Tropical cyclones (TCs) are one of the most destructive natural hazards that pose a serious threat to society around the globe, particularly to those in the coastal regions. In this work, we study the temporal evolution of the regional weather conditions in relation to the occurrence of TCs using climate networks. Climate networks encode the interactions among climate variables at different locations on the Earth's surface, and in particular, time-evolving climate networks have been successfully applied to study different climate phenomena at comparably long time scales, such as the El Ni~no Southern Oscillation, different monsoon systems, or the climatic impacts of volcanic eruptions. Here, we develop and apply a complex network approach suitable for the investigation of the relatively short-lived TCs. We show that our proposed methodology has the potential to identify TCs and their tracks from mean sea level pressure (MSLP) data. We use the ERA5 reanalysis MSLP data to construct successive networks 20 of overlapping, short-length time windows for the regions under consideration, where we focus on the north Indian Ocean and the tropical north Atlantic Ocean. We compare the spatial features of various topological properties of the network, and the spatial scales involved, in the absence and presence of a cyclone. We find that network measures such as degree and clustering exhibit significant signatures of TCs and have striking similarities with their tracks. The study of network topology over time scales relevant to TCs allows us to obtain useful insights into the individual local signature of changes in the flow structure of the regional climate system.