arXiv:2605.23446v1 Announce Type: new Abstract: Graphs with a simple spectrum admit cubic-time isomorphism testing, yet we prove that for every natural number $k$, the $k$-Weisfeiler-Leman ($k$-WL) test cannot distinguish all non-isomorphic graphs with a simple spectrum. As the WL hierarchy upper-bounds the distinguishing power of widely-used Graph Neural Networks (GNNs), this incompleteness applies to all such GNNs, ruling out completeness for every $k$-WL-aligned GNN family. To close this gap, we introduce PRiSM (Partition, Refine, Solve, Match), the first provably complete canonicalization
This research provides a fundamental breakthrough in graph isomorphism testing as the field of AI, particularly GNNs, is rapidly advancing and confronting theoretical limitations.
It addresses a critical theoretical incompleteness in widely used Graph Neural Networks (GNNs), potentially leading to more robust and accurate AI models, vital for various applications from drug discovery to social network analysis.
The introduction of PRiSM offers the first provably complete canonicalization method for simple spectrum graphs, overcoming limitations of the established Weisfeiler-Leman test and GNNs.
- · AI researchers and developers
- · Companies using GNNs in areas like drug discovery and materials science
- · Graph algorithm developers
- · Academic institutions focused on theoretical computer science
- · Developers relying solely on k-WL-aligned GNNs for complex graph problems
- · Research teams that might have over-relied on WL-based completeness assumptions
Improved performance and broader applicability of GNNs in areas currently limited by theoretical incompleteness.
Accelerated development of AI agents and systems that rely on sophisticated graph analysis for decision-making and pattern recognition.
Enhanced AI capabilities across scientific research, potentially leading to breakthroughs in areas that can be modeled as complex graph structures, such as molecular design or network optimization for compute supply chains.
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Read at arXiv cs.LG