  Despite the enormous success of graph neural networks (GNNs), most existing
GNNs can only be applicable to undirected graphs where relationships among
connected nodes are two-way symmetric (i.e., information can be passed back and
forth). However, there is a vast amount of applications where the information
flow is asymmetric, leading to directed graphs where information can only be
passed in one direction. For example, a directed edge indicates that the
information can only be conveyed forwardly from the start node to the end node,
but not backwardly. To accommodate such an asymmetric structure of directed
graphs within the framework of GNNs, we propose a simple yet remarkably
effective framework for directed graph analysis to incorporate such one-way
information passing. We define an incoming embedding and an outgoing embedding
for each node to model its sending and receiving features respectively. We
further develop two steps in our directed GNN model with the first one to
aggregate/update the incoming features of nodes and the second one to
aggregate/update the outgoing features. By imposing the two roles for each
node, the likelihood of a directed edge can be calculated based on the outgoing
embedding of the start node and the incoming embedding of the end node. The
log-likelihood of all edges plays a natural role of regularization for the
proposed model, which can alleviate the over-smoothing problem of the deep
GNNs. Extensive experiments on multiple real-world directed graphs demonstrate
outstanding performances of the proposed model in both node-level and
graph-level tasks.
