# Learning Stable Graph Neural Networks via Spectral Regularization

arXiv: https://arxiv.org/abs/2211.06966 · HTML: https://mlsanity.com/paper/2211.06966/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 28 references that are in the map.
- First posted on arXiv 13 Nov 2022.

## Builds on (most cited first, 18 of 18)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [A Comprehensive Survey on Graph Neural Networks](https://mlsanity.com/paper/1901.00596/) (2019)
- [Graph Signal Processing: Overview, Challenges and Applications](https://mlsanity.com/paper/1712.00468/) (2017)
- [Stability Properties of Graph Neural Networks](https://mlsanity.com/paper/1905.04497/) (2019)
- [Transferability of Spectral Graph Convolutional Neural Networks](https://mlsanity.com/paper/1907.12972/) (2019)
- [Graphs, Convolutions, and Neural Networks: From Graph Filters to Graph Neural Networks](https://mlsanity.com/paper/2003.03777/) (2020)
- [Learning Decentralized Controllers for Robot Swarms with Graph Neural Networks](https://mlsanity.com/paper/1903.10527/) (2019)
- [EdgeNets:Edge Varying Graph Neural Networks](https://mlsanity.com/paper/2001.07620/) (2020)
- [On the Stability of Graph Convolutional Neural Networks under Edge Rewiring](https://mlsanity.com/paper/2010.13747/) (2020)
- [Training Robust Graph Neural Networks with Topology Adaptive Edge Dropping](https://mlsanity.com/paper/2106.02892/) (2021)
- [Stability of Graph Convolutional Neural Networks to Stochastic Perturbations](https://mlsanity.com/paper/2106.10526/) (2021)
- [Wide and Deep Graph Neural Network with Distributed Online Learning](https://mlsanity.com/paper/2107.09203/) (2021)
- [Training Stable Graph Neural Networks Through Constrained Learning](https://mlsanity.com/paper/2110.03576/) (2021)
- [Stability of Graph Neural Networks to Relative Perturbations](https://mlsanity.com/paper/1910.09655/) (2019)
- [Resource Allocation via Graph Neural Networks in Free Space Optical Fronthaul Networks](https://mlsanity.com/paper/2006.15005/) (2020)
- [Robust Graph Neural Networks via Probabilistic Lipschitz Constraints](https://mlsanity.com/paper/2112.07575/) (2021)
- [Wide and Deep Graph Neural Networks with Distributed Online Learning](https://mlsanity.com/paper/2006.06376/) (2020)
- [Decentralized Channel Management in WLANs with Graph Neural Networks](https://mlsanity.com/paper/2210.16949/) (2022)

## Built on by (most cited first, 4 of 4)

- [Stability of Graph Convolutional Neural Networks through the lens of small perturbation analysis](https://mlsanity.com/paper/2312.12934/) (2023)
- [On the Trade-Off between Stability and Representational Capacity in Graph Neural Networks](https://mlsanity.com/paper/2312.02372/) (2023)
- [Learning Lagrangian Interaction Dynamics with Sampling-Based Model Order Reduction](https://mlsanity.com/paper/2407.03925/) (2024)
- [Understanding Learning Dynamics Through Structured Representations](https://mlsanity.com/paper/2508.02126/) (2025)

## Abstract (arXiv)

Stability of graph neural networks (GNNs) characterizes how GNNs react to graph perturbations and provides guarantees for architecture performance in noisy scenarios. This paper develops a self-regularized graph neural network (SR-GNN) solution that improves the architecture stability by regularizing the filter frequency responses in the graph spectral domain. The SR-GNN considers not only the graph signal as input but also the eigenvectors of the underlying graph, where the signal is processed to generate task-relevant features and the eigenvectors to characterize the frequency responses at each layer. We train the SR-GNN by minimizing the cost function and regularizing the maximal frequency response close to one. The former improves the architecture performance, while the latter tightens the perturbation stability and alleviates the information loss through multi-layer propagation. We further show the SR-GNN preserves the permutation equivariance, which allows to explore the internal symmetries of graph signals and to exhibit transference on similar graph structures. Numerical results with source localization and movie recommendation corroborate our findings and show the SR-GNN yields a comparable performance with the vanilla GNN on the unperturbed graph but improves substantially the stability.