Transitivity in friendship graphs has been well known as a key property of social networks. In this paper, we extend the graph transitivity index by introducing a new characteristic quantity of graphs, namely the transitivity matrix. The transitivity matrix measures the microscopic impact that each link has on the global transitivity index of the graph. We argue that the new transitivity metric is synergetic with key aspects of the social science literature on social network theory, and we show that it can be used as a tool for locating bridges as well as redundant links. This work represents a major departure from the traditional graph transitivity index, which provides a coarse measure for how transitive the overall network is.