Graph Automorphism Group Equivariant Neural Networks

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 Sn 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 ℝn. In particular, we find a spanning set of matrices for these layer functions in the standard basis of ℝn. 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.