Local floods induce large-scale abrupt failures of road networks

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 altitude plays a critical role as the third dimension, and the current network-based framework is unsuitable for it. Here we develop a failure model to study the effect of floods on road networks; the result covers 90.6% of road closures and 94.1% of flooded streets resulting from Hurricane Harvey. We study the effects of floods on road networks in China and the United States, showing a discontinuous phase transition, indicating that a small local disturbance may lead to a large-scale systematic malfunction of the entire road network at a critical point. Our integrated approach opens avenues for understanding the resilience of critical infrastructure networks against floods.

. We only consider road intersections in the giant connected component of the origin road network. Simulation results are the outcome of 20 independent global simulation runs and 20 regional simulation runs.

Supplementary Tables
Supplementary    The damage characteristics (behaviors) refer to giant component as a function of the fraction of removed nodes, also known as percolation theory [3].
When the fraction of removed nodes (direct failures) reaches a certain value 1 − p c , it leads to a percolation phase transition where the whole system will be completely fragmented and lose the function. This critical (percolation) threshold p c indirectly reflects the robustness of a road network. The behaviors due to random damage are very different from that due to localized damage on a road network since road networks are spatially k 0 −2 than to random damage p c = 1 (k 0 −1) , where k 0 denotes each node is randomly connected to k 0 other nodes; (3) Scale-free (SF) networks with power law degree distribution P (k) ∼ k −λ are more robust to localized damage than to random damage when λ > 3.825 and the opposite is true when λ < 3.825.

Unique features of flood effect
Floods, as a new and realistic type of network disturbance introduced in this paper, are more locally destructive and has stronger effect on a neighborhood or community than random damage and is not as simple as localized damage since rivers may spread the damages from one location to other locations. Therefore, the destructive effect of floods is somewhere between random damage and localized damage. Three-dimension is a hallmark of some types of network [6]. It is interesting that the 3-dimensional road network demonstrates major differences among flood disturbances and other damages (e.g. random damage, and localized damage). We take road altitudes into account to understand a road network's robustness to flood disturbances. In contrast, we only need to use a 2D road network to analyze its robustness to random damage and localized damage. As shown in Supplementary Figure 10(a), the vulnerability of county group with extremely high population density ([95, 100] percentile) to floods is significantly higher than that to random damage and localized damage. This is indispensable when making investment in infrastructure systems. If we can effectively prevent direct physical damage due to floods, we can efficiently reduce the total population losses resulting from floods in China.
Flood mitigation will be more challenging in China than that in the US.
We usually adopt various measures to control inundation (direct failures) rather than indirect failures in order to mitigate flood risk. However, China will suffer more indirect failures and have more affected population in contrast to the US, as shown in Supplementary Figure 10

Supplementary Note 3: Why CaMa-Flood model?
CaMa-Flood is a creditable global hydrodynamic model. It has been wildly used to simulate region and global river floods [7][8][9][10][11] and validated by various situ and satellite observations [12] and benchmark data sets [13] in major world river basins, such as Amazon Congo, Orinoco, Mississippi, and extreme events, including the 2007 Cyclone Sidr in Bangladesh [14]. This paper focuses on the failures of national scale road networks (of China and the US) due to floods with different intensities. With this purpose, the CaMa-Flood model is applied to produce different floods on large geological scales. More importantly, we attempt to introduce the flood model as the realistic perturbations in a network and develop the corresponding percolation theory, showing novel phase transition phenomena when compared with artificial perturbations. With different research objects, researchers can consider other hydrological models, such as LISFLOOD [15,16] and PCR-GLOBWB [17] to produce different flooding scenarios.