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

TL;DR

This paper unifies message-passing and spectral GNNs under a common framework, showing they are different parametrizations of permutation-equivariant operators, which can accelerate progress in graph learning.

Contribution

It demonstrates that MPNNs and spectral GNNs are largely equivalent in expressive power and advocates for a unified perspective to advance the field.

Findings

Many popular GNN architectures are equivalent in expressive power.

Genuine differences between MPNNs and spectral GNNs occur only in specific regimes.

Unified understanding can accelerate progress in graph learning.

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.