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Schematic diagram of floods and demonstration of the effects of floods, random damage, and localized damage on a road system. a River and road network without flooding. b River and road network when rainfall-induced flooding inundates some road segments. c Sectional view of the boxed region in a. d Sectional view of the boxed region in b. The rainfall forms runoff on the ground. A large amount of runoff converges into a flood and inundates some road segments. The flood follows the river channel and damages infrastructure (e.g., roads) in a river basin. The flood-induced failures have a distinct trajectory. Schematic demonstrations of (e) random damage and (g) localized damage in a sketched network respectively. f 2D and h 3D views of flood disturbance in the sketched network

Schematic diagram of floods and demonstration of the effects of floods, random damage, and localized damage on a road system. a River and road network without flooding. b River and road network when rainfall-induced flooding inundates some road segments. c Sectional view of the boxed region in a. d Sectional view of the boxed region in b. The rainfall forms runoff on the ground. A large amount of runoff converges into a flood and inundates some road segments. The flood follows the river channel and damages infrastructure (e.g., roads) in a river basin. The flood-induced failures have a distinct trajectory. Schematic demonstrations of (e) random damage and (g) localized damage in a sketched network respectively. f 2D and h 3D views of flood disturbance in the sketched network

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Article
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The adverse effect of climate change continues to expand, and the risks of flooding are increasing. Despite advances in network science and risk analysis, we lack a systematic mathematical framework for road network percolation under the disturbance of flooding. The difficulty is rooted in the unique three-dimensional nature of a flood, where altit...

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Context 1
... flood, as a special and realistic type of disturbance, threatens the robustness of infrastructure systems 12,43,44 . As shown in Fig. 1, floods are locally destructive and the situation is similar to localized damage from this perspective. However, a flood can also affect the entire network owing to its wide spatial dispersion through rivers and the altitude of roads with a three-dimensional (3D) network structure, which is similar to the case of random damage. For ...
Context 2
... collected the maximum observed flooding from Dartmouth Flood Observatory data sets 52 . After inputting the flood information to the actual US road network, we use our failure model to identify road intersections that have directly and indirectly failed (as shown in Supplementary Fig. 1). We collected information of road closures reported by TranStar and flooded streets (road segments) reported by public media 53 in Houston and refer to these road closures and flooded streets as reported failures. ...
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... the unimportant nodes become the last straw that breaks the camels back and how these nodes combine with damage patterns to trigger abrupt systematic malfunctions call for more research in the broad field of disaster risk science. Comparing the affected population between China and the US (see Supplementary Fig. 10 and Supplementary Note 2 for details), we find that more highly populated counties are affected by floods more in China than in the US; counties with an extremely high population density ( [95,100] percentile) are more likely to be directly affected by floods in China; and more people are indirectly affected by floods in China. All these findings indicate that flood mitigation will be more challenging in China than in the US. ...
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... shown in Fig. 1, we use this sketch network (jN j ¼ 20) to compare the failures resulting from floods with those resulting from random damage and localized damage on the network. The nodes are removed as a result of any type of disturbance; i.e., the red nodes in Fig. 1e-g. We refer to these removed nodes as direct failures D. When one flood event ...
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... shown in Fig. 1, we use this sketch network (jN j ¼ 20) to compare the failures resulting from floods with those resulting from random damage and localized damage on the network. The nodes are removed as a result of any type of disturbance; i.e., the red nodes in Fig. 1e-g. We refer to these removed nodes as direct failures D. When one flood event occurs, we acquire the inundated nodes (direct failures) in the road network, remove them, and its size is jDj ¼ 5 (Fig. 1e). When comparing the effect of this flood event with the effects of random damage and localized damage, we set the fraction of direct ...
Context 6
... from random damage and localized damage on the network. The nodes are removed as a result of any type of disturbance; i.e., the red nodes in Fig. 1e-g. We refer to these removed nodes as direct failures D. When one flood event occurs, we acquire the inundated nodes (direct failures) in the road network, remove them, and its size is jDj ¼ 5 (Fig. 1e). When comparing the effect of this flood event with the effects of random damage and localized damage, we set the fraction of direct failures in total nodes, 1 À p ¼ 1 4 , for all three types of disturbance in this example. After removing these nodes (direct failures), some nodes are disconnected from the giant connected component of ...
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... nodes (direct failures), some nodes are disconnected from the giant connected component of the road system (i.e., the majority of the road system). This means the vehicles on these nodes cannot reach the majority of nodes in a network. These nodes are referred to as indirect failures and the set of these nodes is denoted I . In the example of Fig. 1, we find the fractions of indirect failures among all nodes are 11 20 , 1 2 , and 2 5 , and the network breaks into three distinct connected components (C), two distinct connected components (C & P), and only one connected component (P) for random damage, floods, and localized damage respectively. The joint set of direct and indirect ...
Context 8
... flood, as a special and realistic type of disturbance, threatens the robustness of infrastructure systems 12,43,44 . As shown in Fig. 1, floods are locally destructive and the situation is similar to localized damage from this perspective. However, a flood can also affect the entire network owing to its wide spatial dispersion through rivers and the altitude of roads with a three-dimensional (3D) network structure, which is similar to the case of random damage. For ...
Context 9
... collected the maximum observed flooding from Dartmouth Flood Observatory data sets 52 . After inputting the flood information to the actual US road network, we use our failure model to identify road intersections that have directly and indirectly failed (as shown in Supplementary Fig. 1). We collected information of road closures reported by TranStar and flooded streets (road segments) reported by public media 53 in Houston and refer to these road closures and flooded streets as reported failures. ...
Context 10
... the unimportant nodes become the last straw that breaks the camels back and how these nodes combine with damage patterns to trigger abrupt systematic malfunctions call for more research in the broad field of disaster risk science. Comparing the affected population between China and the US (see Supplementary Fig. 10 and Supplementary Note 2 for details), we find that more highly populated counties are affected by floods more in China than in the US; counties with an extremely high population density ( [95,100] percentile) are more likely to be directly affected by floods in China; and more people are indirectly affected by floods in China. All these findings indicate that flood mitigation will be more challenging in China than in the US. ...
Context 11
... shown in Fig. 1, we use this sketch network (jN j ¼ 20) to compare the failures resulting from floods with those resulting from random damage and localized damage on the network. The nodes are removed as a result of any type of disturbance; i.e., the red nodes in Fig. 1e-g. We refer to these removed nodes as direct failures D. When one flood event ...
Context 12
... shown in Fig. 1, we use this sketch network (jN j ¼ 20) to compare the failures resulting from floods with those resulting from random damage and localized damage on the network. The nodes are removed as a result of any type of disturbance; i.e., the red nodes in Fig. 1e-g. We refer to these removed nodes as direct failures D. When one flood event occurs, we acquire the inundated nodes (direct failures) in the road network, remove them, and its size is jDj ¼ 5 (Fig. 1e). When comparing the effect of this flood event with the effects of random damage and localized damage, we set the fraction of direct ...
Context 13
... from random damage and localized damage on the network. The nodes are removed as a result of any type of disturbance; i.e., the red nodes in Fig. 1e-g. We refer to these removed nodes as direct failures D. When one flood event occurs, we acquire the inundated nodes (direct failures) in the road network, remove them, and its size is jDj ¼ 5 (Fig. 1e). When comparing the effect of this flood event with the effects of random damage and localized damage, we set the fraction of direct failures in total nodes, 1 À p ¼ 1 4 , for all three types of disturbance in this example. After removing these nodes (direct failures), some nodes are disconnected from the giant connected component of ...
Context 14
... nodes (direct failures), some nodes are disconnected from the giant connected component of the road system (i.e., the majority of the road system). This means the vehicles on these nodes cannot reach the majority of nodes in a network. These nodes are referred to as indirect failures and the set of these nodes is denoted I . In the example of Fig. 1, we find the fractions of indirect failures among all nodes are 11 20 , 1 2 , and 2 5 , and the network breaks into three distinct connected components (C), two distinct connected components (C & P), and only one connected component (P) for random damage, floods, and localized damage respectively. The joint set of direct and indirect ...

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