Graph Neural Networks (GNNs) have achieved significant progress in semi-supervised data classification, with an assumption that a complete graph or accurate structure is available. In this paper, a novel GNN architecture, named discrete-structure-augmentation graph convolutional network (DSA-GCN) is proposed, to apply the GCNs in real-world scenarios where the graphs are noisy and incomplete or even not available. Compared with existing methods, DSA-GCN firstly uses a variational Expectation-Maximization (EM) algorithm to jointly learn graph structure, including a discrete probability distribution on the edges of the graph and label dependency, and the parameters of GCN. Second, DSA-GCN applies novel reconstruction loss in learning discrete dependency structure on graph, together with consistency loss. Third, augmentation strategy is used to derive discrete graph structures with varying sparsity. Extensive experiments demonstrate that DSA-GCN significantly outperforms existing methods under varying levels of edge sparsity.