WEISFEILER-LEMAN

Graph isomorphism via iterative color refinement

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The Weisfeiler-Leman (WL) algorithm is an efficient heuristic for graph isomorphism: each node is assigned a color (label), then iteratively re-colored based on the multiset of its neighbors' colors using a hash function. If two graphs produce different color histograms at any iteration, they are non-isomorphic; identical histograms are a necessary but not sufficient condition. The 1-WL test (also called color refinement) is equivalent to the expressive power of Graph Neural Networks (Xu et al., 2019 — the WL theorem), meaning GNNs cannot distinguish graphs that WL cannot. The k-WL hierarchy for k≥2 uses k-tuples and is strictly more powerful.