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. 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 novel 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. More specifically, the strict injective and metric requirements 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. A mathematical discussion on the relationship between SIR-GCN and widely used GNNs is then laid out to put the contribution into context, establishing SIR-GCN as a generalization of classical GNN methodologies. Experiments on synthetic and benchmark datasets then demonstrate the relative superiority of SIR-GCN, outperforming comparable models in node and graph property prediction tasks.

翻译：近年来，图神经网络（GNNs）因其处理可表示为图的数据的能力而受到广泛关注。这促使多项研究基于图同构任务探索其表示能力。这些工作本质上假设了可数的节点特征表示，可能限制了其适用性。有趣的是，仅有少数研究探讨具有不可数节点特征表示的GNN。本文研究了当节点特征表示空间为不可数时，GNN在节点级、邻域级和图级所有层次上的表示能力的新视角。具体而言，通过在输入空间上采用伪度量距离来构造软单射函数，从而温和地放宽严格的单射性与度量要求，使得不同输入仅在伪度量判定其在某种表示上足够相似时，才可能产生相似输出。基于此，我们提出了一种简单且计算高效的软同构关系图卷积网络（SIR-GCN），该网络通过各向异性和动态的消息函数强调邻域特征表示的上下文化变换。随后通过数学讨论阐述了SIR-GCN与广泛使用的GNN之间的关系，从而将本研究的贡献置于更广阔的背景下，确立SIR-GCN作为经典GNN方法的一般化扩展。在合成数据集和基准数据集上的实验表明，SIR-GCN在节点与图属性预测任务中优于可比模型，展现出相对优越性。