Title: Characterizing the Influence of Graph Elements. Abstract: Influence function, a method from the robust statistics, measures the changes of model parameters or some functions about model parameters with respect to the removal or modification of training instances. It is an efficient and useful post-hoc method for studying the interpretability of machine learning models without the need of expensive model re-training. Recently, graph convolution networks (GCNs), which operate on graph data, have attracted a great deal of attention. However, there is no preceding research on the influence functions of GCNs to shed light on the effects of removing training nodes/edges from an input graph. Since the nodes/edges in a graph are interdependent in GCNs, it is challenging to derive influence functions for GCNs. To fill this gap, we started with the simple graph convolution (SGC) model that operates on an attributed graph, and formulated an influence function to approximate the changes of model parameters when a node or an edge is removed from an attributed graph. Moreover, we theoretically analyzed the error bound of the estimated influence of removing an edge. We experimentally validated the accuracy and effectiveness of our influence estimation function. In addition, we showed that the influence function of a SGC model could be used to estimate the impact of removing training nodes/edges on the test performance of the SGC without re-training the model. Finally, we demonstrated how to use influence functions to effectively guide the adversarial attacks on GCNs.