Neighborhood aggregation algorithms like spectral graph convolutional networks (GCNs) formulate graph convolutions as a symmetric Laplacian smoothing operation to aggregate the feature information of one node with that of its neighbors. While they have achieved great success in semi-supervised node classification on graphs, current approaches suffer from the over-smoothing problem when the depth of the neural networks increases, which always leads to a noticeable degradation of performance. To solve this problem, we present graph convolutional ladder-shape networks (GCLN), a novel graph neural network architecture that transmits messages from shallow layers to deeper layers to overcome the over-smoothing problem and dramatically extend the scale of the neural networks with improved performance. We have validated the effectiveness of proposed GCLN at a node-wise level with a semi-supervised task (node classification) and an unsupervised task (node clustering), and at a graph-wise level with graph classification by applying a differentiable pooling operation. The proposed GCLN outperforms original GCNs, deep GCNs and other state-of-the-art GCN-based models for all three tasks, which were designed from various perspectives on six real-world benchmark data sets..