  Centrality is one of the most fundamental metrics in network science. Despite
an abundance of methods for measuring centrality of individual vertices, there
are by now only a few metrics to measure centrality of individual edges. We
modify various, widely used centrality concepts for vertices to those for
edges, in order to find which edges in a network are important between other
pairs of vertices. Focusing on the importance of edges, we propose an
edge-centrality-based network decomposition technique to identify a hierarchy
of sets of edges, where each set is associated with a different level of
importance. We evaluate the efficiency of our methods using various
paradigmatic network models and apply the novel concepts to identify important
edges and important sets of edges in a commonly used benchmark model in social
network analysis as well as in evolving epileptic brain networks.
