Betweenness Centrality (BC) has proven to be a fundamental metric in many domains to identify the components (nodes) of a system modelled as a graph that are mostly traversed by information flows thus being critical to the proper functioning of the system itself. In the transportation domain, the metric has been mainly adopted to discover topological bottlenecks of the physical infrastructure composed of roads or railways. The adoption of this metric to study the evolution of transportation networks that take into account also the dynamic conditions of traffic is in its infancy mainly due to the high computation time needed to compute BC in large dynamic graphs. This paper explores the adoption of dynamic BC, i.e., BC computed on dynamic large-scale graphs, modeling road networks and the related vehicular traffic, and proposes the adoption of a fast algorithm for ahead monitoring of transportation networks by computing approximated BC values under time constraints. The experimental analysis proves that, with a bounded and tolerable approximation, the algorithm computes BC on very large dynamically weighted graphs in a significantly shorter time if compared with exact computation. Moreover, since the proposed algorithm can be tuned for an ideal trade-off between performance and accuracy, our solution paves the way to quasi real-time monitoring of highly dynamic networks providing anticipated information about possible congested or vulnerable areas. Such knowledge can be exploited by travel assistance services or intelligent traffic control systems to perform informed re-routing and therefore enhance network resilience in smart cities.