Multi-Scale Harmonic Encoding for Feature-Wise Graph Message Passing

TL;DR

MSH-GNN presents a frequency-aware graph neural network that performs feature-wise, node-conditioned propagation using projections derived from the target node, followed by multi-scale harmonic modulation. This design yields a kernel-like view of message functions and, under mild injectivity assumptions, 1-WL-level expressive power. Empirically, it achieves strong performance across molecular, protein, and social graphs, with notable gains on joint structure–frequency discrimination tasks. The framework combines a scalable, differentiable message-passing pipeline with a frequency-aware readout, offering a principled alternative to purely attention- or spectrum-based approaches.

Abstract

Most Graph Neural Networks (GNNs) propagate messages by treating node embeddings as holistic feature vectors, implicitly assuming uniform relevance across feature dimensions. This limits their ability to selectively transmit informative components, especially when graph structures exhibit distinct frequency characteristics. We propose MSH-GNN (Multi-Scale Harmonic Graph Neural Network), a frequency-aware message passing framework that performs feature-wise adaptive propagation. Each node projects incoming messages onto node-conditioned feature subspaces derived from its own representation, enabling selective extraction of frequency-relevant components. Learnable multi-scale harmonic modulations further allow the model to capture both smooth and oscillatory structural patterns. A frequency-aware attention pooling mechanism is introduced for graph-level readout. We show that MSH-GNN admits an interpretation as a learnable Fourier-feature approximation of kernelized message functions and matches the expressive power of the 1-Weisfeiler-Lehman (1-WL) test. Extensive experiments on node- and graph-level benchmarks demonstrate consistent improvements over state-of-the-art methods, particularly in joint structure-frequency analysis tasks.

Figures (5)

• Figure 1: Comparison of neighbor feature aggregation behaviors. Although graphs $G_1$ and $G_2$ have similar average neighbor features, they differ in how these features are composed across feature dimensions. Attention-based aggregation assigns scalar weights to entire neighbor embeddings, which can obscure such differences in feature composition. By contrast, MSH-GNN applies node-conditioned feature-wise projections, enabling these compositional variations to be preserved during aggregation. See Appendix \ref{['appendix:gat-gap']} for further analysis.
• Figure 2: Overview of the MSH-GNN architecture. Incoming neighbor features are first projected into node-conditioned projection components and modulated across multiple harmonic scales to guide feature-wise message passing. The resulting node representations are subsequently pooled in a frequency-aware manner and used for downstream prediction.
• Figure 3: Ablation results on representative TU datasets. Panels (a)--(c) analyze the impact of harmonic encoding, multi-scale sinusoidal modulation, and pooling strategies, respectively, highlighting the effectiveness of each design choice in MSH-GNN.
• Figure 4: Graph-level embedding visualization via t-SNE. Each point represents one graph, colored by structural type (ring / chain / perturbed). MSH-GNN produces more separable clusters, indicating better structure–frequency representation.
• Figure 5: Two graphs $G_1$ and $G_2$ with identical neighbor feature mean but different feature-space composition. GAT cannot distinguish them due to scalar attention over holistic embeddings. MSH-GNN captures directional asymmetry via node-specific harmonic projections.

Theorems & Definitions (9)

• Definition 1: Feature-Wise Projection Message Passing
• Remark 1
• Theorem 1: Sufficient Conditions for 1-WL-Level Expressiveness
• Remark 2
• Remark 3
• Remark 4
• Theorem 2: Sufficient Conditions for 1-WL-Level Expressiveness
• proof
• Remark 5