Graph Theoretical Analysis and Optimization of Urban Metro Rail Networks: A Case Study of Namma Metro

Abstract

Urban metro rail systems are essential components of modern transportation infrastructure, and as such networks grow in scale and complexity, systematic methods for evaluating their efficiency, connectivity, and resilience become increasingly important. This paper presents a graph-theoretical framework for the analysis and optimisation of urban metro rail networks, using the Namma Metro (Bengaluru) as a case study. The network, comprising 83 stations and 82 track segments across the Purple, Green, and Yellow lines (96.1 km) is modelled as a weighted graph G = (V, E, w) with edge weights encoding travel time, distance and passenger flow. Dijkstra's algorithm is applied to determine optimal passenger routes, while degree, closeness, betweenness and eigenvector centrality measures, identify structurally critical stations. A gravity-based model estimates inter-station passenger flow, and a congestion index quantifies station-level demand concentration. A complementary coach-level analysis quantifies within-train passenger imbalance using a variance-based measure and proposes redistribution strategies to reduce it to zero. The results demonstrate that graph theory offers a rigorous and practical foundation for transportation planning, congestion mitigation, and resilience-oriented network design in rapidly growing metropolitan metro systems.