We define a generic class of functions that captures most conceivable aggregations for Message-Passing Graph Neural Networks (MP-GNNs), and prove that any MP-GNN model with such aggregations induces only a polynomial number of equivalence classes on all graphs - while the number of non-isomorphic graphs is doubly-exponential (in number of vertices). Adding a familiar perspective, we observe that merely 2-iterations of Color Refinement (CR) induce at least an exponential number of equivalence classes, making the aforementioned MP-GNNs relatively infinitely weaker. Previous results state that MP-GNNs match full CR, however they concern a weak, 'non-uniform', notion of distinguishing-power where each graph size may required a different MP-GNN to distinguish graphs up to that size. Our results concern both distinguishing between non-equivariant vertices and distinguishing between non-isomorphic graphs.

翻译：我们定义了一类通用函数，这些函数涵盖了消息传递图神经网络（MP-GNNs）中大多数可想象的聚合方式，并证明任何采用此类聚合的MP-GNN模型在所有图上仅诱导出多项式数量的等价类——而非同构图的数量（按顶点数计）却是双指数级的。从一个常见视角来看，我们观察到仅2次迭代的颜色精炼（CR）即可诱导出至少指数数量的等价类，这使得前述MP-GNNs相对而言无限弱。先前的研究结果表明MP-GNNs可以匹配完整的CR，但它们关注的是区分能力的“非均匀”弱概念，即每个图大小可能需要不同的MP-GNN来区分该大小范围以内的图。我们的结果同时涉及区分非等变顶点和区分非同构图。