Graph Learning in 4D: A Quaternion-Valued Laplacian to Enhance Spectral GCNs

We introduce QuaterGCN, a spectral Graph Convolutional Network (GCN) with quaternion-valued weights at whose core lies the Quaternionic Laplacian, a quaternion-valued Laplacian matrix by whose proposal we generalize two widely-used Laplacian matrices: the classical Laplacian (defined for undirected graphs) and the complex-valued Sign-Magnetic Laplacian (proposed within the spectral GCN SigMaNet to handle digraphs with weights of arbitrary sign). In addition to its generality, QuaterGCN is the only Laplacian to completely preserve the (di)graph topology that we are aware of, as it can handle graphs and digraphs containing antiparallel pairs of edges (digons) of different weight without reducing them to a single (directed or undirected) edge as done by other Laplacians. Experimental results show the superior performance of QuaterGCN compared to other state-of-the-art GCNs, particularly in scenarios where the information the digons carry is crucial to successfully address the task at hand.