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Position: Message-passing and spectral GNNs are two sides of the same coin

Antonis Vasileiou, Juan Cervino, Pascal Frossard, Charilaos I. Kanatsoulis, Christopher Morris, Michael T. Schaub, Pierre Vandergheynst, Zhiyang Wang, Guy Wolf, Ron Levie

TL;DR

This work argues that message-passing neural networks (MPNNs) and spectral GNNs are not fundamentally distinct paradigms but different parametrizations of permutation-equivariant operators on graph signals. It shows that under natural assumptions they share expressivity, with genuine gaps emerging mainly when assumptions (e.g., spectrum bounds, universal-approximation guarantees) are relaxed, and highlights complementary strengths: MPNNs for discrete structure and-practical deployment, spectral GNNs for smoothing, bottlenecks, and community detection. The paper also clarifies the role of spectral positional encodings, demonstrates how polynomial spectral filters relate to MPNN depth, and outlines a roadmap for unifying theory, benchmarks, and hybrids to accelerate progress in graph learning. Overall, it advocates a unified, operator-centric framework to analyze and design GNNs rather than treating spectral and spatial approaches as competing paradigms, with practical implications for model selection, stability, and generalization. The proposed view aims to streamline theoretical insights and foster cross-pollination between communities, aiding the development of robust, scalable graph learning methods.

Abstract

Graph neural networks (GNNs) are commonly divided into message-passing neural networks (MPNNs) and spectral graph neural networks, reflecting two largely separate research traditions in machine learning and signal processing. This paper argues that this divide is mostly artificial, hindering progress in the field. We propose a viewpoint in which both MPNNs and spectral GNNs are understood as different parametrizations of permutation-equivariant operators acting on graph signals. From this perspective, many popular architectures are equivalent in expressive power, while genuine gaps arise only in specific regimes. We further argue that MPNNs and spectral GNNs offer complementary strengths. That is, MPNNs provide a natural language for discrete structure and expressivity analysis using tools from logic and graph isomorphism research, while the spectral perspective provides principled tools for understanding smoothing, bottlenecks, stability, and community structure. Overall, we posit that progress in graph learning will be accelerated by clearly understanding the key similarities and differences between these two types of GNNs, and by working towards unifying these perspectives within a common theoretical and conceptual framework rather than treating them as competing paradigms.

Position: Message-passing and spectral GNNs are two sides of the same coin

TL;DR

This work argues that message-passing neural networks (MPNNs) and spectral GNNs are not fundamentally distinct paradigms but different parametrizations of permutation-equivariant operators on graph signals. It shows that under natural assumptions they share expressivity, with genuine gaps emerging mainly when assumptions (e.g., spectrum bounds, universal-approximation guarantees) are relaxed, and highlights complementary strengths: MPNNs for discrete structure and-practical deployment, spectral GNNs for smoothing, bottlenecks, and community detection. The paper also clarifies the role of spectral positional encodings, demonstrates how polynomial spectral filters relate to MPNN depth, and outlines a roadmap for unifying theory, benchmarks, and hybrids to accelerate progress in graph learning. Overall, it advocates a unified, operator-centric framework to analyze and design GNNs rather than treating spectral and spatial approaches as competing paradigms, with practical implications for model selection, stability, and generalization. The proposed view aims to streamline theoretical insights and foster cross-pollination between communities, aiding the development of robust, scalable graph learning methods.

Abstract

Graph neural networks (GNNs) are commonly divided into message-passing neural networks (MPNNs) and spectral graph neural networks, reflecting two largely separate research traditions in machine learning and signal processing. This paper argues that this divide is mostly artificial, hindering progress in the field. We propose a viewpoint in which both MPNNs and spectral GNNs are understood as different parametrizations of permutation-equivariant operators acting on graph signals. From this perspective, many popular architectures are equivalent in expressive power, while genuine gaps arise only in specific regimes. We further argue that MPNNs and spectral GNNs offer complementary strengths. That is, MPNNs provide a natural language for discrete structure and expressivity analysis using tools from logic and graph isomorphism research, while the spectral perspective provides principled tools for understanding smoothing, bottlenecks, stability, and community structure. Overall, we posit that progress in graph learning will be accelerated by clearly understanding the key similarities and differences between these two types of GNNs, and by working towards unifying these perspectives within a common theoretical and conceptual framework rather than treating them as competing paradigms.
Paper Structure (51 sections, 14 theorems, 102 equations, 1 figure)

This paper contains 51 sections, 14 theorems, 102 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background
  3. Message-passing neural networks
  4. Spectral GNNs
  5. Position statements
  6. P1: Spectral GNNs and MPNNs are often equivalent but not the same
  7. Spectral GNNs and MPNNs Are Derived From Separate Symmetries
  8. MPNNs From neighborhood permutations
  9. Spectral GNNs From eigenbasis invariance
  10. Symmetry-preserving universal approximation properties of GNNs
  11. MPNNs and spectral GNNs have similar expressivity properties
  12. Polynomial GNNs are special MPNNs
  13. The role of the GSO in spectral GNNs
  14. MPNNs and spectral GNNs have equivalent expressivity in terms of separation power
  15. MPNNs and spectral GNNs have distinct but closely related universal approximation properties
  16. ...and 36 more sections

Key Result

Lemma 3

Let $\rho=$ReLU be applied element-wise and let $\bm{X}\in\mathbb{R}^{N\times d}$ be the initial node-feature matrix. Then there exists a degree--$0$ polynomial spectral GNN layer that maps $\bm{X}$ to a feature matrix This representation carries the initial node features in the following exact sense: for every matrix $\bm{W}\in\mathbb{R}^{d\times r}$, there exists $\widetilde{\bm{W}}\in\mathbb{R

Figures (1)

  • Figure 1: Overview of our five positions for unifying spectral GNNs and MPNNs in a principled way.

Theorems & Definitions (33)

  • Remark 1
  • Remark 2
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Remark 6
  • Theorem 7: Uniform approximation of MPNNs by spectral GNNs
  • ...and 23 more