Tropical cyclones over the North Indian Ocean, including the Bay of Bengal and the Arabian Sea, pose a large threat to the teeming and low-lying coastal areas. There has long been recognition of the need for a detailed understanding and representation of the spatiotemporal organization and characteristics of the cyclone tracks. This paper uses the International Best Track Archive for Climate Stewardship (IBTrACS) best-track dataset from 1980 through 2025, and utilizes a complex network approach in the representation of the tropical cyclone tracks over the North Indian Ocean. The domain has been divided into 2° x 2° boxes in terms of the latitude-longitude location, and the cyclones treated as a temporally ordered sequence of visits on the boxes. The resulting directed and weighted network has then been assessed using classical network statistical measures, namely the degree, strength, betweenness centrality, clustering coefficient, and the modular communities. The paper specifically aims to provide a conceptual framework in the representation and ultimate understanding of the organized cyclone tracks over the different seasons and multi-decadal periods in an unambiguous manner. The approach aims to present the first detailed and rigorous investigation within the complex network framework, and provides evidence and insights on the hub nodes and paths in the organized cyclones in the middle and northern regions and the northeast Arabian Sea continental shelf, and the changes in the connections in the Arabian Sea over the past decades.