Real-life systems involving interacting objects are typically modeled as graphs and can often grow very large in size. Revealing the community structure of such systems is crucial in helping us better understand their complex nature. However, the ever-increasing size of real-world graphs, and our evolving perception of what a community is, make the task of community detection very challenging. One such challenge, is the discovery of the possibly overlapping communities of a given node in a billion-node graph. This problem is very common in modern large social networks like Facebook and Linkedln. In this paper, we propose a scalable local community detection approach to efficiently unfold the communities of individual target nodes in a given network. Our goal is to reveal the groupings formed around nodes (e.g., users) by leveraging the relations of the different contexts the nodes participate in. Our algorithm, termed Local Dispersion-aware Link Communities or LDLC, measures the similarity of pairs of links in the graph as well as the extent of their participation in multiple contexts. Then, it determines the ordering that we should group the links in order to form communities. Our approach is not affected by constraints existent in previous techniques (e.g., the need for several seed nodes or the need to collapse multiple overlapping communities to one). Our experimental evaluation using ground-truth communities for a wide range of large real-world networks show that LDLC significantly outperforms state-of-the-art methods on both accuracy and efficiency.