Deep learning models for graphs, especially Graph Convolutional Networks (GCNs), have achieved remarkable performance in the task of semi-supervised node classification. However, recent studies show that GCNs suffer from adversarial perturbations. Such vulnerability to adversarial attacks significantly decreases the stability of GCNs when being applied to security-critical applications. Defense methods such as preprocessing, attention mechanism and adversarial training have been discussed by various studies. While being able to achieve desirable performance when the perturbation rates are low, such methods are still vulnerable to high perturbation rates. Meanwhile, some defending algorithms perform poorly when the node features are not visible. Therefore, in this paper, we propose a novel mechanism called influence mechanism, which is able to enhance the robustness of the GCNs significantly. The influence mechanism divides the effect of each node into two parts: introverted influence which tries to maintain its own features and extroverted influence which exerts influences on other nodes. Utilizing the influence mechanism, we propose the Influence GCN (I-GCN) model. Extensive experiments show that our proposed model is able to achieve higher accuracy rates than state-of-the-art methods when defending against non-targeted attacks.