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Recent studies have revealed the vulnerability of graph convolutional networks (GCNs) to edge-perturbing attacks, such as maliciously inserting or deleting graph edges. However, a theoretical proof of such vulnerability remains a big challenge, and effective defense schemes are still open issues. In this paper, we first generalize the formulation of edge-perturbing attacks and strictly prove the vulnerability of GCNs to such attacks in node classification tasks. Following this, an anonymous graph convolutional network, named AN-GCN, is proposed to counter against edge-perturbing attacks. Specifically, we present a node localization theorem to demonstrate how the GCN locates nodes during its training phase. In addition, we design a staggered Gaussian noise based node position generator, and devise a spectral graph convolution based discriminator in detecting the generated node positions. Further, we give the optimization of the above generator and discriminator. AN-GCN can classify nodes without taking their position as input. It is demonstrated that the AN-GCN is secure against edge-perturbing attacks in node classification tasks, as AN-GCN classifies nodes without the edge information and thus makes it impossible for attackers to perturb edges anymore. Extensive evaluations demonstrated the effectiveness of the general edge-perturbing attack model in manipulating the classification results of the target nodes. More importantly, the proposed AN-GCN can achieve 82.7% in node classification accuracy without the edge-reading permission, which outperforms the state-of-the-art GCN.