DRESS: A Continuous Framework for Structural Graph Refinement

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 $\mathcal{O}(n^3)$ or $\mathcal{O}(n^4)$, making them computationally prohibitive for large graphs. In this paper, we start from the Original-DRESS equation (Castrillo, León, and Gómez, 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 $Δ$-DRESS, which runs DRESS on each node-deleted subgraph $G \setminus \{v\}$, connecting the framework to the Kelly--Ulam reconstruction conjecture. Both Motif-DRESS and $Δ$-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 $\mathcal{O}(n^4)$ computational cost.