Network embedding, which learns the low-dimensional representations of nodes, has gained significant research attention. Despite its superior empirical success, often measured by the prediction performance of downstream tasks (e.g., multi-label classification), it is unclear why a given embedding algorithm outputs the specific node representations, and how the resulting node representations relate to the structure of the input network. In this paper, we propose to discern the edge influence as the first step towards understanding skip-gram basd network embedding methods. For this purpose, we propose an auditing framework Near, whose key part includes two algorithms (Near-add and Near-del) to effectively and efficiently quantify the influence of each edge. Based on the algorithms, we further identify high-influential edges by exploiting the linkage between edge influence and the network structure. Experimental results demonstrate that the proposed algorithms (Near-add and Near-del) are significantly faster (up to 2, 000×) than straightforward methods with little quality loss. Moreover, the proposed framework can efficiently identify the most influential edges for network embedding in the context of downstream prediction task and adversarial attacking.