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.

QuaterGCN 是一个具有四元数值权重的谱图卷积网络 (GCN)，其核心是四元数拉普拉斯算子，通过该提案广义化了两个广泛使用的拉普拉斯算子：经典拉普拉斯算子（针对无向图）和复数值标志磁性拉普拉斯算子（用于处理带有任意符号权重的有向图）。实验结果表明，QuaterGCN 的性能优于其他最先进的 GCN，特别是在包含有关重要信息的有向图情景中。

四维图学习：增强谱图GCNs的四元数值拉普拉斯算子