Betweenness centrality metrics usually underestimate the importance of nodes that are close to shortest paths but do not exactly fall on them. In this paper, we reevaluate the importance of such nodes and propose the ρ-geodesic betweenness centrality, a novel metric that assigns weights to paths (and, consequently, to nodes on these paths) according to how close they are to shortest paths. The paths that are just slightly longer than the shortest one are defined as quasi-shortest paths, and they are able to increase or to decrease the importance of a node according to how often the node falls on them. We compare the proposed metric with the traditional, distance-scaled, and random walk betweenness centralities using four network datasets with distinct characteristics. The results show that the proposed metric, besides better assessing the topological role of a node, is also able to maintain the rank position of nodes overtime compared to the other metrics; this means that network dynamics affect less our metric than others. Such a property could help avoid, for instance, the waste of resources caused when data follow only the shortest paths and reduce associated costs.