# Stability of Aggregation Graph Neural Networks

arXiv: https://arxiv.org/abs/2207.03678 · HTML: https://mlsanity.com/paper/2207.03678/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 38 references that are in the map.
- First posted on arXiv 8 Jul 2022.

## Builds on (most cited first, 14 of 14)

- [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)
- [Convolutional Neural Network Architectures for Signals Supported on Graphs](https://mlsanity.com/paper/1805.00165/) (2018)
- [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)
- [A Mathematical Theory of Deep Convolutional Neural Networks for Feature Extraction](https://mlsanity.com/paper/1512.06293/) (2015)
- [Graph Convolutional Neural Networks via Scattering](https://mlsanity.com/paper/1804.00099/) (2018)
- [On the Transferability of Spectral Graph Filters](https://mlsanity.com/paper/1901.10524/) (2019)
- [On the Stability of Graph Convolutional Neural Networks under Edge Rewiring](https://mlsanity.com/paper/2010.13747/) (2020)
- [Learning Decentralized Wireless Resource Allocations with Graph Neural Networks](https://mlsanity.com/paper/2107.01489/) (2021)
- [Algebraic Neural Networks: Stability to Deformations](https://mlsanity.com/paper/2009.01433/) (2020)
- [Sampling of graph signals with successive local aggregations](https://mlsanity.com/paper/1504.04687/) (2015)
- [Graphon Pooling in Graph Neural Networks](https://mlsanity.com/paper/2003.01795/) (2020)
- [Stability of Algebraic Neural Networks to Small Perturbations](https://mlsanity.com/paper/2010.11544/) (2020)

## Built on by (most cited first, 7 of 7)

- [Convolutional Learning on Simplicial Complexes](https://mlsanity.com/paper/2301.11163/) (2023)
- [Sampling and Uniqueness Sets in Graphon Signal Processing](https://mlsanity.com/paper/2401.06279/) (2024)
- [Higher-Order GNNs Meet Efficiency: Sparse Sobolev Graph Neural Networks](https://mlsanity.com/paper/2411.04570/) (2024)
- [Non Commutative Convolutional Signal Models in Neural Networks: Stability to Small Deformations](https://mlsanity.com/paper/2310.03879/) (2023)
- [Alleviating Over-Smoothing via Aggregation over Compact Manifolds](https://mlsanity.com/paper/2407.19231/) (2024)
- [On the Stability of Graph Convolutional Neural Networks: A Probabilistic Perspective](https://mlsanity.com/paper/2506.01213/) (2025)
- [Denoised Recommendation Model with Collaborative Signal Decoupling](https://mlsanity.com/paper/2511.04237/) (2025)

## Abstract (arXiv)

In this paper we study the stability properties of aggregation graph neural networks (Agg-GNNs) considering perturbations of the underlying graph. An Agg-GNN is a hybrid architecture where information is defined on the nodes of a graph, but it is processed block-wise by Euclidean CNNs on the nodes after several diffusions on the graph shift operator. We derive stability bounds for the mapping operator associated to a generic Agg-GNN, and we specify conditions under which such operators can be stable to deformations. We prove that the stability bounds are defined by the properties of the filters in the first layer of the CNN that acts on each node. Additionally, we show that there is a close relationship between the number of aggregations, the filter's selectivity, and the size of the stability constants. We also conclude that in Agg-GNNs the selectivity of the mapping operators is tied to the properties of the filters only in the first layer of the CNN stage. This shows a substantial difference with respect to the stability properties of selection GNNs, where the selectivity of the filters in all layers is constrained by their stability. We provide numerical evidence corroborating the results derived, testing the behavior of Agg-GNNs in real life application scenarios considering perturbations of different magnitude.