Overview
Research introduces an information-complexity property specifically for aggregation functions within Message-Passing Graph Neural Networks (MP-GNNs). It establishes a fundamental expressivity limit, demonstrating that MP-GNN models utilizing aggregation functions possessing this property induce only a polynomial number of equivalence classes across all graphs. This stands in contrast to Color Refinement (CR), where two iterations can generate at least an exponential number of equivalence classes, suggesting a significant relative expressivity disparity.
Research Context
Previous studies often asserted that sum-aggregation MP-GNNs achieve parity with full Color Refinement in terms of distinguishing power. However, this assertion is critiqued as it relies on a 'non-uniform' notion of distinguishing power. Under this non-uniform definition, each distinct graph size might necessitate a different MP-GNN configuration to differentiate graphs of that specific size. The current research addresses both the distinction between non-equivariant vertices and the differentiation of non-isomorphic graphs.
Approach
The research defines an information-complexity property applicable to aggregation functions. This property is described as capturing a vast range of practical aggregation methods. Following this definition, the study proceeds to prove that any MP-GNN model incorporating such aggregations is inherently limited in its expressivity. The proof focuses on the number of equivalence classes induced by these models on graphs.
Findings
- MP-GNN models that utilize aggregation functions conforming to the defined information-complexity property induce only a polynomial number of equivalence classes across all graphs.
- The total number of non-isomorphic graphs is super-exponential with respect to the number of vertices.
- A comparative analysis with Color Refinement (CR) indicates that even a limited application, specifically two iterations of CR, can induce at least an exponential number of equivalence classes.
- This comparison suggests that the aforementioned MP-GNNs are 'infinitely weaker' relative to two iterations of CR, in terms of their ability to distinguish graphs.
- Previous claims regarding sum-aggregation MP-GNNs matching full CR are re-evaluated, noting they relied on a 'non-uniform' distinguishing-power notion, which implies differing MP-GNNs per graph size for distinction.
- The results explicitly concern two distinct distinguishing tasks: distinguishing between non-equivariant vertices and distinguishing between non-isomorphic graphs.
Why This Matters
The findings illuminate a fundamental expressivity limitation inherent in a broad class of Message-Passing Graph Neural Networks employing practical aggregation functions. Understanding this limitation is critical for evaluating the theoretical capabilities of MP-GNNs in tasks requiring fine-grained distinction, such as graph isomorphism testing or identifying unique structural properties within graph data.