Recent studies have revealed the vulnerability of graph convolutional networks (GCNs) to edge-perturbing attacks, such as maliciously inserting or deleting graph edges. However, theoretical proof of such vulnerability remains a big challenge, and effective defense schemes are still open issues. In this article, 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 GCN, named AN-GCN, is proposed to defend against edge-perturbing attacks. In particular, we present a node localization theorem to demonstrate how GCNs locate nodes during their training phase. In addition, we design a staggered Gaussian noise-based node position generator and a spectral graph convolution-based discriminator (in detecting the generated node positions). Furthermore, we provide an optimization method for the designed generator and discriminator. It is demonstrated that the AN-GCN is secure against edge-perturbing attacks in node classification tasks, as AN-GCN is developed to classify nodes without the edge information (making it impossible for attackers to perturb edges anymore). Extensive evaluations verify the effectiveness of the general edge-perturbing attack (G-EPA) 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.