  Recently, the Weisfeiler-Lehman (WL) graph isomorphism test was used to
measure the expressive power of graph neural networks (GNN). It was shown that
the popular message passing GNN cannot distinguish between graphs that are
indistinguishable by the 1-WL test (Morris et al. 2018; Xu et al. 2019).
Unfortunately, many simple instances of graphs are indistinguishable by the
1-WL test.
  In search for more expressive graph learning models we build upon the recent
k-order invariant and equivariant graph neural networks (Maron et al. 2019a,b)
and present two results:
  First, we show that such k-order networks can distinguish between
non-isomorphic graphs as good as the k-WL tests, which are provably stronger
than the 1-WL test for k>2. This makes these models strictly stronger than
message passing models. Unfortunately, the higher expressiveness of these
models comes with a computational cost of processing high order tensors.
  Second, setting our goal at building a provably stronger, simple and scalable
model we show that a reduced 2-order network containing just scaled identity
operator, augmented with a single quadratic operation (matrix multiplication)
has a provable 3-WL expressive power. Differently put, we suggest a simple
model that interleaves applications of standard Multilayer-Perceptron (MLP)
applied to the feature dimension and matrix multiplication. We validate this
model by presenting state of the art results on popular graph classification
and regression tasks. To the best of our knowledge, this is the first practical
invariant/equivariant model with guaranteed 3-WL expressiveness, strictly
stronger than message passing models.
