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Logical Expressiveness of Graph Neural Networks with Hierarchical Node Individualization

Arie Soeteman, Balder ten Cate

TL;DR

This work introduces Hierarchical Ego GNNs (HE-GNNs), an isomorphism-invariant graph learning framework that uses iterative node individualization in a hierarchical, IR-inspired scheme. It establishes strong logical characterizations: HE-GNNs correspond to graded hybrid logic with downarrow (GML(↓)) and extend to GML(↓,W^r) under subgraph restrictions, forming a nesting-depth and radius-based expressiveness hierarchy that relates to the WL and higher-order GNNs. The authors prove that, for graphs of bounded degree, HE-GNNs with depth d capture the same separating power as GML(↓^d) and that HES-GNNs align with GML(↓^d_W^r), yielding a strict, tunable hierarchy that can exceed standard subgraph-GNNs while offering improved practical feasibility. Empirical results on ZINC-12k and strongly regular graphs demonstrate tangible benefits of depth-2 HE-GNNs and depth-2/3 HES-GNNs in distinguishing graphs beyond several baseline models, validating the approach's practical relevance and guiding future design of more expressive, IR-inspired GNNs.

Abstract

We propose and study Hierarchical Ego Graph Neural Networks (HEGNNs), an expressive extension of graph neural networks (GNNs) with hierarchical node individualization, inspired by the Individualization-Refinement paradigm for isomorphism testing. HEGNNs generalize subgraph-GNNs and form a hierarchy of increasingly expressive models that, in the limit, distinguish graphs up to isomorphism. We show that, over graphs of bounded degree, the separating power of HEGNN node classifiers equals that of graded hybrid logic. This characterization enables us to relate the separating power of HEGNNs to that of higher-order GNNs, GNNs enriched with local homomorphism count features, and color refinement algorithms based on Individualization-Refinement. Our experimental results confirm the practical feasibility of HEGNNs and show benefits in comparison with traditional GNN architectures, both with and without local homomorphism count features.

Logical Expressiveness of Graph Neural Networks with Hierarchical Node Individualization

TL;DR

This work introduces Hierarchical Ego GNNs (HE-GNNs), an isomorphism-invariant graph learning framework that uses iterative node individualization in a hierarchical, IR-inspired scheme. It establishes strong logical characterizations: HE-GNNs correspond to graded hybrid logic with downarrow (GML(↓)) and extend to GML(↓,W^r) under subgraph restrictions, forming a nesting-depth and radius-based expressiveness hierarchy that relates to the WL and higher-order GNNs. The authors prove that, for graphs of bounded degree, HE-GNNs with depth d capture the same separating power as GML(↓^d) and that HES-GNNs align with GML(↓^d_W^r), yielding a strict, tunable hierarchy that can exceed standard subgraph-GNNs while offering improved practical feasibility. Empirical results on ZINC-12k and strongly regular graphs demonstrate tangible benefits of depth-2 HE-GNNs and depth-2/3 HES-GNNs in distinguishing graphs beyond several baseline models, validating the approach's practical relevance and guiding future design of more expressive, IR-inspired GNNs.

Abstract

We propose and study Hierarchical Ego Graph Neural Networks (HEGNNs), an expressive extension of graph neural networks (GNNs) with hierarchical node individualization, inspired by the Individualization-Refinement paradigm for isomorphism testing. HEGNNs generalize subgraph-GNNs and form a hierarchy of increasingly expressive models that, in the limit, distinguish graphs up to isomorphism. We show that, over graphs of bounded degree, the separating power of HEGNN node classifiers equals that of graded hybrid logic. This characterization enables us to relate the separating power of HEGNNs to that of higher-order GNNs, GNNs enriched with local homomorphism count features, and color refinement algorithms based on Individualization-Refinement. Our experimental results confirm the practical feasibility of HEGNNs and show benefits in comparison with traditional GNN architectures, both with and without local homomorphism count features.

Paper Structure

This paper contains 24 sections, 49 theorems, 55 equations, 7 figures, 2 tables, 5 algorithms.

Table of Contents

  1. Introduction
  2. Related work
  3. Background and notation
  4. Graphs
  5. Graph neural networks
  6. Graded modal logic
  7. Three-way equivalence
  8. Hierarchical Ego GNNs
  9. Hierarchical Ego GNNs
  10. Graded hybrid logic
  11. Hierarchical Ego GNNs with subgraph restriction
  12. Hierarchical Ego Subgraph-GNNs
  13. Graded hybrid subgraph logic
  14. Comparison with other models
  15. Relationship with Individualization-Refinement
  16. ...and 9 more sections

Key Result

Theorem 2.2

$\rho(\textup{GNN}\xspace) = \rho(\textup{GML}\xspace) = \rho(\textup{WL}\xspace)$

Figures (7)

  • Figure 1: Two non-isomorphic pointed graphs that are WL-indistinguishable.
  • Figure 2: Rooted 5$\times$2-grid (root: $u_1$)
  • Figure 3: Precision on isomorphism classification of strongly regular graphs for four settings of $(v,k,\lambda,\mu)$. Models from left to right: ID-GNN you2021identity, $\mathfrak{F}\textup{-}\textup{GNN}\xspace$ with counts from $C_3, \dots C_{10}$barcelo2021graph, GNN with Tinhofer labelings pellizzoni2024expressivity, HES-GNN with $d=2, r=1$ and HE-GNN with $d=2$.
  • Figure 4: Time (left) and memory (right) per sample of a single forward pass of HES-GNN-($d$,$r$) on ZINC. This does not include gradients. Experiments run with batch size $20$ and feature dimension $256$.
  • Figure 5: Two graphs that are distinguished by a depth $1$HE-GNN, but not by $\textup{WL-IR-$1$}\xspace$. All nodes in both graphs have an empty labeling.
  • ...and 2 more figures

Theorems & Definitions (80)

  • Example 2.1
  • Theorem 2.2
  • Proposition 2.2
  • Proposition 2.2
  • Example 3.1
  • Example 3.2
  • Example 3.3
  • Theorem 3.4
  • Theorem 4.1
  • Theorem 4.2
  • ...and 70 more