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Network Science Based Quantification of Resilience Demonstrated on the Indian Railways Network

Fig 6

Topology of the IRN.

A. A cumulative probability distribution of node degree and strength (as measured by traffic volume) of stations in IRN, on a log-log scale, profile the distributional properties of the stations. The distributions follow truncated power law models, wherein most stations have a small number of connections, with the exception of a few hubs. Hubs are generally geospatially isolated. “k” stands for degree, and “s” stands for strength. Several cities are labeled multiple times as they contain more than one hub. For example, Delhi actually has multiple hub stations, specifically stations named “Hazrat Nizammudin”, “New Delhi” and “Delhi”; Delhi was used for brevity to represent all three. Table 1 details and delineates all stations that have been named identically in this panel. B. A correlation profile of station connectivity of IRN shows the average degree of stations’ nearest network neighbors. K1 and K2 serve to index the degree of any given station. Correlations in connectivity are shown as systematic deviations of the ratio P(K1,K2)/Pr(K1,K2). P(K1,K2) is the likelihood that two stations with connectivity K1 and K2 are connected to each other by the direct link. Pr(K1,K2) is the same value in averaged over a randomized ensemble of 1000 members. Yellow colors in the lower left indicate the tendency of stations with less connectivity to connect to other stations with comparable connectivity, while blue/green colors indicate small likelihood of hubs connecting with one another indicating the IRN’s disassortative nature. This further captures the tendency of the IRN to behave like a collection of relatively independent modules. C. Degree and strength are plotted against betweenness, closeness and Eigenvector centrality. Lines indicate the average for a centrality measure conditional on a particular degree or level strength, respectively, serving to highlight the variability in centrality metrics even for identical levels of connectivity or traffic volume.

Fig 6