# Graphs, Convolutions, and Neural Networks: From Graph Filters to Graph Neural Networks

arXiv: https://arxiv.org/abs/2003.03777 · HTML: https://mlsanity.com/paper/2003.03777/

## Key facts

- Cited by 75 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 Mar 2020.

## Built on by (most cited first, 30 of 75)

- [Graph Filters for Signal Processing and Machine Learning on Graphs](https://mlsanity.com/paper/2211.08854/) (2022)
- [Transferability Properties of Graph Neural Networks](https://mlsanity.com/paper/2112.04629/) (2021)
- [Simplicial Convolutional Neural Networks](https://mlsanity.com/paper/2110.02585/) (2021)
- [Topology-aware Graph Neural Networks for Learning Feasible and Adaptive ac-OPF Solutions](https://mlsanity.com/paper/2205.10129/) (2022)
- [Convolutional Learning on Simplicial Complexes](https://mlsanity.com/paper/2301.11163/) (2023)
- [Tangent Bundle Convolutional Learning: from Manifolds to Cellular Sheaves and Back](https://mlsanity.com/paper/2303.11323/) (2023)
- [Graph Neural Networks Designed for Different Graph Types: A Survey](https://mlsanity.com/paper/2204.03080/) (2022)
- [PowerFlowNet: Power Flow Approximation Using Message Passing Graph Neural Networks](https://mlsanity.com/paper/2311.03415/) (2023)
- [coVariance Neural Networks](https://mlsanity.com/paper/2205.15856/) (2022)
- [Explainable Brain Age Prediction using coVariance Neural Networks](https://mlsanity.com/paper/2305.18370/) (2023)
- [Stability of Graph Convolutional Neural Networks to Stochastic Perturbations](https://mlsanity.com/paper/2106.10526/) (2021)
- [Dimensionless machine learning: Imposing exact units equivariance](https://mlsanity.com/paper/2204.00887/) (2022)
- [Finite Impulse Response Filters for Simplicial Complexes](https://mlsanity.com/paper/2103.12587/) (2021)
- [Wide and Deep Graph Neural Network with Distributed Online Learning](https://mlsanity.com/paper/2107.09203/) (2021)
- [Convolutional Neural Networks on Manifolds: From Graphs and Back](https://mlsanity.com/paper/2210.00376/) (2022)
- [Unsupervised Optimal Power Flow Using Graph Neural Networks](https://mlsanity.com/paper/2210.09277/) (2022)
- [Synthesizing Decentralized Controllers with Graph Neural Networks and Imitation Learning](https://mlsanity.com/paper/2012.14906/) (2020)
- [Graph-Time Convolutional Neural Networks: Architecture and Theoretical Analysis](https://mlsanity.com/paper/2206.15174/) (2022)
- [Deep Statistical Solver for Distribution System State Estimation](https://mlsanity.com/paper/2301.01835/) (2023)
- [Stability to Deformations of Manifold Filters and Manifold Neural Networks](https://mlsanity.com/paper/2106.03725/) (2021)
- [Geometric Graph Filters and Neural Networks: Limit Properties and Discriminability Trade-offs](https://mlsanity.com/paper/2305.18467/) (2023)
- [Graph-Time Convolutional Neural Networks](https://mlsanity.com/paper/2103.01730/) (2021)
- [Convolutional Filtering in Simplicial Complexes](https://mlsanity.com/paper/2201.12584/) (2022)
- [Unsupervised Learning of Sampling Distributions for Particle Filters](https://mlsanity.com/paper/2302.01174/) (2023)
- [Fairness-Aware Graph Filter Design](https://mlsanity.com/paper/2303.11459/) (2023)
- [Redesigning graph filter-based GNNs to relax the homophily assumption](https://mlsanity.com/paper/2409.08676/) (2024)
- [Node-Variant Graph Filters in Graph Neural Networks](https://mlsanity.com/paper/2106.00089/) (2021)
- [Scalable Perception-Action-Communication Loops with Convolutional and Graph Neural Networks](https://mlsanity.com/paper/2106.13358/) (2021)
- [Stability of Aggregation Graph Neural Networks](https://mlsanity.com/paper/2207.03678/) (2022)
- [A Bipartite Graph Neural Network Approach for Scalable Beamforming Optimization](https://mlsanity.com/paper/2207.05364/) (2022)

## Abstract (arXiv)

Network data can be conveniently modeled as a graph signal, where data values are assigned to nodes of a graph that describes the underlying network topology. Successful learning from network data is built upon methods that effectively exploit this graph structure. In this work, we leverage graph signal processing to characterize the representation space of graph neural networks (GNNs). We discuss the role of graph convolutional filters in GNNs and show that any architecture built with such filters has the fundamental properties of permutation equivariance and stability to changes in the topology. These two properties offer insight about the workings of GNNs and help explain their scalability and transferability properties which, coupled with their local and distributed nature, make GNNs powerful tools for learning in physical networks. We also introduce GNN extensions using edge-varying and autoregressive moving average graph filters and discuss their properties. Finally, we study the use of GNNs in recommender systems and learning decentralized controllers for robot swarms.