Cyclones are among the most hazardous extreme weather events on Earth. In certain scenarios, two co-rotating cyclones in close proximity to one another can drift closer and completely merge into a single cyclonic system. Identifying the dynamic transitions during such an interaction period of binary cyclones and predicting the complete merger (CM) event are challenging for weather forecasters. In this work, we suggest an innovative approach to understand the evolving vortical interactions between the cyclones during two such CM events (Noru-Kulap and Seroja-Odette) using time-evolving induced velocity-based unweighted directed networks. We find that network-based indicators, namely, in-degree and out-degree, quantify the changes in the interaction between the two cyclones and are excellent candidates to classify the interaction stages before a CM. The network indicators also help to identify the dominant cyclone during the period of interaction and quantify the variation of the strength of the dominating and merged cyclones. Finally, we show that the network measures also provide an early indication of the CM event well before its occurrence.