Scaling Weisfeiler–Leman Expressiveness Analysis to Massive Graphs with GPUs

Filippo Biondi, Mirco Tribastone, Max Tschaikowski

Keywords: Weisfeiler-Leman Test (1-WL) Computation, Randomized Parallel Algorithm, Linear Algebra Characterization

TL;DR: A novel parallel algorithm computing the 1-WL test (describing GNN expressiveness) that enjoys ~24x speed-ups over established algorithms

Abstract: The Weisfeiler–Leman (WL) test is a cornerstone for analyzing the expressiveness of Graph Neural Networks, but computing its stable coloring at scale has remained a bottleneck. Classical refinement algorithms are inherently sequential and, despite optimal asymptotic complexity, do not exploit modern massively parallel hardware. Moreover, the problem is P-complete, suggesting limited parallelizability in the worst case. We show that these theoretical barriers do not preclude practical scalability. We obtain a linear-algebraic view of stable colorings by reformulating WL refinement as repeated matrix–vector multiplications. Building on this, we introduce two key contributions: (i) a randomized refinement algorithm with tight probabilistic guarantees, and (ii) a batching method that enables the analysis of stable colorings on subgraphs while preserving global correctness. This approach maps directly to GPU-efficient primitives. In numerical experiments, our CUDA implementation delivers up to ~24x speedups 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.

Primary Area: learning on graphs and other geometries & topologies

Submission Number: 12028