  Graph neural networks (GNNs) have gained significant attention in recent
years for their ability to process data that may be represented as graphs. This
success has prompted several studies to explore the representational capability
of GNNs based on the graph isomorphism task. These works inherently assume a
countable node feature representation, potentially limiting their
applicability. Interestingly, only a few theoretical works study GNNs with
uncountable node feature representation. This paper presents a novel
perspective on the representational capability of GNNs across all levels -
node-level, neighborhood-level, and graph-level - when the space of node
feature representation is uncountable. Specifically, it relaxes the injective
requirement in previous works by employing an implicit pseudometric distance on
the space of input to create a soft-injective function. This allows distinct
inputs to produce similar outputs only if the pseudometric deems the inputs to
be sufficiently similar on some representation, which is often useful in
practice. As a consequence, a novel soft-isomorphic relational graph
convolution network (SIR-GCN) that emphasizes non-linear and contextualized
transformation of neighborhood feature representations is proposed. A
mathematical discussion on the relationship between SIR-GCN and widely used
GNNs is then laid out to put the contribution in context, establishing SIR-GCN
as a generalization of classical GNN methodologies. Experiments on synthetic
and benchmark datasets demonstrate the relative superiority of SIR-GCN,
outperforming comparable models in node and graph property prediction tasks.
