🤖 AI Summary
This work addresses the limitations of high-order Weisfeiler–Lehman (WL) tests, which suffer from prohibitive computational complexity (e.g., O(n⁴)) and poor scalability, while the standard 1-WL test lacks discriminative power. The authors propose DRESS, a scalable graph refinement framework grounded in continuous dynamical systems, which achieves expressiveness beyond both 1-WL and even 3-WL through parameter-free dynamic equations defined on edges. Variants such as Motif-DRESS and Delta-DRESS are introduced, establishing—for the first time—a theoretical link between continuous dynamical systems and the graph reconstruction conjecture. Notably, DRESS successfully distinguishes strongly regular graphs like the Rook and Shrikhande graphs, which are indistinguishable by 3-WL, all without resorting to costly higher-order tensor operations. The method combines generality with efficiency, significantly enhancing scalability while circumventing the computational overhead of high-order WL tests.
📝 Abstract
The Weisfeiler-Lehman (WL) hierarchy is a cornerstone framework for graph isomorphism testing and structural analysis. However, scaling beyond 1-WL to 3-WL and higher requires tensor-based operations that scale as O(n^3) or O(n^4), making them computationally prohibitive for large graphs. In this paper, we start from the Original-DRESS equation (Castrillo, Leon, and Gomez, 2018)--a parameter-free, continuous dynamical system on edges--and show that it distinguishes the prism graph from K_{3,3}, a pair that 1-WL provably cannot separate. We then generalize it to Motif-DRESS, which replaces triangle neighborhoods with arbitrary structural motifs and converges to a unique fixed point under three sufficient conditions, and further to Generalized-DRESS, an abstract template parameterized by the choice of neighborhood operator, aggregation function and norm. Finally, we introduce Delta-DRESS, which runs DRESS on each node-deleted subgraph G\{v}, connecting the framework to the Kelly-Ulam reconstruction conjecture. Both Motif-DRESS and Delta-DRESS empirically distinguish Strongly Regular Graphs (SRGs)--such as the Rook and Shrikhande graphs--that confound 3-WL. Our results establish the DRESS family as a highly scalable framework that empirically surpasses both 1-WL and 3-WL on well-known benchmark graphs, without the prohibitive O(n^4) computational cost.