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 to handle digraphs with weights of arbitrary sign). In addition to its generality, our Quaternionic Laplacian is the only Laplacian to completely preserve the topology of a digraph, as it can handle graphs and digraphs containing antiparallel pairs of edges (digons) of different weights without reducing them to a single (directed or undirected) edge as done with 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.

翻译：我们提出QuaterGCN——一种采用四元数权重的谱图卷积网络，其核心是四元数拉普拉斯算子。通过这一四元数值拉普拉斯矩阵的提出，我们推广了两种广泛使用的拉普拉斯矩阵：经典拉普拉斯（定义于无向图）和复值符号磁拉普拉斯（为处理含任意符号权重的有向图而提出）。除了通用性，我们的四元数拉普拉斯是唯一能完整保留有向图拓扑结构的拉普拉斯算子，因为它可以处理包含不同权重反平行边对（双边形）的图和有向图，而无需像其他拉普拉斯算子那样将其简化为单条（有向或无向）边。实验结果表明，QuaterGCN相比其他最先进的图卷积网络具有更优性能，尤其在双边形承载的信息对成功完成目标任务至关重要的场景中表现突出。