A Graph Convolution for Signed Directed Graphs

There are several types of graphs according to the nature of the data. Directed graphs have directions of links, and signed graphs have link types such as positive and negative. Signed directed graphs are the most complex and informative that have both. Graph convolutions for signed directed graphs have not been delivered much yet. Though many graph convolution studies have been provided, most are designed for undirected or unsigned. In this paper, we investigate a spectral graph convolution network for signed directed graphs. We propose a novel complex Hermitian adjacency matrix that encodes graph information via complex numbers. The complex numbers represent link direction, sign, and connectivity via the phases and magnitudes. Then, we define a magnetic Laplacian with the Hermitian matrix and prove its positive semidefinite property. Finally, we introduce Signed Directed Graph Convolution Network(SD-GCN). To the best of our knowledge, it is the first spectral convolution for graphs with signs. Moreover, unlike the existing convolutions designed for a specific graph type, the proposed model has generality that can be applied to any graphs, including undirected, directed, or signed. The performance of the proposed model was evaluated with four real-world graphs. It outperforms all the other state-of-the-art graph convolutions in the task of link sign prediction.