Finding nodes occupying interesting positions in a graph is useful to extract meaningful information from large datasets. While numerous measures have been proposed to evaluate the centrality of nodes, few indicators quantify the capacity of nodes to connect different regions of the graph. Usually, betweenness centrality is used for this purpose, but we show here that it gives equal scores to “local” centers (i.e. nodes of high degree central to a single region) and to “global” bridges, which connect different regions. This distinction is important because the roles of these nodes are quite diverse. For example, in networks of scientific collaborations, local centers correspond to nodes which are important for a single sub-discipline, while bridges correspond to nodes which connect different sub-disciplines, leading to interdisciplinary collaborations. We show that a new measure of network topology, the bridgeness, is able to discriminate between local centers and global bridges, in synthetic and real networks.