  Various classes of Graph Neural Networks (GNN) have been proposed and shown
to be successful in a wide range of applications with graph structured data. In
this paper, we propose a theoretical framework able to compare the expressive
power of these GNN architectures. The current universality theorems only apply
to intractable classes of GNNs. Here, we prove the first approximation
guarantees for practical GNNs, paving the way for a better understanding of
their generalization. Our theoretical results are proved for invariant GNNs
computing a graph embedding (permutation of the nodes of the input graph does
not affect the output) and equivariant GNNs computing an embedding of the nodes
(permutation of the input permutes the output). We show that Folklore Graph
Neural Networks (FGNN), which are tensor based GNNs augmented with matrix
multiplication are the most expressive architectures proposed so far for a
given tensor order. We illustrate our results on the Quadratic Assignment
Problem (a NP-Hard combinatorial problem) by showing that FGNNs are able to
learn how to solve the problem, leading to much better average performances
than existing algorithms (based on spectral, SDP or other GNNs architectures).
On a practical side, we also implement masked tensors to handle batches of
graphs of varying sizes.
