Assessing criticality in subway networks for risk prevention through topological performance indices

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Department of Civil, Environmental and Territorial Engineering Master of thesis in Civil Engineering for Risk Mitigation

A

SSESSING CRITICALITY IN SUBWAY

NETWORKS FOR RISK PREVENTION

THROUGH TOPOLOGICAL PERFORMANCE

INDICES

Master thesis of:

Hiva Viseh

Supervisors:

Prof. Lorenzo Mussone

Prof. Roberto Notari

March 2020 March 2020 March 2020

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DEDICATION

I dedicate my dissertation work to my loving parents Tahereh and Hamid. A special feeling of gratitude to my dear brother Sina, who has encouraged and supported me all the time, and has never

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ACKNOWLEDGEMENTS

I would like to express my deep gratitude to my master thesis advisors, Prof. Lorenzo Mussone and Prof. Roberto Notari, for their countless hours of reflecting, reading, encouraging, and most of all patience throughout the entire process. Their excitement and willingness to provide feedback made

the completion of this research an enjoyable experience and I have learned many things since I became their student. Also, I wish to thank my committee members who were more than generous

with their expertise and precious time.

Hiva Viseh

Politecnico di Milano

Department of Civil, Environmental and Territorial Engineering Lecco, Italy

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Abstract

Proper functioning of a city relies on how optimal all its critical infrastructure systems operate. One of the most important infrastructure systems, especially for metropolitan cities are transportation networks which are the backbone of all critical infrastructures.

Transportation systems behave differently in the face of disruptions due to a “Random Failure or Incident”, “Intentional Attack” and “Natural Disasters”. In addition, the response of each mode of transportation in case of the same disruptive event can be totally different (e.g. comparing a node disruption in an aviation network with a subway network which leads to totally different economic consequences). As a result, we should be aware of all kinds of risks, which threaten each mode of transportation network to avoid any devastating scenario.

Increasing the resiliency of different transportation networks has even become more vital in recent years because of two reasons: 1. uncertainty in the context of future climate change and concern about increases in the frequency and severity of natural hazards; 2. countering terrorism is a worldwide top national security priority.

By boosting the resiliency of the networks, we will be able to minimize the costs of a disruptive event from the economical, social and operational point of view. In order to do so, we have to define the critical components of networks and simulate the response of networks when those components stop working. Graphs are among the most commonly used tools for studying different types of transportation systems. Through graph analysis, we are able to find and modify the most vulnerable parts of the network to increase its general robustness.

In this thesis, subway networks were analyzed by settling measures on graphs. Thirty-four subway networks all around the world were studied through some known indices (the four centrality indices) and three new ones (Ishortest, Icentr, and Pathrank) specifically developed for this study. The most crucial components of these networks (through the four centrality indices, Icentr and Pathrank) as well as the stations whose malfunctions have the highest effects on the performance of the whole system (through Ishortest), were defined.

The obtained results will help us to decrease the vulnerability of existing subway networks by increasing the robustness of defined stations. The methodology which has been used in this thesis can be extended to other modes of transportation with minimal effort.

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List of Tables

Table 1. Natural Hazards and their effects on different transportation modes. ... 5

Table 2. Flood accidents of subway stations around the world. ... 6

Table 3. Cases of subway networks which were affected by the earthquake. ... 7

Table 4. Some recent terrorist attacks and their consequences on different transportation modes. ... 9

Table 5. Subway terrorism attacks around the world. ... 10

Table 6. Underground fire incidents around the world. ... 12

Table 7. Adjacency matrix of the graph G which is represented in Fig 5. a. ... 19

Table 8. Calculation of the betweenness centrality of node F in graph G. ... 24

Table 9. Calculation of the Bonacich power centrality for each node in graph G by considering different values for β parameter. ... 28

Table 10. Calculation the nodes’ Importance of the graph by considering two scenarios. ... 30

Table 11. Indices used for analyzing network status. ... 35

Table 12. Icentr’s results for graph G in figure 25. ... 37

Table 13. Pathrank’s results for graph G in Fig. 31 (by considering logit as weighting function and paths up to length 7. ... 38

Table 14. Ishortest results for the graph in figure 32. ... 39

Table 15. List of subway networks studied in this paper, a) Alphabetically ordered by city name, b) Ordered by network size. ... 41

Table 16. Degree Centrality values of Toronto subway network. ... 45

Table 17. Eigenvector Centrality values of Toronto subway network. ... 45

Table 18. Betweenness Centrality values of Toronto subway network. ... 45

Table 19. Closeness Centrality values of Toronto subway network. ... 46

Table 20. Icentr’s values for Toronto’s subway network (no weight). ... 49

Table 21. Icentr’s values for Toronto’s subway network (considering the number of stations as weight). ... 49

Table 22. Icentr’s values for Toronto’s subway network (considering the inverse of distances as weight). ... 50

Table 23. Toronto’s subway network Pathrank’s values for path length equal to eleven. ... 51

Table 24. Ishortest values of Toronto’s subway network (no weight). ... 52

Table 25. Ishortest values of Toronto’s subway network (considering the number of stations as weight). ... 52

Table 26. Ishortest values of Toronto’s subway network (considering the inverse of number of stations as weight). ... 53

Table 27. Ishortest values of Toronto’s subway network (considering the distances as weight). ... 53

Table 28. Degree Centrality values of Washington DC subway network. ... 56

Table 29. Eigenvector Centrality values of Washington DC subway network. ... 56

Table 30. Betweenness Centrality values of Washington DC subway network. ... 56

Table 31. Closeness Centrality values of Washington DC subway network. ... 57

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Table 33. Icentr values for Washington DC’s subway network (considering the number of stations as

weight). ... 60

Table 34. Icentr values for Washington DC’s subway network (considering the inverse of distances as weight). ... 60

Table 35. Pathrank values of Washington DC’s subway network, for maximum path length equal to twelve. ... 62

Table 36. Ishortest values of Washington DC’s subway network (no weight). ... 62

Table 37. Ishortest values of Washington DC’s subway network (considering the number of stations as weight). ... 63

Table 38. Ishortest values of Washington DC’s subway network (considering the inverse of number of stations as weight). ... 63

Table 39. Ishortest values of Washington DC’s subway network (considering the distances as weight). ... 63

Table 40. Degree Centrality values of Beijing subway network. While considering the adjacency matrix, all nodes with degree 4 are selected as nodes with highest degree centrality. Here, we report only some of them. ... 67

Table 41. Eigenvector Centrality values of Beijing subway network. ... 68

Table 42. Betweenness Centrality values of Beijing subway network. ... 68

Table 43. Closeness Centrality values of Beijing subway network. ... 68

Table 44. Icentr values for Beijing subway network (no weight). ... 71

Table 45. Icentr values for Beijing subway network considering the number of stations as weight. ... 71

Table 46. Icentr values for Beijing subway network, considering the inverse of distances as weight. ... 72

Table 47. Pathrank values of Beijing subway network. ... 75

Table 48. Ishortest values of Beijing subway network (no weight). ... 77

Table 49. Ishortest values of Beijing subway network (considering the number of stations as weight). ... 77

Table 50. Ishortest values of Beijing subway network (considering the inverse of number of stations as weight). ... 77

Table 51. Ishortest values of Beijing subway network (considering the distances as weight). ... 77

Table 52. Highest index value of Centrality, Icentr, PathRank and Ishortest for the thirty-four metro networks. The value in parenthesis for the Ishortest index, is the number of subgraphs created in case of disruption at that station and the amount of loss in case of cancelling that node. ... 81

List of Figures

Figure 1. Proportion of attack tactics in underground systems. ... 9

Figure 2. Proportion of fire causes. ... 12

Figure 3. Atypical recovery curve. ... 14

Figure 4. A comparison between three different recovery curves. ... 15

Figure 5.a. Graph G with 7 nodes and 8 edges. b. A bar chart which represents the degree centrality of each node in graph G. ... 19

Figure 6.a. An undirected graph. b. vertices which are directly connected to vertex A. c. the only direct link between neighbors of point A. ... 20

Figure 7. Graph G with 4 nodes and 3 edges. ... 20

Figure 8.a. Graph G with 7 nodes and 8 edges. b. Values of closeness centrality for each node in graph G... 22

Figure 10. A star shape graph with 7 nodes and 6 edges. ... 24

Figure 11. The value of the eigenvector centrality of each node indicates that points C and E are the most important nodes of the network. ... 25

Figure 13. The normalized closeness for each node in graph G. ... 27

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Figure 15. The normalized betweenness for each node in graph G. ... 27

Figure 16. The Eigenvector centrality for each node in graph G. ... 28

Figure 17. The more important nodes of graph G according to Bonacich power centrality. ... 28

Figure 19. Degree of each node in graph G. ... 29

Figure 20. Betweenness of each node in graph G. ... 30

Figure 22. Calculation of the Laplacian matrix in the above graph. ... 32

Figure 23. Graph G consists of 8 nodes and 12 edges, 𝐴𝐶𝐵𝑎𝑠𝑒 = 1.2841, 𝐺𝐶𝐶𝐵𝑎𝑠𝑒 = 8. ... 32

Figure 24. Graph G after removal of first edge, 𝐴𝐶𝐸, 1 = 1.1206, 𝐴𝐶𝑁𝐴𝐶𝑁𝑏𝑎𝑠𝑒 = 1.12061.2841 = 0.87 𝐺𝐶𝐶𝐸, 1 = 8. ... 32

Figure 25. Graph G after removal of second edge, 𝐴𝐶𝐸, 2 = 0.9304, 𝐴𝐶𝑁𝐴𝐶𝑁𝑏𝑎𝑠𝑒 = 0.93041.2841 = 0.72 𝐺𝐶𝐶𝐸, 2 = 8. ... 33

Figure 26. Graph G after removal of first node, 𝐴𝐶𝑁, 1 = 0.9636, 𝐴𝐶𝑁𝐴𝐶𝑁𝑏𝑎𝑠𝑒 = 0.96361.2841 = 0.75 𝐺𝐶𝐶𝑁, 1 = 7. ... 33

Figure 27. Graph G after removal of second node, 𝐴𝐶𝑁, 1 = 0.6571, 𝐴𝐶𝑁𝐴𝐶𝑁𝑏𝑎𝑠𝑒 = 0.65711.2841 = 0.51 𝐺𝐶𝐶𝐵𝑎𝑠𝑒 = 6. ... 33

Figure 28. Graph F consists of 7 nodes and 8 edges, 𝐴𝐶𝐵𝑎𝑠𝑒 = 0.2679, 𝐺𝐶𝐶𝐵𝑎𝑠𝑒 = 7. ... 34

Figure 29. Graph F after removal of its central node, 𝐴𝐶𝑁, 1 = -0.0000, 𝐴𝐶𝑁𝐴𝐶𝑁𝑏𝑎𝑠𝑒 = −0.00000.2679 = 0 , 𝐺𝐶𝐶𝑁, 1 = 3. ... 34

Figure 30. a. Icentr procedure for vertex 1(step 1). b. Icentr procedure for vertex 1(step 2). ... 36

Figure 32.a. We place a node at Vieux-Port (marked in black) to distinguish the blue from the red path. b. The fictitious vertex 14 is added as terminus to avoid degree two... 42

Figure 33. Toronto subway network map... 44

Figure 34. Graph of Toronto metro network. ... 44

Figure 35. Graphs represent the node with the highest centrality value in Toronto’s subway network (no weight). ... 46

Figure 36. Graphs represent the node with the highest centrality value in Toronto’s subway network (considering the number of stations as weight). ... 47

Figure 37. Graphs represent the node with the highest centrality value in Toronto’s subway network (considering the inverse of distances as weight). ... 47

Figure 38. Marked nodes show stations with highest centrality values in Toronto subway network. ... 48

Figure 39. Marked nodes show stations with highest Icentr rank considering eigenvector (a), closeness (b), and betweenness (c), in Toronto subway network (no weight). ... 50

Figure 40. Marked nodes show stations with highest Icentr rank considering eigenvector (a), closeness (b), and betweenness (c), in Toronto subway network (considering the number of stations as weight). ... 50

Figure 41. Marked nodes show stations with highest Icentr rank considering eigenvector (a), closeness (b), and betweenness (c), in Toronto subway network (considering the inverse of distances as weight). ... 51

Figure 42. Marked nodes show stations with highest Pathrank’s values considering no weight (a), the number of stations as weight (b), and the inverse of distances as weight (c), in Toronto subway network. ... 52

Figure 43. Marked nodes show the costliest stations (increasing the travel distance) to Toronto subway network if disrupted considering no weight. a. For a non-separated network. b. For a separated network. .... 53

Figure 44. Marked nodes show the costliest stations (increasing the travel distance) to Toronto subway network if disrupted considering the number of stations as weight. a. For a non-separated network. b. For a separated network. ... 54

Figure 45. Marked nodes show the costliest stations (increasing the travel distance) to Toronto subway network if disrupted considering the inverse of number of stations as weight. a. For a non-separated network. b. For a separated network. ... 54

Figure 46. Marked nodes show the costliest stations (increasing the travel distance) to Toronto subway network if disrupted considering the inverse of distances as weight. a. For a non-separated network. b. For a separated network. ... 54

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Figure 48. Graph of Washington DC metro network. ... 55 Figure 49. Graphs represent the nodes with the highest centrality values in Washington DC’s subway

network, considering no weight. ... 57 Figure 50. Graphs represent the node with the highest centrality value in Washington DC’s subway network, considering the number of stations as weight. ... 58 Figure 51. Graphs represent the nodes with the highest centrality values in Washington DC’s subway

network, considering the inverse of distances as weight. ... 58 Figure 52. Marked nodes show stations with highest centrality values in Washington DC subway network according to the analysis of the centrality indices. ... 59 Figure 53. Marked nodes show stations with highest Icentr rank considering eigenvector (a), closeness (b), and betweenness (c), in Washington DC subway network (no weight). ... 61 Figure 54. Marked nodes show stations with highest Icentr rank considering eigenvector (a), closeness (b), and betweenness (c), in Washington DC subway network (considering the number of stations as weight). .. 61 Figure 55. Marked nodes show stations with highest Icentr rank considering eigenvector (a), closeness (b), and betweenness (c), in Washington DC subway network (considering the inverse of distances as weight). . 61 Figure 56. Marked nodes show stations with highest PathRank’s values considering adjacency (a), station (b), and inverse of distance matrix (c), in Washington DC subway network. ... 62 Figure 57. Marked nodes show the costliest stations (increasing the travel distance) to Washington DC subway network if disrupted considering no weight. a. For a non-separated network. b. For a separated network. ... 64 Figure 58. Marked nodes show the costliest stations (increasing the travel distance) to Washington DC subway network if disrupted considering the number of stations as weight. a. For a non-separated network. b. For a separated network. ... 64 Figure 59. Marked nodes show the costliest stations (increasing the travel distance) to Washington DC subway network if disrupted considering the inverse of number of stations as weight. a. For a non-separated network. b. For a separated network. ... 64 Figure 60. Marked nodes show the costliest stations (increasing the travel distance) to Washington DC subway network if disrupted considering the distances as weight. a. For a non-separated network. b. For a separated network. ... 65 Figure 61. Beijing subway network map. ... 66 Figure 62. Graph of Beijing subway network. ... 67 Figure 63. Graphs represent the nodes with the highest centrality values in Beijing’s graph, considering no weight. ... 69 Figure 64. Graphs represent the nodes with the highest centrality values in Beijing’s graph, considering the number of stations as weight. ... 69 Figure 65. Graphs represent the nodes with the highest centrality values in Beijing’s graph, considering the inverse of distances as weight. ... 70 Figure 66. Marked nodes show stations with highest centrality values in Beijing subway network according to centrality analysis. ... 70 Figure 67. Marked nodes show stations with highest Icentr rank considering eigenvector (a), closeness (b), and betweenness (c), in Beijing subway network (no weight). ... 73 Figure 68. Marked nodes show stations with highest Icentr rank considering eigenvector (a), closeness (b), and betweenness (c), in Beijing subway network (considering the number of stations as weight). ... 74 Figure 69. Marked nodes show stations with highest Icentr rank considering eigenvector (a), closeness (b), and betweenness (c), in Beijing subway network (considering the inverse of distances as weight). ... 75 Figure 70. Marked nodes show stations with highest PathRank values considering no weight (a), the number of stations as weight (b), and inverse of distances as weight (c) in Beijing subway network. ... 76 Figure 71. Marked nodes show the costliest stations (increasing the travel distance) to Beijing subway network if disrupted considering no weight. a. For a non-separated network. b. For a separated network. .... 78 Figure 72. Marked nodes show the costliest stations (increasing the travel distance) to Beijing subway network if disrupted considering the number of stations as weight. a. For a non-separated network. b. For a separated network. ... 78

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Figure 73. Marked nodes show the costliest stations (increasing the travel distance) to Beijing subway network if disrupted considering the inverse of number of stations as weight. a. For a non-separated network. b. For a separated network. ... 78 Figure 74. Marked nodes show the costliest stations (increasing the travel distance) to Beijing subway network if disrupted considering the distances as weight. a. For a non-separated network. b. For a separated network. ... 79 Figure 75. Three networks with different structure. a. Circular. b. Rectangular. c. Polygon. ... 81

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Section 1: Consequences of natural and

manmade hazards threatening variant modes

of transportation network

Introduction

The successful operation of a city is highly dependent on the smooth functioning of the different critical infrastructure systems including transportation network, communication network, and information technology network. If any of these sectors fails, cities face a risk of cascading failures from socio economic aspect. In addition, since these infrastructure networks are interdependent, a disruption in capacity or service level in one network during and after any kinds of incidents, may greatly affect other networks and therefore may lead to devastating results in wellbeing of society, economic security and social welfare.

One of the most important infrastructure systems, especially for metropolitan cities are transportation networks. Transportation systems behave differently in the face of disruptions due to a “Random Failure or Incident”, “Intentional Attack” and “Natural Disasters”. In some modes of transportation networks such as aviation networks, a large scale disruptive event in an airport (a node disruption) leads to major perturbation and huge economical loss. Since we expect the variation in the different transportation network mode’s performance in the face of different disruptions, we should be aware of all kinds of risks which threaten a transport network to avoid any devastating scenario.

Disruption due to “Natural Disasters”

Transportation networks are the backbone of critical infrastructures since they provide accessibility to the other systems and rescue operations, immediately after a disaster strikes and during restoration processes. Natural disasters could impact the network devastatingly and therefore reducing the speed and quality of sending aid and supplies to the victims. Moreover, the cost associated with the disruption of economic activities, is a signiﬁcant component of disaster impact. The impact mainly depends on three factors:1) the nature and extent of diffusion of disasters; 2) the extent of exposure of population and infrastructures and 3) the extent of vulnerability of population and infrastructures [1].

On the other hand, due to uncertainty in the context of future climate change, there is concern about increases in the frequency and severity of natural hazards. Therefore, it would be worthwhile to mention different natural hazards and their possible consequences on each mode of transport.

Transportation’s

Mode Direct losses Indirect socio-economic losses

Avalanche Roads The road is interrupted a few days in the _{year because of avalanche warnings} Reduction in accessibility, travel _{delay and costlier routes}

Rock Fall Roads and Railways

• Blockages of roads and railways • Destruction of road and avalanche

gallery

• Traffic disruption generation

• Reduction in accessibility, travel delay and costlier routes

• Economic loss due to reconstructing and cleaning up the roads

Landslide Roads

• Destruction of road, avalanche gallery and rock gallery.

• Generation of large gaps in the bedrock around the activation area of the rockslide

• Economic loss due to reconstructing and cleaning up the roads

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2 Marine

Transportation

Landslides and bank failures could block channels. Debris from fallen trees and other materials could hinder navigation

Volcano

Rail Transit

• Fine ash can enter engines and cause increased in friction on all moving parts, also blockage of air filters • Track switches may be blocked by

fallen ash

• Reduction of traction on rail line caused by fallen ash

• Electric power trains may experience flashover (short-circuiting) on their overhead power lines following ash accumulation and light rain

• Electronic signal equipment may also experience flashover of signal equipment

Temporary suspension of rail services or the delay in normal schedules

Road and Highways

• Road closure (this depends on the ash fall depth and characteristics, road gradient and local weather conditions) • Covering the road marking by ashfall

with depth larger than 1mm

• Reduction in traction by ash. Wet ashfall with depth larger than 50 mm may make roads impassable

• Extra loading on bridges in case of very thick ashfall, especially when wet

Travel delays and decreased transportation efficiency due to: 1. Short-term speed restrictions of

15-30 km per hour or less (10-20 miles per hour)

2. Closure of highways to limit the likelihood of motorists to become stranded before reaching their destination

3. Closure of roads in urban areas to facilitate clean up and prevent stirring up the ash

Marine Transportation

• Volcanic ash can block air intake filters in a matter of minutes, crippling airflow to vital machinery. Ash particles are very abrasive, and if they get into the moving parts of an engine, they quickly cause damages

• Ashfall can be corrosive to metal, or other exposed shipboard equipment • Certain types of volcanic ash do not

disperse easily in water, instead they clump on the surface of water in pumice rafts. These rafts can clog salt-water intake strainers very quickly, which can result in overheating of shipboard machinery dependent on seawater service cooling.

• Ashfalls can reduce visibility to less than 1/2 mi (800 m), which is a hazard to navigation

Falling and deposited ash can result in adverse conditions and delays for marine transportation

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Aviation

Volcanic ash ejected into the atmosphere by explosive eruptions has known damaging effects on aircraft. Ash particles can abrade forward-facing surfaces, including windscreens, fuselage surfaces, and compressor fan blades. Ash contamination also can lead to failure of critical navigational and operational instruments. Moreover, the melting temperature of the glassy silicate material in an ash cloud is lower than combustion temperatures in modern jet engines; consequently, ash particles sucked into an engine can melt quickly and accumulate as re-solidified deposits in cooler parts, degrading engine performance even to the point of in-flight compressor stall and loss of thrust power

Temporary operational disruption, ranging from flight cancellations to airport closures lasting from hours to weeks

Flood

Railway

• Floodwater can damage trains, cause electrical faults and destabilize the lines due to ballast washing away

• Cut off a region in the country since road flooding also have occurred at the same time

• Reduction in accessibility, travel delay and costlier routes due to suspension of rail transport

• Economic loss due to reconstruction of the railways and cleaning up the debris after flood, also suspension of the service during the recovery time

• Increasing in the roads' traffic

• Some roads became impassable • Hours of slow-moving traffic

• Cancellation of the majority of public transport

• Drivers were forced to abandon their cars either due to road closures or running out of fuel after severe hold ups

• Increases emissions into the environment caused by slow moving traffic

• Economic loss due to reconstruction of the roads and cleaning up the debris after flood

Underground

• Flooding of the metro stations

• Flooding of the metro tunnels causes line closure

• Damage to the ventilation system because of sent down water

• Economic loss due to cleaning up the debris after flood

• Economic loss caused by damaging the ventilation system and other facilities in the stations

Aviation

Airports experience disruptions due to flooding of the terminal buildings and damages to essential electrical and IT installations

Disrupting operations for several days whilst repairs are carried out

Earthquake Railway

• Vertical and horizontal track misalignments

• Tunnel misalignments and failure of tunnel linings

• Overturning of rail cars and locomotives

• Structural damage to railroad buildings • Releasing and spilling of hazardous

materials from overturned tank rail cars because of the earthquake

• Railway's tunnel collapse • Failure of traffic signal

• After the earthquake, railway bridges will have to be inspected and repaired

• Large economic loss due to dysfunction of the source of transportation for moving of bulk goods across the region

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• Roadways' surface displacements • Flooding roads and highways because

of broken levees caused by earthquake • Cracking the pavements due to ground

failure

• Major to destructive damage to bridges and overpasses

• Slope instability

• Devastating socioeconomic consequences, including hindered access to hospitals, evacuation areas, emergency relief centers, and fire departments in first hours and days after striking an earthquake and other impacts due to closed mass-transit facilities and the inability to get to or from work till after recovery time and coming back to primary level of service

• Economic loss due to reconstruction of the damaged infrastructures

• Economic loss due to repairing and replacing broken bridges. Also, the indirect costs due to bridge closures during the reconstruction time

• Restrain the navigability of the rivers and canals because of the failure of dams, levees and reservoirs due to ground shaking and liquefaction • Changing in channel depth or the river

depth caused by uplift and subsidence • Channels blocking due to collapse of

bridges

• Fire following earthquakes at port facilities

• Damaging port facilities

• Increasing in lateral flows that could block channels caused by liquefaction

• Direct and long-term consequences for the national economy if rivers, used for transport are no longer navigable after the earthquake • Economic impacts due to loss of

functionality of port facilities

Aviation

• Damaging of the airport facilities, especially runways, power, communication, and radar components • Damaging of terminals and control

towers

• Fire and explosion as a result of rupture to the tanks which are used for storing the airplanes fuel and underground pipelines which transfer the fuel to airplane gate areas

Airports and heliports which are located near the epicenter of the earthquake may be inoperable for weeks after the earthquake.

Consequently, they will not be available for incoming and outgoing flights in the immediate response phase aftermath of the disaster

Hurricane Railway

• Destruction of railroad bridges • Because of capability for accurate

forecasting hurricane, generally the railroads are prepared for the hurricane. Many railroads evacuate their staff and equipment prior to the storm. Also, they put contingency measures in place, rerouting trains and stopping traffic on lines in the path of the hurricane. Although, strong hurricane causes significant impact on the railroad facilities and bridges

• Requiring the rerouting of traffic and putting increased strain on other rail segments

• Economic cost to local businesses, industrial sectors, or the national economy

• Problems related to transporting if railroads connect with the ports in the region and are used for carrying international cargo shipments and local commodities • Reconstruction of the rail

infrastructure in severe hurricane will take weeks or months. Although, overall losses of operating revenue will be reduced in case of redundant rail network which allows trains to be routed through other gateways

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• Devastating impact on most of the transportation infrastructure

• Damage to highway road surfaces in the beachfront region

• Disruption of key crossings which leads to restraining travel between different areas for an extended period

• Damage to bay and river crossings throughout the region by the storm surge and wave action

• Displacement or misalignment of bridges' spans by wave action due to hurricane

• Although the effects are limited in some locations and damage is repaired within days, in some coastal sections, prominent elements of the transportation network will remain closed many months after the storm

• Sometimes the major crossings remain completely closed and construction is expected to prolong

• After the storm, all traffic will be rerouted. This causes extensive delays to traffic moving

• Removing many feet of sand from the road surface in some places will take weeks

• Significant damage to the ports due to their low-lying locations. The ports are susceptible of damages because of strong winds, heavy rains and storm surge. These affect port operations and cause long term power outages and waterway hindrances

• Missing, relocating and destroying the coast guard’s navigation aids (the system of buoys, beacons, and lighthouses that facilitate safe navigation)

• Network of ports, harbors, and navigable waterways is vitally important to the regional and national economy. Prolonging the reconstruction and reopening the ports leads to devastating economic loss

• Ports serve as the main points for waterborne transportation of freight to different regions, and their transportation activities have a big role in economic benefits of each country. Ports are the primary gateway in most of the countries for imports and is the main transshipment point for different commodities. It will take nearly weeks to clear mud and debris from port facilities, and to fully evaluate and restart necessary machineries

Aviation

• Significant damage to airports (passenger terminals, maintenance facilities, and navigational equipment) due to strong winds, flooding rains and tornadoes associated with hurricanes. Furthermore, power outages take air traffic control facilities off-line and obscure night-time runway. As a result, some airports will be closed for even weeks

• Spreading debris across the runways by strong winds

Disrupting operations for several days whilst repairs were carried out

Table 1. Natural Hazards and their effects on different transportation modes.

• Flood impact on subway network

As an example, we consider in more detail an underground system in case of flooding. Since subway stations and rails are located underground, the water can flow into the stations and tunnels throughout the station accesses and tunnels if the waterproof facilities are not working properly. Among the many consequences of a flooded underground, we highlight:

1. Cancellation of some trips because either the origin station or the destination station is ﬂooded.

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3. Significant delay in several trips that occur due to the ﬂooding. This may happen either because travelers are forced to take indirect routes from origin to destination to avoid flooded links, or as a result of increased strain on passable links.

According to [2], 13 flood incidents which occurred in subway networks, are reported in Table 2.

Date City Cause Consequence

2001 Taipei (Taiwan) Typhoon Water flew into stations and shopping area via tunnel 2002 Prague Rainfall About 20 km of lines and 19 stations were soaked 2004 New York, (USA) Hurricane Water flew into stations and several subway lines were _closed 2007 New York, (USA) Rainstorm The subway system heavily flooded

2008 Beijing, China Rainstorm Line 2 out of service

2011 Beijing, China Rainstorm Cable was soaked in rainwater and caused a short circuit 2011 Nanjing, China Rainstorm Water flew into the tunnel

2012 Beijing, China Rainstorm Water caused the induction board work improperly 2012 New York, (USA) Hurricane Many subway tunnels were inundated with flood water 2014 Busan, South Korea's Rainstorm Some sections of line1, line 2 and line 4 were flooded 2014 New York, (USA) Rainstorm Some subway stations flooded

2016 Guangzhou, China Rainstorm Water flow into Changban station through entrance 2016 Wuhan, China Rainstorm Several stations of line 4 were flooded

Table 2. Flood accidents of subway stations around the world.

• Earthquake impact on subway network

Tectonic activity is the reason of the most serious geophysical disasters. Earthquakes are outstanding forms of geophysical threats due to the fact that they are unpredictable. Fortunately, they are focused in vicinity to boundaries of tectonic plates. Tsunamis, which are generally provoked by earthquake, are also considered an emerging risk. The 2011 Tohoku earthquake in Japan is one of the largest recorded earthquakes. However, its associated tsunamis caused the most extensive damage to Japanese transport infrastructure [1]. Earthquakes could be a threat for the safety of subway stations, tunnels and passengers. Although, engineers generally believe that the earthquake has limited impact on the underground structures [2]. Earthquakes with different magnitude influence the subway systems in different ways. Usually a smaller earthquake may only force the subway train slow down or stop. But a larger earthquake can cause the overturn of the station’s columns, or even the destruction of the whole station. As an example, in the case of an earthquake in Los Angeles County, Metro’s Rail Operations Center quickly notifies train operators that there has been an earthquake. All trains are stopped at the next station (if not already in one) and then they are put on restricted speed to allow the train operator to inspect the tunnel for any signs of trouble or damage. In addition, operation staff ensures all the signaling and electronic systems work properly. Once metro operation staff ensures that no damage has taken, trains can return to their normal speeds. In case of a larger magnitude earthquake, additional steps may be taken. These could include suspending service to allow structural engineers to thoroughly inspect tracks and tunnels [3].

According to [2], some cases of subway networks which were affected by earthquakes are listed in table 3.

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Date Location Magnitude _(Richter) Consequence

1985 Mexico City, _(Mexico) 8.1 disruption occurred. Passengers were evacuated safely. All lines of metro were out of service. Some power Subway used to transport rescue personnel 1989 Loma Prieta, (USA) 6.9 No damage to tunnels. Subway served as a lifeline _structure 1994 Northridge, (USA) 6.7 No damage

1995 Kobe, (Japan) 7.3 5 stations damaged, about 3km tunnel collapsed 2002 Taipei, (Taiwan) 6.8 No damage

2003 Sendai, (Japan) 7 Sendai Subway were out of service

2010 Chile 8.8 Running next day. Some damages occurred at entrance _{of the stations} 2014 Hualien, (Taiwan) 5.9 Taipei MRT were out of service

2015 Kanto, (Japan) 5.6 Tokyo subway were out of service 2015 Afghanistan 7.8 Delhi subway out of service, some people trapped

Table 3. Cases of subway networks which were affected by the earthquake.

Disruption due to “Intentional Attacks”

Intentional attacks are defined as targeted destructions caused by outside artificial forces. Targets of intentional attacks are in general, very important stations or trucks so to disturb the functionality of the entire network for a long time.

Countering terrorism (domestic and transnational terrorism) has become, without any doubt, the top national security priority for the world nowadays. According to the report of the Institute of Economics and Peace (2016) (28), the global economic impact of terrorism reached US$89.6 billion in 2015. The economic impact has decreased by 15% from its 2014 level, because of the overall reduction in the number of people killed during terrorist attacks. However, the 2015 economic impact of terrorism was still at its second highest level since 2000. The economic costs resulting from terrorism have grew approximately eleven-fold during the last 15 years. Transportation systems are one of the sectors of the economy that suffers the most because of terrorism. The economic impact of terrorism is calculated using IEP’s (Institute for Economics and Peace) cost of violence methodology. The model considers both the direct and indirect costs such as the lost life-time earnings, cost of medical treatments and property destruction from incidents of terrorism. The direct costs include those borne by the victims of the terrorist act and associated costs, such as medical spending. The indirect costs include lost productivity and earning as well as the psychological trauma to the victims, their families and friends. In Table 4, some recent severe terrorist attacks to different modes of transportation system are reported.

Place Year Transportation’

s Mode Type Consequences on the network

Saint Petersburg,

Russia 2017 Metro Explosive Device

St Petersburg’s subway system carries 2 million passengers a day and is busy most of the time. The whole system was closed in the aftermath of the blast, and several streets at ground level were also shut off, as medical helicopters landed at the scene to evacuate the injured. By evening, several metro lines had reopened. The driver of the train won praise for deciding to continue to the next station, Technologichesky Institute, rather than stopping in the tunnel, a move that investigators said probably saved lives and made it easier for

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rescuers to reach the injured individual.

Orly International

Airport, Paris 2017 Airport

Gun Shooting

This attack led to temporary closure of the airport and the evacuation of about 3,000 people. All flights to and from Orly were suspended and some were diverted to the larger Charles de Gaulle Airport east of Paris. The airport's west terminal was fully reopened by early afternoon, but air movements at the south terminal remained partly suspended with only incoming flights being permitted.

London Bridge,

London, England 2017 Road Ramming

Vehicle-All buildings within the vicinity of London Bridge were evacuated, and London Bridge, Borough and Bank Underground stations were closed at the request of the police (On 3 June 2017). The mainline railway stations at London Bridge, Waterloo East, Charing Cross and Cannon Street were also closed. London Bridge mainline railway and Underground stations remained closed throughout 4 June, while Borough tube station reopened that evening. A cordon was established around the scene of the attack. London Bridge station reopened at 05:00 on Monday 5 June. New security measures were implemented on eight central London bridges following the attack to reduce the likelihood of further vehicle attacks, with concrete barriers installed. The barriers caused severe congestion in cycle lanes during peak hours.

Brussels Airport in Zavent

em, Belgium 2016 Airport

Suicide Bombings

The airport was closed on 22 March. On 29 March, an operational test was performed. A post-reopening target of 800–1,000 passengers per hour was projected, compared to pre-bombing traffic of 5,000 passengers per hour. The delay in the reopening was attributed to extensive damage to the building's infrastructure. A temporary terminal was planned for use after the reopening. When the airport reopened, only Brussels Airlines would serve the airport, but other airlines would be allowed to return later. Airport businesses were affected. Cargo flights resumed on 23 March. Car rental offices were also closed. On 30 March, plans to reopen the airport were cancelled again due to a strike by airport police over a dispute over inadequate security. The dispute was resolved, and the airport was later scheduled to be reopened on 3 April. On that day, a Brussels Airlines flight left for Faro and a flight to Athens and another flight to Turin was scheduled for the same day. Upon reopening, only passengers could enter a temporary departure hall and security checkpoints were implemented at the roadway to the airport. Only car and taxi traffic could enter but public transit remained suspended. Hotel business revenue in Brussels had been cut in half since the airport closure. On 1 May, the departure hall of Brussels Airport, which had sustained the most damage during the bombings, partially reopened.

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Maalbeek metro station in

central Brussels 2016 Metro

Suicide Bombing

The station was closed for over a month following the attacks. On 25 April 2016, the Maalbeek station reopened again.

Table 4. Some recent terrorist attacks and their consequences on different transportation modes.

• Impact of intentional attack on subway network

As an example, we thoroughly study underground networks in case of terrorist attack and/ or setting intentional fire. Reports on known cases show that underground systems are one of the preferred targets of terrorists. It is primarily due to the potential for disruption, destruction, openness, accessibility, lack of passenger identiﬁcation, and weakness of security caused by the significant number of passengers using the network on a daily basis [4]. The result of such attacks on a hub station would be devastating since they can become very crowded especially during peak hours. According to the statistics of the attack cases, placing explosives, suicide bombings and release poisonous gases are the three main types of underground attack tactics used by terrorist. The proportion of each kind of attack is presented in Fig. 1[2].

Figure 1. Proportion of attack tactics in underground systems.

According to [2], some of the terrorist attacks occurred in underground systems from 1976 to 2016 are listed in Table 5.

Date City Tactics Deaths Injuries

1976.01.12 New York, (USA) Explosives 0 0 1977.01.08 Moscow, (Russia) Explosives 7 37 1986.09.04 Paris, (France) Explosives 0 0 1990.07.06 London, (England) Explosives 0 0 1994.03.19 Baku, (Azerbaijan) Suicide Bombings 14 49 1994.06.03 Baku, (Azerbaijan) Remote-Detonated Explosive 13 42 1995.03.20 Tokyo, (Japan) Deadly Gas Sarin 12 near 6000 1995.07.25 Paris, (France) Explosives 8 80 1995.08.26 Lyon, (France) Explosives 0 0

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Table 5. Subway terrorism attacks around the world.

• Fire due to intentional attack or random failure in the subway network

In comparison with other means of transportation, underground systems usually have good safety records. However, because of the large number of persons potentially involved in the case of a ﬁre incident, the possibility of damage is high. Kings Cross (London, 1987, 31 fatalities), Baku metro (Azerbaijan, 1995, 286 fatalities), Kaprun funicular tunnel (Austria, 1996, 155 fatalities), and Daegu metro (South Korea, 2003, 192 fatalities) are examples which demonstrate the high damage possibility in case of severe metro accidents, particularly for large ﬁres involving several trains [5]. Some incidents caused by fire among the many ones are reported in Table 6, whose content is extracted from [2].

1995.10.17 Paris, (France) Explosives 0 29 1998.01.01 Moscow, (Russia) Remote-Detonated Explosive 0 3 1998.06.03 Caracas, (Venezuela) Explosives 0 0 1998.07.02 Caracas, (Venezuela) Explosives 0 0 1999.02.17 Istanbul, (Turkey) Fire or Firebomb 0 0 1999.04.11 Moscow, (Russia) Explosives 0 0 2000.08.20 Caracas, (Venezuela) Explosives 0 0 2000.08.23 Caracas, (Venezuela) Explosives 0 0 2001.08.02 Caracas, (Venezuela) Explosives 0 0

2001.08.14 Rome, (Italy) Explosives 0 0

2001.09.02 Montréal, (Canada) Poisonous Gas 0 over 40 2002.03.02 Medellin, (Colombia) Explosives 0 9 2002.05.12 Milan, (Italy) Gas Canister (for the kitchen) 0 0 2002.06.02 Lyubertsy, (Russia) Remote-Detonated Explosive 0 1 2002.12.01 Caracas, (Venezuela) Explosives 0 0 2003.07.16 Moscow, (Russia) Explosives 0 0 2004.02.06 Moscow, (Russia) Suicide Bombings 40 120 2004.03.11 Madrid, (Spain) Explosives 192 2050 2004.08.31 Moscow, (Russia) Explosives 10 50 2005.07.07 London, (England) Suicide Bombings 56 784 2006.07.11 Bombay, (India) Explosives 209 over 700 2008.11.25 St. Petersburg, (Russia) Explosives 3 2 2010.03.29 Moscow, (Russia) Suicide Bombings 40 102 2011.04.11 Minsk, (Belarus) Explosives 15 204 2014.06.25 Cairo, (Egypt) Explosives 7 over 100 2014.09.08 Santiago, (Chile) Explosives 0 14 2015.12.01 Istanbul, (Turkey) Explosives 2 1 2016.03.22 Brussels, (Belgium) Suicide Bombings 35 340 2016.09.28 Tokyo, (Japan) Unknown Gas 0 9

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Date City Cause Deaths Injuries

1969 Beijing, (China) Electric locomotive failure 6 200

1973 Paris, (France) Arson 2 0

1974 Montréal, (Canada) Short circuit 0 0 1975 Boston, (USA) Lighting line failure in tunnel 0 0 1976 Lisbon, (Portugal) Locomotive failure 0 0

1976 Toronto, (Canada) Arson 0 0

1978 Cologne, (Germany) Cigarette butts 0 8 1979 San Francisco, (USA) Short circuit 1 56 1979 Paris, (France) Short circuit of car 0 26 1979 Philadelphia, (USA) Transformer failure 0 178

1979 New York, (USA) Cigarette butts 0 4

1980 Hamburg, (Germany) Seat catch fire 0 4 1980 London, (England) Cigarette butts 1 0 1981 Moscow, (Russia) Circuit failure 7 0 1981 Bonn, (Germany) Staff operation mistake 0 0 1982 New York, (USA) Transmission equipment failure 0 86

1982 New York, (USA) Arson 0 0

1982 London, (England) Short circuit 0 15 1983 Nagoya, (Japan) Rectifier failure 3 3 1983 Munich, (Germany) Circuit failure 0 7 1984 Hamburg, (Germany) Seat catch fire 0 1 1984 London, (England) Warehouse catch fire 0 18

1985 Paris, (France) Garbage on fire 0 6

1985 Tokyo, (Japan) Locomotive bearing failure 0 0 1987 London, (England) Technical failure 32 over 100 1991 Zurich, (Switzerland) Short circuit of subway locomotive 0 58

1991 Berlin, (Germany) Arson 0 18

1994 Taipei, (Taiwan) Substation catch fire 0 3 1995 Baku, (Azerbaijan) Caused by electrical malfunction 289 270

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1995 Daegu, (South Korea) Gas pipeline leakage 103 230

2001 Sao Paulo, (Brazil) Unknown 1 27

2003 Daegu, (South Korea) Arson in the subway train 192 148 2003 Shanghai, (China) The fire of neighboring mall 0 0

2004 Hong Kong Arson in the car 0 14

2005 Beijing, (China) Unknown 0 0

2006 New York, (USA) Unknown 0 15

2012 Kiev, (Ukraine) Chandelier line catch fire 0 0

2013 Moscow, (Russia) Short circuit 0 52

2014 Nanjing, (China) Flammable material brought by passenger 0 0

2015 Washington, (USA) Unknown 1 2

2016 Shanghai, (China) Neighboring shop catch fire 0 0 2016 Tokyo, (Japan) Something burns at vent 0 0

Table 6. Underground fire incidents around the world.

The proportion of different causes for each typology of fire accident is reported in Fig. 2. As shown in the figure, power line failure, mechanical equipment failure and arson are among the top three causes of fire in the underground system, with a proportion of 26.19%, 16.67% and 14.29% respectively [2].

Figure 2. Proportion of fire causes.

Electrical devices/sensors or power lines installed in the tunnel, station hall, platform and train are other fire hazards. Any malfunction such as short circuit, overload or any other equipment failure could lead to a devastating fire disaster. To prevent such incidents, regular inspection of the equipment and power lines should be carried out. To avoid fire caused by arson, security check is a viable solution to keep flammable material out of a subway station [2]. Since subway systems are

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used by a large number of travelers every day, security checks for both passengers and luggage are necessarily limited but can be increased in some special days or circumstances.

Disruption due to “Random Failure”

Since it is almost impossible to specify the corresponding destructive power for each random failure, therefore, we describe a random failure as dysfunction of a network due to failure on one or several nodes with random probability of occurrence. Due to linkages between the components of a transportation network, the failure of a component may affect the normal functionality of other components or even the whole system.

Technical malfunctions such as power failure, gear failure, brake failure, operational mistakes by staff or drivers, temporary suspension of service for special activity or maintenance or safety inspection, are examples of random failure.

Definition of some integrated concepts to adverse

transportation events

Any kinds of unfavorable transportation event can be analyzed from three aspects. Hazard, risk and resilience or its inverse which is vulnerability. Therefore, it is worth briefly explaining each of these concepts.

Hazard

A hazard in transportation field can be considered as a source or a situation with the potential for harm in terms of fatality or human injury, damage to public or private property, damage to the transportation network or its surrounding environment, or a combination of these. We can name different environmental threats such as flood, landslide, fog or a truck which carries dangerous goods as examples of transportation system hazards. During the study of a transportation event through hazard analysis, all kinds of threats to a transportation network, its users, and its surrounding environment, people and properties should be defined.

Risk

Risk is the chance of occurring an adverse event that will have negative effects on transportation system and its users. The level of risk demonstrates: the chance of the occurrence of an unwanted event as well as the potential consequences of the unwanted event on the network and its users.

Resilience

Resiliency can be defined in several ways. As an example, one definition is the ability of a transportation network to absorb disruptive events agilely and return itself to a level of service equal to or greater than the pre-disruption level of service within an acceptable time frame [6]. Therefore, resiliency is the ability to minimize the costs of a disruptive event from economic, social and operational point of view.

Engineering analysis of resiliency in a transportation system involves the use of quantitative metrics to define the state of the system prior to a disruption, the extent of the disruption, the rate of recovery and the time it will take for the system to be fully recovered to normal functionality. Defining the resilience of a network is crucial for effective mitigation planning before the disruption and optimum resource allocation in the aftermath of the disruption [7].

A comprehensive information about the resiliency of a system can be achieved from its recovery curve which represents the functionality of the system prior, during and after disruption.

Researchers who have studied the resilience of networks have proposed many quantitative measures such as reduced capacity, speed, cost, delay and travel time. However, a single metric is not able to present all the characteristics of a recovery curve. Therefore, resilience metric(s) must involve all the relevant characteristics of the recovery curve to measure the resilience of the system accurately.

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Fig. 3 shows a typical recovery curve and the resilience of the system which is typically quantified as a function of the shaded area [9]. An external shock (e.g. an intentional attack) at time 𝑡_𝐼 causes an instantaneous decline in the system state Q(t), (e.g. the system functionality). The intensity of the event and the characteristics and state of the network prior to the event are factors which define the residual system state, 𝑄𝑟𝑒𝑠.

Q (𝑡_𝐿) is the desired system state at the end of the recovery process. Q (𝑡_𝐿) could be either equal or higher than the original functionality. The accessibility of damaged components of the network, prepared recovery plans, weather condition during recovery and the availability of resources needed for the recovery process are examples of influential factors on network resiliency. Therefore, the impact of these resilience influencing factors, will be demonstrated through the recovery curve shape and the recovery time, 𝑇_𝑅 = 𝑡_𝐿 − 𝑡_𝐼.

According to [9], the typical resilience metric is defined by Eq.1. R = ∫ 𝑄(𝑡) 𝑑𝑡 𝑡𝐿 𝑡𝐼 𝑇𝑅 = ∫₀𝑇𝑅𝑄̂(𝜏) 𝑑𝜏 𝑇𝑅 (1) Where 𝜏 = t - 𝑡_𝐼 and 𝑄̂(𝜏) = 𝑄(𝑡)

The problem about Eq.1 is that it gives the same value of resilience R for different combinations of Q(t) and 𝑇𝑅. For example, we consider the three possible recovery curves in Fig. 4 [9]. These three

different curves correspond to different resilience level although the obtained R for the three recovery curves are all equal to 0.75.

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Figure 4. A comparison between three different recovery curves.

Different types of measures may be used with respect to different types of disruptive events, modeling/analytical technique, transportation modes, and the study objectives. Therefore, there is no comprehensive metric to represent network’s resiliency at the present day. Some researchers [8] suggested to use a metric of multiple measures. The proposed model to measure resilience by [8] is defined by Eq.2.

𝑅_𝑒 = 𝑇𝑖 + 𝐹𝛥𝑇𝑓 + 𝑅𝛥𝑇𝑟

𝑇𝑖 + 𝛥𝑇𝑓 + 𝛥𝑇𝑟

(2) where for any failure event (f), the corresponding failure F is defined as:

F = ∫ 𝑓𝑑𝑡 𝑡𝑓 𝑡𝑖 ∫𝑡𝑓𝑄𝑑𝑡 𝑡𝑖 (3)

And recovery proﬁle R is defined as the following: R = ∫ 𝑟𝑑𝑡 𝑡𝑟 𝑡𝑓 ∫𝑡𝑟𝑄𝑑𝑡 𝑡𝑓 (4) 𝑡_𝑖 = Time to incident 𝑡_𝑓 = Time to failure 𝑡_𝑟 = Time to recovery 𝛥𝑇_𝑑 = Disruption duration 𝛥𝑇_𝑓 = Failure duration 𝛥𝑇𝑟 = Recovery duration

The failure proﬁle value(F) can be considered as a measure of robustness and redundancy and the recovery proﬁle value (R) can be considered as a measure of resourcefulness and rapidity.

In [10], instead, the authors propose to form an index, which is a combination of several measures. According to [10], a set of variables that indicates infrastructure characteristics and user behavior of a transportation system was identified via analysis of previous research and nine of them were selected. They are: Road Available Capacity, Road Density, Alternate Infrastructure Proximity, Level of Intermodality, Average Delay, Average Speed Reduction, Personal Transport Cost,

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Commercial/Industrial Transport Cost and Network Management. Therefore, any change in discussed matrices will influence Transportation Network Resiliency Index.

Reliability and Robustness

Another aspect that need to be considered is network reliability. According to [12], ‘‘A network is reliable if the expected trip costs are acceptable even when users are extremely pessimistic about the state of the network’’. In [12], the author enlightens that reliability has two aspects: network connectivity and performance reliability. When analyzing network connectivity, the more sparsely connected the network, the more diﬃcult it may be for travelers to arrive at their destinations on schedule in the case of segment blockages or failure. However, the evaluation of reliability is onerous, as it depends on both the physical infrastructure and the behavioral responses of travelers. Survivability, or robustness of a transportation network, measures the performance of a network in various accident scenarios. Robustness is defined in terms of the alternative routes which the network offers under fluctuations in traffic demand. According to [13]: “Robustness indicates that the system will maintain its functionality intact (remain unchanged or nearly unchanged) when exposed to disturbance.”

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Section 2: Review of existing deﬁnitions and

metrics

In order to avoid the high socio-economic costs that are associated to the previously illustrated failures, we have to define the critical components of networks, and simulate the response of networks when those components stop working.

Graphs are among the most commonly used tools for representing the topology of different types of transportation systems, where nodes are stations (in a transit network) or airports (in an aviation networks), and edges are links in a transit network or ﬂight connections between two airports. At a first stage of analysis, any kind of disruption in network can be represented as the cancelation of a node or an edge in the associated graph. As mentioned in [14], the impact of the failure can be computed by comparing the new graph with the baseline one. Also, through graphs analysis we can find and modify the most vulnerable parts of the correspondence network to increase its general robustness.

Graph definition

Here, it is worthwhile to mention some basic concepts about graphs, where a graph is a couple of two sets, V and E. The set V contains the vertices of the graph, while the set E contains the edges, that is to say, unordered pairs of distinct vertices. Since we do not allow edges going from a vertex to itself, there is no loop in graphs we consider. Furthermore, we do not consider multiple edges between the same two vertices, and edges have no orientation. Hence, with graph, we mean simple graphs, as they are called in literature.

If we assume that V contains N vertices, it is known that there are at most N(N-1)/2 edges.

Lastly, G is connected if for every pair of vertices u, v there is a path from u to v, where a path is a sequence of pairwise different vertices 𝑣₀, 𝑣₁, . . . , 𝑣_𝑟 such that u = 𝑣₀, v = 𝑣_𝑟, and( 𝑣_𝑖, 𝑣_𝑖+1)is an edge.

Sometimes, it can be useful to consider weighted graphs, where a weight on an edge (on a node, respectively) is a number associated with that edge (with that node, respectively).

It will be advantageous to represent a graph through its adjacency matrix. Given a graph G = (V, E), we number its vertices as 1, 2, …, N. The adjacency matrix A(G) of G is a N x N matrix whose elements are:

𝐴(𝐺)𝑖𝑗={

1 𝑖𝑓 {𝑖, 𝑗} ∈ 𝐸 0 𝑖𝑓 {𝑖, 𝑗} ∉ 𝐸

When we have an undirected graph, then 𝑎_𝑖𝑗=𝑎_𝑗𝑖 and the adjacency matrix will be symmetric. In a graph with no loops, all the diagonal entries of the adjacency matrix will be 0. Since we consider simple graphs, A(G) is a symmetric matrix with zeros along its main diagonal. If the edges of G have a weight, then 𝑎_𝑖𝑗 is equal to the weight of the edge, if {𝑖, 𝑗} is an edge, 0 otherwise.

Let G be a graph, and let v be a vertex. Another vertex u is a neighbor of v if the pair u, v is an edge. The degree of a vertex v is the number of its neighbors. If G has N vertices and M edges, then

2 M = ∑_𝑣∈𝑉𝑑𝑒𝑔(𝑣) (5)

We call graph G dense, if the number of edges of G is close to 𝑁 × 𝑙𝑜𝑔(𝑁), where N is the number of vertices. Otherwise, we call it a sparse graph.

As mentioned in [14], a transportation network can have both one-way links, and two-way links. When the network has one-way links, it should be represented as a directed graph. However, underground networks have (in most cases) two-ways links only, and so we can represent them as graphs in the sense we adopt. Notice that also the air transport network can be represented as a

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graph in our sense. The main difference between subway and air transport networks is that the first network can be represented as a planar graph, because crossings are associated to stations, in general.

Since many real networks are significantly heterogeneous in terms of different properties of their links and nodes, it would be better to represent them as weighted graphs. In a graph representing a transportation network, possible weights can be the number of passengers, time, speed or cost of traversing each link, geographical distances, or it could be a function of two or more of the above possible weights, depending on the amountof details taken into consideration.

Indices

In order to assess the consequences of a disruption or failure in a component of a network, we could consider a random disruption or a targeted one. In a random disruption scenario, every component has the same probability of malfunctioning, while in a targeted disruption the most crucial components of the network will be chosen to fail. In order to find the most crucial components of a network, and to evaluate the effects of malfunctions on the performances of networks, we have to settle measures on graphs.

Centrality indices

Centrality is a fundamental concept in network analysis. “Centrality measurements” are used to locate the most important components of the network and can help in explaining the network’s dynamics. Various indices have been proposed to measure Centrality. In this thesis, we report on five centrality indices: Degree, Closeness, Betweenness, Eigenvector and Bonacich centrality are reported.

Degree Centrality

In degree centrality, the importance, or centrality, of a node is measured by the number of nodes it immediately influences, and therefore, it is strongly related to the degree of a node. As previously said, the degree of a node is defined as the number of its neighbors. Nodes with a high degree are often called hubs. In contrary with peripheral nodes, which have limited impact on the dynamics of the network, nodes with high degree centrality or hubs have a major effect on the functionality of the whole network. As a result, perturbations in hubs cause broader damages to networks.

As shown in Eq. 6, degree centrality of a node in a graph G, can be computed as the marginals of the adjacency matrix A of graph G [27]:

𝐶_𝑖𝐷𝐸𝐺=∑ 𝑎𝑗 𝑖𝑗 (6)

In the following example, the degree of node C is the sum of the elements on the third row or third column of the graph G’s adjacency matrix.

Assessing criticality in subway networks for risk prevention through topological performance indices

A

SSESSING CRITICALITY IN SUBWAY

NETWORKS FOR RISK PREVENTION

THROUGH TOPOLOGICAL PERFORMANCE

INDICES

Hiva Viseh

Prof. Lorenzo Mussone

Prof. Roberto Notari

DEDICATION

ACKNOWLEDGEMENTS

Abstract

Contents

List of Tables

List of Figures

Section 1: Consequences of natural and

manmade hazards threatening variant modes

of transportation network

Introduction

Disruption due to “Natural Disasters”

• Flood impact on subway network

• Earthquake impact on subway network

Disruption due to “Intentional Attacks”

• Impact of intentional attack on subway network

• Fire due to intentional attack or random failure in the subway network

Disruption due to “Random Failure”

Definition of some integrated concepts to adverse

transportation events

Hazard

Risk

Resilience

Reliability and Robustness

Section 2: Review of existing deﬁnitions and

metrics

Graph definition

Indices

Centrality indices

Degree Centrality