweighting scheme for GCNs, theoretically supported by the mean field theory. A method, known as DrGCN, for reweighting the different dimensions of the node representations in graph convolutional networks by reducing variance between dimensions.