  Graph Neural Networks (GNNs) have recently been applied to graph learning
tasks and achieved state-of-the-art (SOTA) results. However, many competitive
methods run GNNs multiple times with subgraph extraction and customized
labeling to capture information that is hard for normal GNNs to learn. Such
operations are time-consuming and do not scale to large graphs. In this paper,
we propose an efficient GNN framework called Geodesic GNN (GDGNN) that requires
only one GNN run and injects conditional relationships between nodes into the
model without labeling. This strategy effectively reduces the runtime of
subgraph methods. Specifically, we view the shortest paths between two nodes as
the spatial graph context of the neighborhood around them. The GNN embeddings
of nodes on the shortest paths are used to generate geodesic representations.
Conditioned on the geodesic representations, GDGNN can generate node, link, and
graph representations that carry much richer structural information than plain
GNNs. We theoretically prove that GDGNN is more powerful than plain GNNs. We
present experimental results to show that GDGNN achieves highly competitive
performance with SOTA GNN models on various graph learning tasks while taking
significantly less time.
