  We study and compare different Graph Neural Network extensions that increase
the expressive power of GNNs beyond the Weisfeiler-Leman test. We focus on (i)
GNNs based on higher order WL methods, (ii) GNNs that preprocess small
substructures in the graph, (iii) GNNs that preprocess the graph up to a small
radius, and (iv) GNNs that slightly perturb the graph to compute an embedding.
We begin by presenting a simple improvement for this last extension that
strictly increases the expressive power of this GNN variant. Then, as our main
result, we compare the expressiveness of these extensions to each other through
a series of example constructions that can be distinguished by one of the
extensions, but not by another one. We also show negative examples that are
particularly challenging for each of the extensions, and we prove several
claims about the ability of these extensions to count cliques and cycles in the
graph.
