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Graph Convolutional Networks (GCN) is a pioneering model for graph-based semi-supervised learning. However, GCN does not perform well on sparsely-labeled graphs. Its two-layer version cannot effectively propagate the label information to the whole graph structure (i.e., the under-smoothing problem) while its deep version over-smoothens and is hard to train (i.e., the over-smoothing problem). To solve these two issues, we propose a new graph neural network called GND-Nets (for Graph Neural Diffusion Networks) that exploits the local and global neighborhood information of a vertex in a single layer. Exploiting the shallow network mitigates the over-smoothing problem while exploiting the local and global neighborhood information mitigates the under-smoothing problem. The utilization of the local and global neighborhood information of a vertex is achieved by a new graph diffusion method called neural diffusions, which integrate neural networks into the conventional linear and nonlinear graph diffusions. The adoption of neural networks makes neural diffusions adaptable to different datasets. Extensive experiments on various sparsely-labeled graphs verify the effectiveness and efficiency of GND-Nets compared to state-of-the-art approaches.