Community detection is a classical method for understanding the structure of relational data. In this paper, we study the problem of identifying mixed-membership community structure. We argue that it is beneficial to perform this task on the line graph, which can be constructed from an input graph by encoding the relationship between its edges. Here, we propose a curvature-based algorithm for mixed-membership community detection on the line graph. Our algorithm implements a discrete Ricci curvature flow under which the edge weights of a graph evolve to reveal its community structure. We demonstrate the performance of our approach in a series of benchmark experiments.