Network embedding (NE) aims to encode the relations of vertices into a low-dimensional space. After NE, we can obtain the learned vectors of vertices that preserve the proximity of network structures for subsequent applications, e.g., vertex classification and link prediction. In existing NE models, they usually exploit the skip-gram with a negative sampling method to optimize their objective functions. Generally, this method learns the vertex representation only from the local connectivity of vertices (i.e., neighbors). However, there is a larger scope of vertex connectivity in real-world scenarios: a vertex may have multifaceted aspects and should belong to overlapping communities. Taking a social network as the overlapping example, a user may subscribe to the channels of politics, economy, and sports simultaneously, but the politics share more common attributes with the economy and less with the sports. In this article, we propose an adversarial learning approach (ACNE) for modeling overlapping communities of vertices. Specifically, we map the association between communities and vertices into an embedding space. Moreover, we take further research on enhancing our ACNE with the following two operations. First, in the initialization stage, we adopt a walking strategy with perception to obtain paths containing more possible boundary vertices to improve overlapping community detection. Then, after representation learning with ACNE, we use soft community assignments from a simple classifier as supervision to update the weights of ACNE. This self-training mechanism referred to as ACNE-ST can help ACNE to achieve better performance. Experimental results demonstrate that the proposed methods, including ACNE and ACNE-ST, can outperform the state-of-the-art models on the subsequent tasks of vertex classification and overlapping community detection.