Vectorial Graph Convolutional Network

Abstract

Graph Convolutional Networks (GCN) have drawn considerable attention recently due to their outstanding performance in processing graphstructured data. However, GCNs still limited to the undirected graph because they theoretically require a symmetric matrix as the basis for the Laplacian transform. This causes the isotropic problem of the operator and reduced sensitivity in response to different information. In order to solve the problem, we generalize the spectral convolution operator to directed graphs by field extension, which improves the edge representations from scalars to vectors. Therefore, it brings in the concept of direction. That is to say, and even homogeneous information can become distinguishable by its differences in directions. In this paper, we propose the Vectorial Graph Convolutional Network(VecGCN) and the experimental evidence showing the advantages of a variety of directed graph node classification and link prediction tasks.



However, the vast majority of these researches are based on undirected graphs, and even the original graphs are naturally directed. This phenomenon will take the risk of discarding potentially important information Kawamoto et al. (2018); Zhang et al. (2021) . For example, you may have heard of a celebrity, but he/she doesn't know you. From the GATs' perspective, it is easy to understand that the attention values from node i to node j and node j to node i are not necessarily equal, which means the information is not symmetric on the edges. Adjacency matrix A is the topological edge set. Unless graph G is undirected, A is not symmetric. Unfortunately, GCNs are developed from spectral theory Kipf & Welling (2016); Xu et al. (2018a); Gilmer et al. (2017) and limited to symmetric convolutional kernels Beaini et al. (2021) , the object matrix of the kernels needs to be positive semi-definite and symmetric because the decomposition of a such matrix is orthogonal that can be taken as Fourier transform basis. It, in turn, requires the graph to be undirected to satisfy the above two conditions, or the eigenvalues of A can not be solved in the real number field. Thus, extending spectral methods to directed graphs is not straightforward Zhang et al. (2021) .



a ubiquitous data structure where entities are vertices and edges are their pairwise relationships. Most Graph Neural Networks(GNNs) fall into one of two categories: spectral Defferrard et al. (2016); Kipf & Welling (2016) or spatial networks Hamilton et al. (2017a); Veličković et al. (2017); Backstrom & Leskovec (2011). Spatial approaches are based on a localized averaging operator with learnable weights that iteratively traverse the entire graph. Spectral approaches based on eigen-decomposition of graph Laplacian and smooth those signals through Fourier transform Zhou et al. (2020); Wu et al. (2020). The application domains ranging from social networks Chen et al. (2012) to quantum chemistry Liao et al. (2019) and text classification Yao et al. (2019), etc. One of the key techniques is Graph Convolutional Networks (GCNs) Defferrard et al. (2016); Kipf & Welling (2016); Xu et al. (2018a), it's the variant of Convolutional Neural Networks (CNNs) Mallat (2016) on graphs, that learns the representations from both vertices and edges. It is particularly important to apply representations to downstream tasks Hamilton et al. (2017b), e.g., node classification and link prediction Hu et al. (2020).

