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DRESS: A Continuous Framework for Structural Graph Refinement

Eduar Castrillo Velilla

TL;DR

The 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.

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 $\mathcal{O}(n^3)$ 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.

DRESS: A Continuous Framework for Structural Graph Refinement

TL;DR

The 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.

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 or , 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 , 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 -DRESS, which runs DRESS on each node-deleted subgraph , connecting the framework to the Kelly--Ulam reconstruction conjecture. Both Motif-DRESS and -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 computational cost.
Paper Structure (17 sections, 2 theorems, 5 equations)

This paper contains 17 sections, 2 theorems, 5 equations.

Table of Contents

  1. Introduction
  2. Isomorphism Test.
  3. Related Work.
  4. Original-DRESS
  5. Original-DRESS Distinguishes Graphs That 1-WL Cannot
  6. Remark.
  7. Motif-DRESS
  8. Properties
  9. Expressiveness
  10. Proof Sketch of Convergence
  11. Generalized-DRESS
  12. $\Delta$-DRESS
  13. Connection to the Reconstruction Conjecture
  14. Properties
  15. Expressiveness
  16. ...and 2 more sections

Key Result

Theorem 1

There exist graph pairs that 1-WL cannot distinguish but Original-DRESS can. In particular, DRESS distinguishes the prism graph ($C_3 \square K_2$) from the complete bipartite graph $K_{3,3}$, a pair that 1-WL provably cannot separate.

Theorems & Definitions (4)

  • Theorem 1: DRESS distinguishes beyond 1-WL
  • proof
  • Theorem 2: Convergence of Motif-DRESS
  • proof : Proof sketch