Recently, graph neural networks (GNNs) have become an important and active research direction in deep learning. It is worth noting that most of the existing GNN-based methods learn graph representations within the Euclidean vector space. Beyond the Euclidean space, learning representation and embeddings in hyper-complex space have also shown to be a promising and effective approach. To this end, we propose Quaternion Graph Neural Networks (QGNN) to learn graph representations within the Quaternion space. As demonstrated, the Quaternion space, a hyper-complex vector space, provides highly meaningful computations and analogical calculus through Hamilton product compared to the Euclidean and complex vector spaces. Our QGNN obtains state-of-the-art results on a range of benchmark datasets for graph classification and node classification. Besides, regarding knowledge graphs, our QGNN-based embedding model achieves state-of-the-art results on three new and challenging benchmark datasets for knowledge graph completion. Our code is available at: \url{https://github.com/daiquocnguyen/QGNN}.
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