Scaling Weisfeiler-Leman Expressiveness Analysis to Massive Graphs with GPUs

2026-07-01

AI Summary

Traditional 1-WL stable coloring algorithms struggle to scale to web-scale graphs due to their inherently sequential nature and high memory overhead. This work addresses this limitation by leveraging the linear-algebraic formulation of 1-WL coloring to propose the first randomized refinement algorithm with rigorous probabilistic correctness guarantees. Furthermore, we introduce a provably correct graph batching strategy that preserves coloring accuracy while enabling efficient mapping of computations onto GPU primitives. By integrating randomized algorithms, graph partitioning, and CUDA-based parallelism, our approach successfully achieves stable coloring on web-scale graphs with over 30 billion edges—attaining up to two orders of magnitude speedup over CPU baselines, whereas conventional methods either time out or fail entirely at this scale.

Abstract

The stable coloring of the Weisfeiler-Leman (1-WL) test is a cornerstone of Graph Neural Networks because it provides an upper bound to the expressive power of message-passing architectures. Unfortunately, computing it presents two fundamental bottlenecks. First, classic algorithms are inherently sequential and cannot exploit modern massively parallel hardware. Second, these are \emph{global} algorithms, i.e., they require availability in memory of the full graph, severely limiting applicability to real-world instances. We leverage a linear-algebraic interpretation of 1-WL stable coloring and introduce two key contributions: (i)~a randomized refinement algorithm with tight probabilistic guarantees and (ii)~a correctness-preserving batching scheme that decomposes the graph into independently processable subgraphs while provably returning a stable coloring of the original graph. This approach maps directly to GPU-efficient primitives. In numerical experiments, our CUDA implementation delivers speedups up to two orders of magnitude over classical CPU-based partition refinement and, for the first time, successfully computes stable colorings on web-scale graphs with over 30 billion edges, where CPU baselines time out or fail.

Problem

• Weisfeiler-Leman
• graph neural networks
• stable coloring
• massive graphs
• parallel computing

Innovation

• Weisfeiler-Leman
• GPU acceleration
• randomized algorithm
• graph partitioning
• scalable graph coloring