  From the perspective of expressive power, this work compares multi-layer
Graph Neural Networks (GNNs) with a simplified alternative that we call
Graph-Augmented Multi-Layer Perceptrons (GA-MLPs), which first augments node
features with certain multi-hop operators on the graph and then applies an MLP
in a node-wise fashion. From the perspective of graph isomorphism testing, we
show both theoretically and numerically that GA-MLPs with suitable operators
can distinguish almost all non-isomorphic graphs, just like the
Weifeiler-Lehman (WL) test. However, by viewing them as node-level functions
and examining the equivalence classes they induce on rooted graphs, we prove a
separation in expressive power between GA-MLPs and GNNs that grows
exponentially in depth. In particular, unlike GNNs, GA-MLPs are unable to count
the number of attributed walks. We also demonstrate via community detection
experiments that GA-MLPs can be limited by their choice of operator family, as
compared to GNNs with higher flexibility in learning.
