Keskeiset käsitteet
The Weisfeiler-Leman (WL) test is a powerful method for verifying graph isomorphism, and its connection to the expressive capabilities of graph neural networks has sparked significant interest in understanding the specific graph properties that the WL test can effectively distinguish. This paper provides a precise characterization of the WL-dimension of labeled graph motif parameters, which unifies the study of subgraph counting and induced subgraph counting problems.
Tiivistelmä
The paper explores the expressive power of the Weisfeiler-Leman (WL) test, a widely-recognized method for verifying graph isomorphism, and its connection to the capabilities of graph neural networks (GNNs).
Key highlights:
The paper provides a precise characterization of the WL-dimension of labeled graph motif parameters, which encompasses both subgraph counting and induced subgraph counting problems.
For subgraph counting, the WL-dimension is shown to be precisely the maximum treewidth of the homomorphic images of the pattern graph.
For induced subgraph counting, the WL-dimension is shown to be the number of vertices in the pattern graph minus 1.
The paper demonstrates that if the kWL test can distinguish graphs with different numbers of occurrences of a graph motif Γ, then the exact number of occurrences of Γ can be computed uniformly using only local information from the last layer of a corresponding GNN.
The paper presents polynomial-time algorithms for determining the WL-dimension of subgraph counting and counting k-graphlets, resolving open questions from previous work.
The results unify the study of subgraph counting and induced subgraph counting problems, providing a comprehensive understanding of the expressive power of the WL test in the context of graph motif parameters.