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The classical Weisfeiler-Leman algorithm aka color refinement is fundamentalfor graph learning with kernels and neural networks. Originally developed forgraph isomorphism testing, the algorithm iteratively refines vertex colors. Onmany datasets, the stable coloring is reached after a few iterations and theoptimal number of iterations for machine learning tasks is typically evenlower. This suggests that the colors diverge too fast, defining a similaritythat is too coarse. We generalize the concept of color refinement and propose aframework for gradual neighborhood refinement, which allows a slowerconvergence to the stable coloring and thus provides a more fine-grainedrefinement hierarchy and vertex similarity. We assign new colors by clusteringvertex neighborhoods, replacing the original injective color assignmentfunction. Our approach is used to derive new variants of existing graph kernelsand to approximate the graph edit distance via optimal assignments regardingvertex similarity. We show that in both tasks, our method outperforms theoriginal color refinement with only a moderate increase in running timeadvancing the state of the art.

Gradual Weisfeiler-Leman: Slow and Steady Wins the Race