wl expressivity control

This paper critically examines the overreliance on the Weisfeiler–Lehman (WL) test in Graph Neural Network (GNN) expressivity research, identifying three fundamental mismatches: semantic (structural equivalence ≠ functional expressivity), feature aggregation capability, and local computability. Method: It introduces the CONGEST model from distributed computing as a rigorous theoretical foundation for GNN expressivity analysis, employing communication complexity to quantify message-passing constraints. Contribution/Results: The analysis proves that simulating one WL iteration requires Ω(n) communication bandwidth—refuting its assumed local computability. It clarifies the actual impact of virtual nodes/edges, corrects widespread misconceptions regarding “pre-computation–enhanced expressivity,” and establishes a more rigorous, computationally grounded, and architecture-aware expressivity framework tailored to practical GNN designs.

This paper establishes a tight lower bound on the minimum number of iterations required by the $k$-dimensional Weisfeiler–Leman ($k$-WL) algorithm to decide graph isomorphism. Addressing the long-standing open question—posed in 2001—of whether $k$-WL iteration complexity can exceed linear time, the authors construct a family of CFI graphs and combine WL coloring dynamics analysis with logical characterizations of graphs. They prove that, for general $n$-vertex graphs, $Omega(n^{k/2})$ iterations are necessary and sufficient for $k$-WL to achieve full coarsest equitable partition refinement. This resolves the open problem definitively. Moreover, the result yields the first quantitative connections between WL iteration depth and fundamental concepts in theoretical computer science: definability in first-order logic with counting, expressive power of graph neural networks, and computational complexity. The work thus provides a foundational theoretical basis for graph representation learning.

This work investigates the expressive limits of global attention-based graph foundation models in representing mixed-integer linear programming (MILP) problems, with a focus on their ability to distinguish between non-isomorphic graph instances that are 1-WL equivalent. By integrating the 1-dimensional Weisfeiler–Leman (1-WL) graph isomorphism test, analysis of symmetric multiset functions, and various graph encoders—including Graphormer and GraphGPS—augmented with random walk positional encodings, the study reveals for the first time that prevailing global attention architectures are universally constrained by 1-WL equivalence when encoding MILP structures. The authors introduce an encoder-agnostic diagnostic framework and validate it across ten diverse architectures, demonstrating that all produce identical embeddings for 1-WL-equivalent non-isomorphic graphs, while the incorporation of positional information partially alleviates this limitation.

This work investigates the completeness of the Weisfeiler–Lehman (WL) test for isometry discrimination of point clouds in Euclidean space ℝᵈ, modeled as complete graphs with pairwise Euclidean distance labels. Using high-dimensional WL tests and isometry-invariance analysis, we establish tight theoretical bounds: the (d−1)-WL test achieves completeness in three iterations for distinguishing any non-isometric point clouds in ℝᵈ, whereas the d-WL test attains completeness in a single iteration. Notably, the 2-WL test is complete for point clouds in ℝ³ but fails to distinguish coplanar configurations and is incomplete in ℝ⁶. These results fully resolve the long-standing question of WL completeness for 3D point clouds and provide asymptotically tight bounds for general d-dimensional settings. The findings fundamentally characterize the expressive limits of graph neural networks—when applied to geometric data via distance-labeled graphs—in capturing intrinsic spatial structure and isometric invariance.

Formal verification of graph algorithms—particularly those involving intricate pointer manipulations—in separation logic remains challenging due to the difficulty of abstracting low-level heap structures while preserving semantic precision. Method: This paper introduces an algebraic modeling approach: it is the first to represent graph structures as partial commutative monoids and, grounded in category theory, designs structure-preserving higher-order morphisms and combinators that jointly encode pointer-operation semantics and abstract graph reasoning. Contribution/Results: The method enables natural divide-and-conquer verification and significantly simplifies separation logic proof rules. Applied to the Schorr–Waite graph-marking algorithm, it yields a concise, modular, and reusable formal verification—eliminating reliance on complex loop invariants and fine-grained pointer tracking. This advances the abstraction level, scalability, and engineering practicality of graph algorithm verification.

研究通过隐式线图高阶Weisfeiler-Leman算法（ILG-k-WL）探讨了线图与其根图之间固定维度WL表达力的关系，对于k=3时，证明了从线图到根图的反向包含关系。

Traditional message-passing networks and their simplicial complex extensions struggle to distinguish mesh structures with identical connectivity but differing geometric embeddings, due to a lack of geometric awareness. This work proposes the Geometric Simplicial Weisfeiler–Lehman (GSWL) test, which for the first time incorporates vertex coordinates into the color refinement process, thereby establishing a geometric-aware simplicial message-passing framework. By integrating an Euler characteristic transform, the framework’s expressive power is fully characterized. We prove that GSWL is equivalent to geometric-aware message passing over families of finite geometric simplicial complexes, thus constructing a hierarchical theory bridging combinatorial and geometric expressivity. Experiments on both synthetic and real-world mesh data demonstrate that the proposed approach substantially enhances expressive power, clearly revealing the modeling advantages conferred by explicit geometric information.

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.

Existing k-Weisfeiler-Leman (k-WL) tests fail to distinguish all non-isomorphic simple spectral graphs, thereby limiting the expressive power of graph neural networks (GNNs) on this class of graphs. This work proposes PRiSM, the first method to achieve provably complete canonical labeling for simple spectral graphs. PRiSM integrates spectral graph theory with a partition-refine-solve-match mechanism derived from eigendecomposition, and unifies it within a framework combining DeepSets and Transformer architectures. The approach not only offers theoretical guarantees of completeness but also achieves performance that significantly surpasses or matches current spectral canonicalization methods across graph classification, regression, and expressivity benchmarks. In doing so, PRiSM resolves a long-standing open problem concerning isomorphism testing for simple spectral graphs and the associated limitations in GNN expressiveness.

This study addresses the exponential growth in computational cost associated with the dimensionality required by the Weisfeiler-Leman (WL) algorithm to distinguish non-isomorphic graphs, which fundamentally limits graph isomorphism testing efficiency. By leveraging rank decomposition structures over the F2 field, this work establishes a theoretical connection between the WL dimension and the maximum cut-rank and cross-loading of graphs. Incorporating the Cai-Fürer-Immerman construction, it proposes sufficient dimensional bounds determined by these parameters and proves their tightness up to constant factors. The primary contribution reveals that graphs with bounded rank-width can be completely distinguished in low dimensions; for instance, graphs of linear rank-width k require only 2k+2 dimensions for isomorphism testing. This provides a tight theoretical optimization framework for high-dimensional WL computations.