Understanding overlapping community structures is crucial for network analysis and prediction. AGM (Affiliation Graph Model) is one of the favorite models for explaining the densely overlapped community structures. In this paper, we thoroughly re-investigate the assumptions made by the AGM model on real datasets. We find that the AGM model is not sufficient to explain several empirical behaviors observed in popular real-world networks. To our surprise, all our experimental results can be explained by a parameter-free hypothesis, leading to more straightforward modeling than AGM which has many parameters. Based on these findings, we propose a parameter-free Jaccard-based Affiliation Graph (JAG) model which models the probability of edge as a network specific constant times the Jaccard similarity between community sets associated with the individuals. Our modeling is significantly simpler than AGM, and it eliminates the need of associating a parameter, the probability value, with each community. Furthermore, JAG model naturally explains why (and in fact when) overlapping communities are densely connected. Based on these observations, we propose a new community-driven friendship formation process, which mathematically recovers the JAG model. JAG is the first model that points towards a direct causal relationship between tight connections in the given community with the number of overlapping communities inside it. Thus, \emph{the most effective way to bring a community together is to form more sub-communities within it.} The community detection algorithm based on our modeling demonstrates a significantly simple algorithm with state-of-the-art accuracy on six real-world network datasets compared to the existing link analysis based methods.