Graph Automorphism Group Equivariant Neural Networks

TL;DR

This paper develops neural networks that are equivariant to the automorphism group of a graph, capturing the true symmetries of the data rather than the symmetric group, enabling more accurate learning from graph-structured data.

Contribution

It provides a full characterization of Aut(G)-equivariant linear functions between tensor power layers, including a spanning set of matrices in the standard basis.

Findings

Characterized Aut(G)-equivariant functions for graph neural networks.

Derived a spanning set of matrices for these functions.

Showed that any finite group can be realized as a graph automorphism group.

Abstract

Permutation equivariant neural networks are typically used to learn from data that lives on a graph. However, for any graph $G$ that has $n$ vertices, using the symmetric group $S_n$ as its group of symmetries does not take into account the relations that exist between the vertices. Given that the actual group of symmetries is the automorphism group Aut$(G)$, we show how to construct neural networks that are equivariant to Aut$(G)$ by obtaining a full characterisation of the learnable, linear, Aut$(G)$-equivariant functions between layers that are some tensor power of $\mathbb{R}^{n}$. In particular, we find a spanning set of matrices for these layer functions in the standard basis of $\mathbb{R}^{n}$. This result has important consequences for learning from data whose group of symmetries is a finite group because a theorem by Frucht (1938) showed that any finite group is isomorphic to the automorphism group of a graph.