# Transferability of Spectral Graph Convolutional Neural Networks

arXiv: https://arxiv.org/abs/1907.12972 · HTML: https://mlsanity.com/paper/1907.12972/

## Key facts

- Cited by 92 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 30 Jul 2019.

## Built on by (most cited first, 30 of 92)

- [MagNet: A Neural Network for Directed Graphs](https://mlsanity.com/paper/2102.11391/) (2021)
- [Not too little, not too much: a theoretical analysis of graph (over)smoothing](https://mlsanity.com/paper/2205.12156/) (2022)
- [Equivariant and Stable Positional Encoding for More Powerful Graph Neural Networks](https://mlsanity.com/paper/2203.00199/) (2022)
- [GFT: Graph Foundation Model with Transferable Tree Vocabulary](https://mlsanity.com/paper/2411.06070/) (2024)
- [Theory of Graph Neural Networks: Representation and Learning](https://mlsanity.com/paper/2204.07697/) (2022)
- [Generalization Analysis of Message Passing Neural Networks on Large Random Graphs](https://mlsanity.com/paper/2202.00645/) (2022)
- [Scalable and Adaptive Graph Neural Networks with Self-Label-Enhanced training](https://mlsanity.com/paper/2104.09376/) (2021)
- [Transferability of Graph Neural Networks: an Extended Graphon Approach](https://mlsanity.com/paper/2109.10096/) (2021)
- [Transferability Properties of Graph Neural Networks](https://mlsanity.com/paper/2112.04629/) (2021)
- [OOD Link Prediction Generalization Capabilities of Message-Passing GNNs in Larger Test Graphs](https://mlsanity.com/paper/2205.15117/) (2022)
- [Towards Graph Foundation Models: Learning Generalities Across Graphs via Task-Trees](https://mlsanity.com/paper/2412.16441/) (2024)
- [Generalised Implicit Neural Representations](https://mlsanity.com/paper/2205.15674/) (2022)
- [A graphon-signal analysis of graph neural networks](https://mlsanity.com/paper/2305.15987/) (2023)
- [MSGNN: A Spectral Graph Neural Network Based on a Novel Magnetic Signed Laplacian](https://mlsanity.com/paper/2209.00546/) (2022)
- [On the Universality of Graph Neural Networks on Large Random Graphs](https://mlsanity.com/paper/2105.13099/) (2021)
- [A Survey on Spectral Graph Neural Networks](https://mlsanity.com/paper/2302.05631/) (2023)
- [Relational Deep Learning: Challenges, Foundations and Next-Generation Architectures](https://mlsanity.com/paper/2506.16654/) (2025)
- [Learning by Transference: Training Graph Neural Networks on Growing Graphs](https://mlsanity.com/paper/2106.03693/) (2021)
- [Tangent Bundle Convolutional Learning: from Manifolds to Cellular Sheaves and Back](https://mlsanity.com/paper/2303.11323/) (2023)
- [HoloNets: Spectral Convolutions do extend to Directed Graphs](https://mlsanity.com/paper/2310.02232/) (2023)
- [Bridging the Gap between Spatial and Spectral Domains: A Unified Framework for Graph Neural Networks](https://mlsanity.com/paper/2107.10234/) (2021)
- [Understanding Non-linearity in Graph Neural Networks from the Bayesian-Inference Perspective](https://mlsanity.com/paper/2207.11311/) (2022)
- [Stability of Graph Convolutional Neural Networks to Stochastic Perturbations](https://mlsanity.com/paper/2106.10526/) (2021)
- [Overcoming Oversmoothness in Graph Convolutional Networks via Hybrid Scattering Networks](https://mlsanity.com/paper/2201.08932/) (2022)
- [MentorGNN: Deriving Curriculum for Pre-Training GNNs](https://mlsanity.com/paper/2208.09905/) (2022)
- [When to Pre-Train Graph Neural Networks? From Data Generation Perspective!](https://mlsanity.com/paper/2303.16458/) (2023)
- [Limits, approximation and size transferability for GNNs on sparse graphs via graphops](https://mlsanity.com/paper/2306.04495/) (2023)
- [Subgraph Pooling: Tackling Negative Transfer on Graphs](https://mlsanity.com/paper/2402.08907/) (2024)
- [Generalization Bounds for Message Passing Networks on Mixture of Graphons](https://mlsanity.com/paper/2404.03473/) (2024)
- [A Manifold Perspective on the Statistical Generalization of Graph Neural Networks](https://mlsanity.com/paper/2406.05225/) (2024)

## Abstract (arXiv)

This paper focuses on spectral graph convolutional neural networks (ConvNets), where filters are defined as elementwise multiplication in the frequency domain of a graph. In machine learning settings where the dataset consists of signals defined on many different graphs, the trained ConvNet should generalize to signals on graphs unseen in the training set. It is thus important to transfer ConvNets between graphs. Transferability, which is a certain type of generalization capability, can be loosely defined as follows: if two graphs describe the same phenomenon, then a single filter or ConvNet should have similar repercussions on both graphs. This paper aims at debunking the common misconception that spectral filters are not transferable. We show that if two graphs discretize the same "continuous" space, then a spectral filter or ConvNet has approximately the same repercussion on both graphs. Our analysis is more permissive than the standard analysis. Transferability is typically described as the robustness of the filter to small graph perturbations and re-indexing of the vertices. Our analysis accounts also for large graph perturbations. We prove transferability between graphs that can have completely different dimensions and topologies, only requiring that both graphs discretize the same underlying space in some generic sense.