This paper extends the concept of betweenness centrality based on Kirchhoff’s law for electric circuits from centrality of nodes to centrality of edges. It is shown that this new measure admits analytical definition for some classes of networks such as bipartite graphs, with computation for larger networks. This measure is applied for detecting community structure within networks. The results of numerical experiments for some examples of networks, in particular, Math-Net.ru (a Web portal of mathematical publications) are presented, and a comparison with PageRank is given.