Abstract: Graph neural networks (GNNs) and message passing neural networks (MPNNs) have been proven to be expressive for subgraph structures in many applications.

Some applications in heterogeneous graphs require explicit edge modeling, such as subgraph isomorphism counting and matching.

However, existing message passing mechanisms are not designed well in theory.

In this paper, we start from a particular edge-to-vertex transform and exploit the isomorphism property in the edge-to-vertex dual graphs.

We prove that searching isomorphisms on the original graph is equivalent to searching on its dual graph.

Based on this observation, we propose dual message passing neural networks (DMPNNs) to enhance the substructure representation learning in an asynchronous way for subgraph isomorphism counting and matching as well as unsupervised node classification.

Extensive experiments demonstrate the robust performance of DMPNNs by combining both node and edge representation learning in synthetic and real heterogeneous graphs.