  Invariant and equivariant networks have been successfully used for learning
images, sets, point clouds, and graphs. A basic challenge in developing such
networks is finding the maximal collection of invariant and equivariant linear
layers. Although this question is answered for the first three examples (for
popular transformations, at-least), a full characterization of invariant and
equivariant linear layers for graphs is not known.
  In this paper we provide a characterization of all permutation invariant and
equivariant linear layers for (hyper-)graph data, and show that their
dimension, in case of edge-value graph data, is 2 and 15, respectively. More
generally, for graph data defined on k-tuples of nodes, the dimension is the
k-th and 2k-th Bell numbers. Orthogonal bases for the layers are computed,
including generalization to multi-graph data. The constant number of basis
elements and their characteristics allow successfully applying the networks to
different size graphs. From the theoretical point of view, our results
generalize and unify recent advancement in equivariant deep learning. In
particular, we show that our model is capable of approximating any message
passing neural network
  Applying these new linear layers in a simple deep neural network framework is
shown to achieve comparable results to state-of-the-art and to have better
expressivity than previous invariant and equivariant bases.
