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Generalization of Graph Neural Networks through the Lens of Homomorphism

Shouheng Li, Dongwoo Kim, Qing Wang

TL;DR

The paper tackles the challenge of understanding GNN generalization by introducing an entropy-based perspective on graph homomorphisms. By linking homomorphism counts to information-theoretic measures and incorporating a Wasserstein-margin viewpoint, it derives non-vacuous, data-dependent bounds that apply across $1$-WL, $k$-WL, HI-GNN, and SI-GNN variants within a unified $\\mathcal{F}$-MPNN framework. The approach yields graph- and node-level generalization bounds that reflect pattern- and depth-related effects on GNN expressivity, and the authors validate these bounds against both real and synthetic datasets. Overall, the results provide a principled, structure-aware tool for predicting and understanding generalization gaps in GNNs, with practical implications for pattern selection and model design.

Abstract

Despite the celebrated popularity of Graph Neural Networks (GNNs) across numerous applications, the ability of GNNs to generalize remains less explored. In this work, we propose to study the generalization of GNNs through a novel perspective - analyzing the entropy of graph homomorphism. By linking graph homomorphism with information-theoretic measures, we derive generalization bounds for both graph and node classifications. These bounds are capable of capturing subtleties inherent in various graph structures, including but not limited to paths, cycles and cliques. This enables a data-dependent generalization analysis with robust theoretical guarantees. To shed light on the generality of of our proposed bounds, we present a unifying framework that can characterize a broad spectrum of GNN models through the lens of graph homomorphism. We validate the practical applicability of our theoretical findings by showing the alignment between the proposed bounds and the empirically observed generalization gaps over both real-world and synthetic datasets.

Generalization of Graph Neural Networks through the Lens of Homomorphism

TL;DR

The paper tackles the challenge of understanding GNN generalization by introducing an entropy-based perspective on graph homomorphisms. By linking homomorphism counts to information-theoretic measures and incorporating a Wasserstein-margin viewpoint, it derives non-vacuous, data-dependent bounds that apply across -WL, -WL, HI-GNN, and SI-GNN variants within a unified -MPNN framework. The approach yields graph- and node-level generalization bounds that reflect pattern- and depth-related effects on GNN expressivity, and the authors validate these bounds against both real and synthetic datasets. Overall, the results provide a principled, structure-aware tool for predicting and understanding generalization gaps in GNNs, with practical implications for pattern selection and model design.

Abstract

Despite the celebrated popularity of Graph Neural Networks (GNNs) across numerous applications, the ability of GNNs to generalize remains less explored. In this work, we propose to study the generalization of GNNs through a novel perspective - analyzing the entropy of graph homomorphism. By linking graph homomorphism with information-theoretic measures, we derive generalization bounds for both graph and node classifications. These bounds are capable of capturing subtleties inherent in various graph structures, including but not limited to paths, cycles and cliques. This enables a data-dependent generalization analysis with robust theoretical guarantees. To shed light on the generality of of our proposed bounds, we present a unifying framework that can characterize a broad spectrum of GNN models through the lens of graph homomorphism. We validate the practical applicability of our theoretical findings by showing the alignment between the proposed bounds and the empirically observed generalization gaps over both real-world and synthetic datasets.
Paper Structure (37 sections, 19 theorems, 43 equations, 3 figures, 3 tables)

This paper contains 37 sections, 19 theorems, 43 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Generalization Bounds
  4. Generalization in GNNs
  5. Homomorphism and Subgraph GNNs
  6. Preliminaries
  7. Graph Homomorphism and Entropy
  8. Homomorphic Image and Spasm
  9. $\mathcal{F}$-pattern Trees
  10. Graph Neural Networks
  11. $\mathcal{F}$-MPNN as a Unified GNN Framework
  12. Generalization Bounds through Homomorphism
  13. Analysis Setup.
  14. Graph-level Generalization Bound by Homomorphism Entropy
  15. Extending to Node-level Generalization Bound
  16. ...and 22 more sections

Key Result

Corollary 4.1

Let $\widetilde{D}_{KL}(\mathcal{F}, S, \tilde{S}) = D_{KL}\left( X_{\mu_S, T_L(\mathcal{F})}\parallel X_{\mu_{\tilde{S}}, T_L(\mathcal{F})} \right)$. Given $m$ i.i.d graph samples, with probability at least $1-\delta > 0$. we have

Figures (3)

  • Figure 1: Accuracy gap and bound value using different homomorphism pattern sets, obtained from GCN of 4 layers.
  • Figure 2: Accuracy gap and bound value using different homomorphism pattern sets
  • Figure 3: Bounds of homomorphism-injected and subgraph-injected models of 4 layers.

Theorems & Definitions (34)

  • Definition 3.1: Entropy of Homomorphism
  • Definition 3.2: $\mathcal{F}$-MPNN
  • Corollary 4.1: Expectation Bound for Graph Classification
  • Lemma 4.1: Data-dependent Bound for Graph Classification
  • Corollary 4.2: Expectation Bound for Node Classification
  • Lemma 4.2: Data-dependent Bound for Node Classification
  • Lemma 5.1
  • Corollary 5.1
  • Definition 3.0: $p$-Wasserstein Distance Chuang2021-ik
  • Remark 3.1
  • ...and 24 more