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

arXiv: https://arxiv.org/abs/2505.15015 · HTML: https://mlsanity.com/paper/2505.15015/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 38 of its 49 references that are in the map.
- First posted on arXiv 21 May 2025.

## Builds on (most cited first, 30 of 38)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Graph Attention Networks](https://mlsanity.com/paper/1710.10903/) (2017)
- [Inductive Representation Learning on Large Graphs](https://mlsanity.com/paper/1706.02216/) (2017)
- [How Powerful are Graph Neural Networks?](https://mlsanity.com/paper/1810.00826/) (2018)
- [Neural Message Passing for Quantum Chemistry](https://mlsanity.com/paper/1704.01212/) (2017)
- [A Comprehensive Survey on Graph Neural Networks](https://mlsanity.com/paper/1901.00596/) (2019)
- [Predict then Propagate: Graph Neural Networks meet Personalized PageRank](https://mlsanity.com/paper/1810.05997/) (2018)
- [How Attentive are Graph Attention Networks?](https://mlsanity.com/paper/2105.14491/) (2021)
- [Recipe for a General, Powerful, Scalable Graph Transformer](https://mlsanity.com/paper/2205.12454/) (2022)
- [DropEdge: Towards Deep Graph Convolutional Networks on Node Classification](https://mlsanity.com/paper/1907.10903/) (2019)
- [Adaptive Universal Generalized PageRank Graph Neural Network](https://mlsanity.com/paper/2006.07988/) (2020)
- [Masked Label Prediction: Unified Message Passing Model for Semi-Supervised Classification](https://mlsanity.com/paper/2009.03509/) (2020)
- [Rethinking Graph Transformers with Spectral Attention](https://mlsanity.com/paper/2106.03893/) (2021)
- [Provably Powerful Graph Networks](https://mlsanity.com/paper/1905.11136/) (2019)
- [Invariant and Equivariant Graph Networks](https://mlsanity.com/paper/1812.09902/) (2018)
- [Improving Graph Neural Network Expressivity via Subgraph Isomorphism Counting](https://mlsanity.com/paper/2006.09252/) (2020)
- [Diffusion-Convolutional Neural Networks](https://mlsanity.com/paper/1511.02136/) (2015)
- [Weisfeiler and Lehman Go Topological: Message Passing Simplicial Networks](https://mlsanity.com/paper/2103.03212/) (2021)
- [Weisfeiler and Lehman Go Cellular: CW Networks](https://mlsanity.com/paper/2106.12575/) (2021)
- [Representing Long-Range Context for Graph Neural Networks with Global Attention](https://mlsanity.com/paper/2201.08821/) (2022)
- [Pure Transformers are Powerful Graph Learners](https://mlsanity.com/paper/2207.02505/) (2022)
- [Exphormer: Sparse Transformers for Graphs](https://mlsanity.com/paper/2303.06147/) (2023)
- [GaAN: Gated Attention Networks for Learning on Large and Spatiotemporal Graphs](https://mlsanity.com/paper/1803.07294/) (2018)
- [Graph Neural Tangent Kernel: Fusing Graph Neural Networks with Graph Kernels](https://mlsanity.com/paper/1905.13192/) (2019)
- [A Survey on Graph Kernels](https://mlsanity.com/paper/1903.11835/) (2019)
- [NAGphormer: A Tokenized Graph Transformer for Node Classification in Large Graphs](https://mlsanity.com/paper/2206.04910/) (2022)
- [Directional Graph Networks](https://mlsanity.com/paper/2010.02863/) (2020)
- [Accurate Learning of Graph Representations with Graph Multiset Pooling](https://mlsanity.com/paper/2102.11533/) (2021)
- [Specformer: Spectral Graph Neural Networks Meet Transformers](https://mlsanity.com/paper/2303.01028/) (2023)

## Built on by (most cited first, 1 of 1)

- [Cheeger--Hodge Contrastive Learning for Structurally Robust Graph Representation Learning](https://mlsanity.com/paper/2604.26301/) (2026)

## Abstract (arXiv)

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.