Fig 1.
Node removal and recovery process in the representative network with N = 6.
Nodes X and Y are selected randomly for removal at time, T = 1 and T = 2, respectively. A. (i) The SCF = 1 at step T = 0 (pre-hazard). Node X (red) is selected for removal at step T = 1. (ii) Removal of node X results in reduction of the size of the Giant Component (GC), which sets SCF = 0.5. Dashed nodes (edges) means that nodes (edges) gets detached from the GC and hence incapacitated. Node Y (blue) is selected for removal at step T = 2 (f = 1/6, meaning one out of the six nodes is targeted for removal). (iii) The GC ceases to exist after the removal of node Y. B. To highlight the asymmetric nature of recovery process, nodes are restored to their full functionality in the same order these were removed (i.e. node X followed by node Y) from the network. (iv) Node X (yellow) is selected for restoration to full functionality in the first step of the recovery process. (v) This results in the recovery of the node X to full functionality (f’ = 1/6, meaning one out of the six nodes is fully functional). As a result, three nodes directly connected to X gain at least one edge and the GC grows, making SCF = 0.67. Then, node Y (green) is selected for recovery in step (vi). Recovery of node Y to its full functionality result in restoration of the SCF of the network to 1 as shown in (vi).
Fig 2.
IRN and resilience curves under complete failure and recovery.
A. The IRN is displayed. The 12-largest communities, each of which map to a color, capture the vast majority (91.6%) of the stations. Stations are sized by traffic volume. B. (Left) We quantify the robustness of IRN as it responds to random versus intentional attacks, where intentional attacks are motivated by either railway station connectivity (degree) or traffic volume (strength). For intentional attacks, approximately 20% of stations must be disrupted for the full IRN to lose all critical functionality, as measured with SCF. (Right) Using the same metric SCF, recovery strategies that propose alternative prioritizations for recovery of stations are compared, using an N = 1000 ensemble of randomly generated sequences as a baseline. Number of connections (degree) and traffic volume (strength) are used as intuitive measures for generating recovery sequences, and the results are plotted. In addition, betweenness, Eigenvector, and closeness centrality are used, and the results are plotted.
Fig 3.
Resilience curves for the IRN’s two largest communities.
The same as Fig 2B is demonstrated for the two largest communities shown in Fig 2A. A. The same as Fig 2B is displayed but for the largest community (in South India, Community ID 1 from Fig 1B). B. The same as Fig 2B is shown but for the second largest community (in North India, Community ID 2 from Fig 2A) with the inset showing that, at different levels of partial recovery (e.g., SCF ~ = 0.4), it is not always clear which metric is most effective for prioritizing stations.
Fig 4.
Simulating the impact of realistic natural and man-made hazards.
The top row schematically illustrates portions of the IRN initially impacted by realistic natural and cyber or cyber-physical threat scenarios, all with the same initial network topology as shown in Fig 2A, where 752 stations reside in the largest giant component (SCF = 1). The bottom row displays the community structure post hazard in each case. A. The impact of a disaster with properties similar to that of the December 2004 Indian Ocean tsunami is displayed. As suggested by the insight that communities are relatively independent as obtained from Fig 2, the regional nature of the hazard (shaded blue, top) significantly impacts the Southeastern coast, removing 28 stations. The number of communities increases from 49 to 75. Yet, the structure of the remainder of the network remains relatively intact (SCF = 0.903, where 679 stations remain in the giant component, see Methods and Data). B. For a simulated cyber or cyber-physical attack scenario, where 19 stations are perhaps maliciously targeted based on traffic volume (nodes shaded grey, top) and removed, the network structure is fractured significantly (SCF = 0.890, where 669 stations remain in the giant component). The number of communities increases from 49 to 96. C. A scenario based on a cascade from the power grid, similar to the 2012 blackout (shaded grey, top) is also simulated. The impact is significant, removing 39 stations, but the degradation of the IRN remains regionally contained (SCF = 0.852, where 641 stations remain in the giant component). The number of communities increases from 49 to 102. Note that differences that appear relatively in the SCF can have significant practical implications with a large network. In the case of the IRN given TF = 752, an SCF dropping by about 0.01 means 10 less stations are part of the giant component.
Fig 5.
Recovery from simulated natural and man-made hazards.
A. Recovery curves after the simulated tsunami are displayed. As a baseline, the gray shaded interval represents the 99% bounds of N = 1000 randomly generated recovery sequences. At each step, the 99% bounds are the 5th and 995th largest SCF values from the N = 1000 member ensemble. The SCF begins at 0.903. B. The same as A is displayed but for the simulated cyber-physical attack. Here, the SCF begins at 0.852. C. The same as A is displayed but for the simulated power grid failure cascade. Here the SCF begins at 0.890. D. For the tsunami, at each step of the recovery curve, the percentage of ensemble members that a given metric is larger than in terms of SCF is plotted. This is repeated for each metric (connectivity, traffic volume, betweenness centrality, Eigenvector centrality, and closeness centrality). E. The same as D but for the cyber-physical attack recovery curve. F. The same as D but for the power grid failure cascade. In D-F, in some cases, lines overlap each other; when this is the case, one line is thicker than the other to enable visibility of both.
Fig 6.
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.
Table 1.
Network summary statistics on 25 most connected (highest degree) stations.