# Generalized Permutants and Graph GENEOs

arXiv: https://arxiv.org/abs/2206.14798 · HTML: https://mlsanity.com/paper/2206.14798/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 46 references that are in the map.
- First posted on arXiv 29 Jun 2022.

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

- [Open Graph Benchmark: Datasets for Machine Learning on Graphs](https://mlsanity.com/paper/2005.00687/) (2020)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [Geometric deep learning: going beyond Euclidean data](https://mlsanity.com/paper/1611.08097/) (2016)
- [Deep Learning on Graphs: A Survey](https://mlsanity.com/paper/1812.04202/) (2018)
- [On the Generalization of Equivariance and Convolution in Neural Networks to the Action of Compact Groups](https://mlsanity.com/paper/1802.03690/) (2018)
- [Harmonic Networks: Deep Translation and Rotation Equivariance](https://mlsanity.com/paper/1612.04642/) (2016)
- [Explainable Machine Learning for Scientific Insights and Discoveries](https://mlsanity.com/paper/1905.08883/) (2019)
- [Scalars are universal: Equivariant machine learning, structured like classical physics](https://mlsanity.com/paper/2106.06610/) (2021)
- [Understanding Deep Convolutional Networks](https://mlsanity.com/paper/1601.04920/) (2016)
- [Geometric Deep Learning and Equivariant Neural Networks](https://mlsanity.com/paper/2105.13926/) (2021)
- [Universal Approximation Theorems for Differentiable Geometric Deep Learning](https://mlsanity.com/paper/2101.05390/) (2021)
- [Towards a topological-geometrical theory of group equivariant non-expansive operators for data analysis and machine learning](https://mlsanity.com/paper/1812.11832/) (2018)
- [On Invariance and Selectivity in Representation Learning](https://mlsanity.com/paper/1503.05938/) (2015)
- [On the finite representation of group equivariant operators via permutant measures](https://mlsanity.com/paper/2008.06340/) (2020)
- [Graphs in machine learning: an introduction](https://mlsanity.com/paper/1506.06962/) (2015)
- [Steady and ranging sets in graph persistence](https://mlsanity.com/paper/2009.06897/) (2020)
- [Machines of finite depth: towards a formalization of neural networks](https://mlsanity.com/paper/2204.12786/) (2022)

## Built on by (most cited first, 5 of 5)

- [Mathematical Foundation of Interpretable Equivariant Surrogate Models](https://mlsanity.com/paper/2503.01942/) (2025)
- [A novel approach to graph distinction through GENEOs and permutants](https://mlsanity.com/paper/2406.08045/) (2024)
- [GENEOnet: Statistical analysis supporting explainability and trustworthiness](https://mlsanity.com/paper/2503.09199/) (2025)
- [Reconstruction of SINR Maps from Sparse Measurements using Group Equivariant Non-Expansive Operators](https://mlsanity.com/paper/2507.19349/) (2025)
- [An Algebraic Representation Theorem for Linear GENEOs in Geometric Machine Learning](https://mlsanity.com/paper/2601.03910/) (2026)

## Abstract (arXiv)

In this paper we establish a bridge between Topological Data Analysis and Geometric Deep Learning, adapting the topological theory of group equivariant non-expansive operators (GENEOs) to act on the space of all graphs weighted on vertices or edges. This is done by showing how the general concept of GENEO can be used to transform graphs and to give information about their structure. This requires the introduction of the new concepts of generalized permutant and generalized permutant measure and the mathematical proof that these concepts allow us to build GENEOs between graphs. An experimental section concludes the paper, illustrating the possible use of our operators to extract information from graphs. This paper is part of a line of research devoted to developing a compositional and geometric theory of GENEOs for Geometric Deep Learning.