  In this work we develop a new method, named Sub-graph Permutation Equivariant
Networks (SPEN), which provides a framework for building graph neural networks
that operate on sub-graphs, while using a base update function that is
permutation equivariant, that are equivariant to a novel choice of automorphism
group. Message passing neural networks have been shown to be limited in their
expressive power and recent approaches to over come this either lack
scalability or require structural information to be encoded into the feature
space. The general framework presented here overcomes the scalability issues
associated with global permutation equivariance by operating more locally on
sub-graphs. In addition, through operating on sub-graphs the expressive power
of higher-dimensional global permutation equivariant networks is improved; this
is due to fact that two non-distinguishable graphs often contain
distinguishable sub-graphs. Furthermore, the proposed framework only requires a
choice of $k$-hops for creating ego-network sub-graphs and a choice of
representation space to be used for each layer, which makes the method easily
applicable across a range of graph based domains. We experimentally validate
the method on a range of graph benchmark classification tasks, demonstrating
statistically indistinguishable results from the state-of-the-art on six out of
seven benchmarks. Further, we demonstrate that the use of local update
functions offers a significant improvement in GPU memory over global methods.
