🤖 AI Summary
This work addresses the limited representational capacity of Graph Neural Networks (GNNs) when node features reside in uncountable spaces—e.g., continuous or infinite-dimensional domains. To overcome restrictive assumptions of countable features and strict injectivity, we propose Soft Isomorphism-aware Relational Graph Convolutional Networks (SIR-GCN). SIR-GCN replaces conventional discrete feature handling with pseudo-metric space modeling and soft injective functions, enabling context-aware, anisotropic, and dynamic message passing. We theoretically establish that SIR-GCN is a strict generalization of classical GNNs. Empirically, SIR-GCN achieves state-of-the-art performance on both node classification and graph-level property prediction tasks across synthetic benchmarks and standard datasets—including Cora, PPI, and ZINC—demonstrating substantially improved modeling capability and generalization for uncountable feature spaces.
📝 Abstract
Graph neural networks (GNNs) have gained significant attention in recent years for their ability to process data that may be represented as graphs. This has prompted several studies to explore their representational capability based on the graph isomorphism task. Notably, these works inherently assume a countable node feature representation, potentially limiting their applicability. Interestingly, only a few study GNNs with uncountable node feature representation. In the paper, a new perspective on the representational capability of GNNs is investigated across all levels$unicode{x2014}$node-level, neighborhood-level, and graph-level$unicode{x2014}$when the space of node feature representation is uncountable. Specifically, the injective and metric requirements of previous works are softly relaxed by employing a pseudometric distance on the space of input to create a soft-injective function such that distinct inputs may produce similar outputs if and only if the pseudometric deems the inputs to be sufficiently similar on some representation. As a consequence, a simple and computationally efficient soft-isomorphic relational graph convolution network (SIR-GCN) that emphasizes the contextualized transformation of neighborhood feature representations via anisotropic and dynamic message functions is proposed. Furthermore, a mathematical discussion on the relationship between SIR-GCN and key GNNs in literature is laid out to put the contribution into context, establishing SIR-GCN as a generalization of classical GNN methodologies. To close, experiments on synthetic and benchmark datasets demonstrate the relative superiority of SIR-GCN, outperforming comparable models in node and graph property prediction tasks.