  Graph neural network (GNN) has been demonstrated powerful in modeling
graph-structured data. However, despite many successful cases of applying GNNs
to various graph classification and prediction tasks, whether the graph
geometrical information has been fully exploited to enhance the learning
performance of GNNs is not yet well understood. This paper introduces a new
approach to enhance GNN by discrete graph Ricci curvature. Specifically, the
graph Ricci curvature defined on the edges of a graph measures how difficult
the information transits on one edge from one node to another based on their
neighborhoods. Motivated by the geometric analogy of Ricci curvature in the
graph setting, we prove that by inserting the curvature information with
different carefully designed transformation function $\zeta$, several known
computational issues in GNN such as over-smoothing can be alleviated in our
proposed model. Furthermore, we verified that edges with very positive Ricci
curvature (i.e., $\kappa_{i,j} \approx 1$) are preferred to be dropped to
enhance model's adaption to heterophily graph and one curvature based graph
edge drop algorithm is proposed. Comprehensive experiments show that our
curvature-based GNN model outperforms the state-of-the-art baselines in both
homophily and heterophily graph datasets, indicating the effectiveness of
involving graph geometric information in GNNs.
