  Graph Neural Networks (GNNs) have shown remarkable success in learning from
graph-structured data. However, their application to directed graphs (digraphs)
presents unique challenges, primarily due to the inherent asymmetry in node
relationships. Traditional GNNs are adept at capturing unidirectional relations
but fall short in encoding the mutual path dependencies between nodes, such as
asymmetrical shortest paths typically found in digraphs. Recognizing this gap,
we introduce Commute Graph Neural Networks (CGNN), an approach that seamlessly
integrates node-wise commute time into the message passing scheme. The
cornerstone of CGNN is an efficient method for computing commute time using a
newly formulated digraph Laplacian. Commute time information is then integrated
into the neighborhood aggregation process, with neighbor contributions weighted
according to their respective commute time to the central node in each layer.
It enables CGNN to directly capture the mutual, asymmetric relationships in
digraphs.
