  Message Passing Neural Networks (MPNNs) are a widely used class of Graph
Neural Networks (GNNs). The limited representational power of MPNNs inspires
the study of provably powerful GNN architectures. However, knowing one model is
more powerful than another gives little insight about what functions they can
or cannot express. It is still unclear whether these models are able to
approximate specific functions such as counting certain graph substructures,
which is essential for applications in biology, chemistry and social network
analysis. Motivated by this, we propose to study the counting power of Subgraph
MPNNs, a recent and popular class of powerful GNN models that extract rooted
subgraphs for each node, assign the root node a unique identifier and encode
the root node's representation within its rooted subgraph. Specifically, we
prove that Subgraph MPNNs fail to count more-than-4-cycles at node level,
implying that node representations cannot correctly encode the surrounding
substructures like ring systems with more than four atoms. To overcome this
limitation, we propose I$^2$-GNNs to extend Subgraph MPNNs by assigning
different identifiers for the root node and its neighbors in each subgraph.
I$^2$-GNNs' discriminative power is shown to be strictly stronger than Subgraph
MPNNs and partially stronger than the 3-WL test. More importantly, I$^2$-GNNs
are proven capable of counting all 3, 4, 5 and 6-cycles, covering common
substructures like benzene rings in organic chemistry, while still keeping
linear complexity. To the best of our knowledge, it is the first linear-time
GNN model that can count 6-cycles with theoretical guarantees. We validate its
counting power in cycle counting tasks and demonstrate its competitive
performance in molecular prediction benchmarks.
