Graph Learning in 4D: a Quaternion-valued Laplacian to Enhance Spectral
  GCNs
Stefano Fiorini Stefano Coniglio Michele Ciavotta Enza Messina
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.