Vertices with high betweenness and closeness centrality represent influential entities in a network. An important problem for time varying networks is to know a-priori, using minimal computation, whether the influential vertices of the current time step will retain their high centrality, in the future time steps, as the network evolves. In this paper, based on empirical evidences from several large real world time varying networks, we discover a certain class of networks where the highly central vertices are part of the innermost core of the network and this property is maintained over time. As a key contribution of this work, we propose novel heuristics to identify these networks in an optimal fashion and also develop a two-step algorithm for predicting high centrality vertices. Consequently, we show for the first time that for such networks, expensive shortest path computations in each time step as the network changes can be completely avoided; instead we can use time series models (e.g., ARIMA as used here) to predict the overlap between the high centrality vertices in the current time step to the ones in the future time steps. Moreover, once the new network is available in time, we can find the high centrality vertices in the top core simply based on their high degree.