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The authors have declared that no competing interests exist.

Infrastructure systems are the structural backbone of cities, facilitating the flow of essential services. Because those systems can be disrupted by natural hazards, risk management has been the prevailing approach for assessing the consequences and expected level of damage. Although this may be a valuable metric, the practice of risk assessment does not represent how hazards affect a network of assets on a larger scale. In contrast, network topology metrics are useful because they evaluate the performance of network infrastructures by looking at the system as a whole. As described here, we began this study to improve our understanding of how flooding events affect the topological properties of road networks, in this case, the urban road infrastructure of Zurich, Switzerland. Using maps of flooding risk, we developed a procedure to extract the damaged networks and analyze the centrality metrics for peak water levels on the surface of the city. Our approach modelled roads as edges and junctions between roads as nodes. The betweenness centrality metric characterizes the importance of nodes or edges for any type of exchange within a network, whereas the closeness centrality metric measures the accessibility of a specific node to all the other nodes. This investigation produced three main findings. First, descriptive analyses showed that the characteristics and patterns of nodes and edges changed under the flooding events. Second, the distribution function of centrality metrics became heavier in the tails as the flood magnitude increased. Third, the associated strain shifted critical nodes to areas in which those nodes would not be important under normal conditions. These findings are essential for identifying crucial locations and devising plans to address risks. Future projects could expand our approach by including traffic flow to move the analysis closer to real-world flows, and by studying the accessibility under emergency conditions at local levels.

Along with clustering and the aggregation of assets in space, urbanization has been a driving force of global change. Within those clusters, the values at risk are continually increasing and–when exposed to patterns of natural hazards–have resulted in a tremendous rise in the costs associated with expected damage. Risk assessment is a method used to analyze the consequences to single assets or small sets of assets that are exposed to the pattern of a specific hazard. However, that approach does not take into account that assets are connected and that a triggering hazardous event can cause such damage to spread in a cascading manner through an entire network. Therefore, the question becomes how does one characterize the response of that network of assets when subjected to natural hazards. Network science views real systems as being formed by interacting parts that represent infrastructures. Information about their geographical locations and the relationships among roads is used to detect the robustness of roads. However, little is known about how natural hazards affect the topological properties of a network. The scientific literature has presented three research streams for addressing the response of real road networks to alterations in their topological structure. The first method examines scalar changes in centrality metrics to determine patterns of variation. Such research began during the second half of the 20^{th} Century, within a social systems context. For example, Bavelas [

Despite all of these advances, a coherent means is still lacking for characterizing changes in network properties under major disruptions. Taking up this challenge, we designed this present study to obtain a logical methodology for comparing the pre- versus post-event conditions of large-scale road networks. Our investigation included a descriptive analysis of centrality metrics and assessed the variability of those metrics as well as the spatial arrangement of critical network components. Out of necessity this project involved only a single geographical area, i.e., the city of Zurich, and focused only on flooding events.

Our study area was the city of Zurich, in northern Switzerland. This city represents an urban region, as delimited by administrative boundaries set in 1934. Its 12 districts cover approximately 88 Km^{2}, with a population of approximately 415,682 in 2016 [

We relied upon road data for Zurich from the Swiss Federal Office of Topography (Swisstopo), as well as flood hazard maps and digital elevation models (DEMs). For the road information, we used the swissTLM3D 1.1 data model for Year 2012 [

After pre-processing to convert geographical data into a network representation, we analyzed the processed networks with centrality metrics. Distributions were examined with ArcMap 10.4 software on the ArcGIS environment for the geographical analyses, the Arcpy and Igraph packages in Python for topological analysis, and R packages for the statistical analysis.

A network is defined as a set of points in a space connected by a set of edges that are links between node pairs. Modelling of network representations is commonly used to study systems having different natures. Here, we designated edges as roads and nodes as road junctions. As the primal representation, it is the opposite of a dual representation, where nodes would be roads, and road junctions, edges. The entity of a road can be examined from different perspectives [

Planners classify road types according to their widths, which characterize their physical form [

For this project, we developed three types of centrality analysis. In general, nodes and edges are the basic components of a network, and network topology is defined by the relationships between the total number of nodes

We defined normalized node betweenness centrality as (_{st} is the number of shortest paths going from a source node _{st}(i) is the number of shortest paths going from node

Normalized edge betweenness centrality is defined as (_{st}(e) is the number of shortest paths going from node

We calculated the normalized closeness centrality per (_{ij} is the shortest-path length between node

Our descriptive analysis characterized the changes in the numbers of nodes and edges and in the centrality values from baseline conditions to the two flooding events. During those events, the closure of a certain set of roads may have altered the centrality values at a single node/edge location. We identified four classes of changes: 1) node or edge flooded and, therefore, closed in the system under a flood scenario; 2) increase in the centrality value at a single node or edge from the baseline; 3) decrease in that value; or 4) no impact from flooding, so that the centrality value did not change from the baseline level. We calculated the differences between normalized centrality values under baseline conditions as well as in the flooding scenario, examining the entire network system (total number of roads in Zurich) because we wanted to investigate the overall trend among centrality values.

The distribution functions are used to systematically characterize the variability of the variable of interest, which, in this case, is the betweenness centrality. If we are interested in the tail of the distribution, which has been traditionally the case for risk management, we have to use specific models for accurately representing this tail. The extreme value distributions are used in a traditional risk management approach, whereas the power law distributions are popular in the field of complexity science. Power law distributions are defined as ^{−α} (4) when _{min}. Although the power laws cannot characterize the “left-hand” part of the distribution, they constitute a straightforward approach to characterize the upper tail using two parameters, the exponent alpha and the lower bound _{min} at which the power law distribution has to be cut off. The Kolmogorov–Smirnov (KS) statistic is a goodness-of-fit measure used in power law analyses [

The spatial analysis was conducted to visualize the centrality results on maps. After importing the results from the centrality calculations of Python on the ArcGIS environment, we joined those results with the node and edge layers so that each node/edge was associated with its centrality value. This produced shapefiles relative to each centrality value, which we could then display, as quantitative results, on maps or shapefiles that corresponded to actual geographical locations. For our purposes, high centrality values were critical because they referred to the most important nodes/edges. To identify their particular locations, we selected the 0.99-quantiles from the centrality results and presented only those on the map of Zurich.

After obtaining a descriptive characterization of network properties, we analyzed the statistical distributions of the centrality metrics to understand their variability in response to flooding events. Finally, we investigated the spatial distributions of the centrality results.

The descriptive analysis yielded node and edge characteristics, patterns for the two flooding scenarios (100 and 300 years), and the three centrality metrics. Although flooding considerably altered the topological properties of the network, a small percentage of nodes and edges could not be characterized by those scenarios. The overall flow capacity by the network was represented by betweenness centrality metrics that, when compared with baseline conditions, showed changes mostly at the nodes/edges. The baseline conditions included 6704 nodes and 9931 edges. The number of nodes decreased to 6676 (decline of 0.4%) for the 100-year flood and to 6472 (–3%) for the 300-year event. Meanwhile, the number of edges decreased to 9796 (–1%) for the 100-year peak and to 9381 (–6%) when compared with the baseline level. These results indicated that the flooded nodes and edges represented a small part of the network, and that major disruptions emerged during the 300-year peak event. Centrality metrics characterize the importance of nodes and edges for overall flow capacity. The transition from ‘normal’ to ‘flooded’ conditions results in one of four responses by single nodes/edges. As shown in

Node-Edge Transition | |||||
---|---|---|---|---|---|

100 | 2933 (44%) | 2886 (43%) | 857 (13%) | 28 (0.4%) | |

300 | 2453 (37%) | 3183 (48%) | 836 (12%) | 232 (3.0%) | |

100 | 4704 (47%) | 5091 (51%) | 1 (0%) | 135 (1%) | |

300 | 4496 (45%) | 4885 (49%) | 0 (0%) | 550 (6%) | |

100 | 75 (1%) | 6601 (98%) | 0 | 28 (0.4%) | |

300 | 75 (1%) | 6397 (95%) | 0 | 232 (3.0%) |

Four causes of change in nodes/edges were possible: 1) increase in centrality values, 2) decrease in values, 3) values maintained at the same levels, or 4) exclusion from the network because those nodes/edges were flooded. Changes were calculated as the differences between normalized values in a flood scenario and those under baseline conditions. Percentages indicate the ratio of the number of nodes (edges) to the total number of nodes (edges) when compared with baseline conditions, i.e., 6704 nodes and 9931 edges.

The empirical distribution functions of the centrality metrics for our three scenarios differed significantly in their tails. For the BC results, this demonstrated an increase in the heaviness of the tail at higher flood magnitudes. A heavier tail meant that either the significance of critical edges needed to maintain greater network performance was enhanced or the failure of some particular roads had a larger impact on the network. When we considered systems composed only of roads at least 3 m wide, we found that the 100-year flood distribution was greater than that of the 300-year flood. Our closeness centrality results also showed a decline in distribution values.

From a baseline of 4795 nodes and 6470 edges, the scenario featuring a 100-year flood event showed a decrease in node and edge numbers to 4773 (drop of 0.5%) and 6396 (drop of 1.1%), respectively, whereas the 300-year flood resulted in respective reductions in node and edge numbers (4572, down by 5%; and 5999, drop of 7%) when compared with baseline conditions. Here, we calculated NBC and EBC values by using the width capacities as network weights. Basic statistical analyses of the normalized centralities values showed that, under baseline conditions, the maximum values were 0.08 for the node betweenness centrality and 0.07 for the edge betweenness centrality. For both metrics, the maximum values were 0.12 for the 100-year scenario and 0.13 for the 300-year scenario. The standard deviation of NBC was approximately 8.4·10^{−3} for the baseline, 9.7·10^{−3} for the 300-year flood, and 1.3·10^{−2} for the 100-year scenario. For EBC, the standard deviation was approximately 6.8·10^{−3} for the baseline, 8.1·10^{−3} for the 300-year flood, and 1.1·10^{−2} for the 100-year scenario. For the node-BC, the 0.5-quantiles were approximately 9.4·10^{−4} for the baseline and it decreased to 8·10^{−5} in the 100-year flood, while in the 300-year flood to 4.4·10^{−5}. For the edge-BC, the 0.5-quantiles were approximately 6.8·10^{−4} under baseline conditions, but those decreased to approximately 6·10^{−4} for the 100-year flood and to 3.6·10^{−5} for the 300-year flood. While the 0.99-quantile of node betweenness centrality was around 4.3·10^{−2} in the baseline, and it increased to 5.6·10^{−2} in the 300-year scenario and to 7.1·10^{−2} in the 100-year scenario. While, when we compared the 0.99-quantiles of edge betweenness centrality, it was 3.4·10^{−2} in the baseline, 4.7·10^{−2} in the 300-year scenario and to 6.5·10^{−2} in the 100-year scenario. These results indicated that the values of the distributions increased in variability as flooding intensified, and that they became more broadly distributed during the 100-year flood.

The results of the normalized closeness betweenness centrality showed that the maximum values and the standard deviation decreased from the baseline condition to the 300-year flood, respectively from around 74 to 6 for the maximum and the standard deviation from around 8 to 2. As in the previous analysis, the quantiles always decreased values from the baseline to the 300-year peak. ^{−2}. Beyond that point, a switch occurred in the baseline and 100-year curves, with the latter showing higher probabilities than the former. A second switch in the distributed values occurred at approximately 2·10^{−2}, where the probabilities of the 300-year flood centrality values became higher than the probabilities of the baseline. The 100-year flood values had higher probabilities than the baseline or the 300-year flood scenarios. These findings meant that, when considering the network created to support vehicle flows, the critical locations associated with the 100-year scenario became more important than either the baseline conditions or the 300-year flood scenario, even though the degree of disruption was greater for the latter than for the 100-year flood.

Edge and node betweenness centralities were calculated using average road widths as network weights.

The parametric characterization of the upper tail of the NBC distribution functions yielded alpha values of 3, 6.4, and 1.9 for the baseline as well as the 100-year and 300-year flood scenarios, respectively. The corresponding values for the EBC distributions were approximately 2.5, 2.2, and 1.9, which were higher than those observed in other German cities [_{min} ranged from 0.003 to 0.06 for NBC and from 0.01 to 0.02 for EBC. The KS statistic is the maximum distance between the data distribution function and the power-law-fitted distribution. In case of NBC, the KS statistic was 0.07 in the baseline and 0.06 in the two flooding scenarios, whereas it was 0.05 in the baseline and 0.04 in the two flooding scenarios for EBC. Because the lowest KS values were observed in the flooding scenarios, BCs became closer to a power law distribution under the flood strain when compared with that in the baseline condition. We tested the hypothesis that the empirical distribution function is equal to the theoretical power law distribution function above the threshold value using the KS test. The critical distances indicated that the p-values were lower than 0.01, indicating that we should reject the hypothesis that the empirical distribution and corresponding power law distributions are obtained from the same population. The test of other distributions, exponential and lognormal, and the empirical distribution were from the same population had to be rejected, too.

In the third part of our analysis, the spatial pattern of nodes/edges with high relevance to overall network performance shifted from the baseline scenario to the two flooding scenarios.

During hazardous periods, emergency managers must identify which sequence of roads is critical for managing traffic flows inside a city. To test this, we first analyzed edge betweenness centrality and the spatial distribution of the edges at the tails, which corresponded to edges having normalized BC values equal to or higher than the 0.99-quantiles. We also examined how the pattern of critical edges changed for roads wider than 3 m. As shown in

For each network edge, EBC was evaluated by dividing number of shortest paths through edge by total number of shortest paths within network. Values of edge betweenness were normalized with (N)(N-1), where N was number of nodes under baseline conditions. Figure presents values equal to or larger than 0.99-quantiles. Reprinted from National Map 1:25000 on sheet 1091 under a CC BY license, with permission from the Federal Office of Topography Swisstopo (original copyright 2019).

As the second step in this test, we studied the spatial pattern of normalized NBC results, selecting nodes with values equal to or above the 0.99-quantiles, which represented the critical junctions between roads (

Results were calculated for system of roads at least 3 m wide. Betweenness centrality was evaluated for each network node by dividing number of shortest paths through nodes by total number of shortest paths within network. Values of betweenness were normalized with (N-1)(N-2), where N was number of nodes under baseline conditions. Figure presents values equal to or larger than 0.99-quantiles. Reprinted from National Map 1:25000 on sheet 1091 under a CC BY license, with permission from the Federal Office of Topography Swisstopo (original copyright 2019).

Finally, we considered the spatial distributions for CC (

Values for CC were calculated for each node, dividing by total sum of shortest-path lengths within network, and were normalized with (N-1), where N is number of nodes for given scenario. Figure presents values equal to or larger than 0.99-quantiles. Reprinted from National Map 1:25000 on sheet 1091 under a CC BY license, with permission from the Federal Office of Topography Swisstopo (original copyright 2019).

We used descriptive analysis, variability of the metrics, and spatial distributions on the entire road system structure of Zurich to determine how centrality metrics might be altered by an historic flooding event. Both the characteristics and patterns of nodes and edges changed in response to 100- and 300-year events. Here, the betweenness centrality metric characterized the importance of exchange within a network, whereas the closeness centrality measured the accessibility. However, when compared with the total number of nodes and edges existing under baseline (pre-flood) conditions, only a small percentage of them were affected due to the unique geography associated with this city. Flooding was restricted to areas near water bodies in the central and western areas—in particular, the alluvial fan of the Sihl River—whereas the eastern and northern areas were at higher elevations. A survey of nodes and edges indicated that changes were greater for the betweenness centrality metrics than for closeness centrality, and values for the latter mainly showed decreases. This was a result of flood-related road closures, which meant that the shortest-path lengths between node pairs could only increase from baseline conditions, and therefore, they have less influence on closeness centrality than on betweenness centralities. Most research has tended to focus on the unique influence BC can have when characterizing changes to road networks either during flooding events [

Our analysis of BC and the empirical distribution function also showed that graphed tails grew heavier as the flood magnitude increased. This response was reflected by the rises noted for standard deviations and the 0.99-quantiles. Furthermore, the KS statistic in case of power law distributions decreased in flooding scenarios, indicating that the tails were closer to the power law under the flood strain when compared with that in the baseline condition. As such, in the flooding scenarios, nodes or edges with high BC values were relatively more important than those calculated for the baseline network. In particular, the 100-year flood showed the heaviest distribution when we looked only at roads at least 3 m wide. Furthermore, closeness centrality distributions decreased as the flood magnitude increased. The most critical BC values were those closest to the greatest extremes because they referred to the nodes and edges with the highest frequency of shortest paths passing through, i.e., locations with the most abundant city traffic. Our results were also supported by those previously reported from studies of BC distributions for datasets in other cities. For example, Lämmer et al. [

For our third point, we conducted a spatial analysis of centrality metrics to show that the spatial pattern of centralities with values equal to or larger than the 0.99-quantiles shifted locations in the three scenarios. This meant that sites with a high impact on system performance changed their geographical positions during our test scenarios. The betweenness centrality results also proved that those sites were located on the main paths of Zurich under baseline conditions. This result confirmed those of [

Our findings have implications for planners, policy-makers, and scientists. Maps of flood risk provide information about tangible losses from a hazard event, with costs commonly being quantified according to the level of economic damage or the number of people affected. Our use of topological properties is another quantitative tool that can evaluate the impacts of a hazard on the capacity for traffic to move within a road network. Here, we relied upon changes in centrality to account for how the road system as a whole was altered. Planners can use those results to rank the relative importance of individual roads or, more generally, determine which areas will be most affected. From that, policy-makers can utilize the information to derive preventative actions for minimizing the impact of future flood hazards. For scientists, our study demonstrated that centrality metrics contribute to realizing the influence a distributed hazard has on a road network. Here, the analysis of BC distribution functions over time revealed that the effect of the two flooding events on the tails was similar to that seen as a road system ages. Therefore, topological metrics alone can help us detect any shift in properties as such a system changes.

This research project had some limits–first, because we used only Zurich as our case study. Because topological properties depend upon the particular geography of a city, different results might be obtained when centrality metrics are evaluated elsewhere. Furthermore, we did not use any traffic data but instead employed only data relative to the road structure. This approach meant that we omitted any information about congestion or the most common commuter routes within Zurich.

Future work might compare the effects of flooding on the topology of road systems characterized by different urban plans or geographical constraints. Incorporating traffic information would also aid in producing a more realistic view of conditions.

Finally, follow-up investigations could be used to characterize the spatial variability of centralities, especially the closeness centrality, in limited areas of the network. This approach can be applied by setting a radius from a selected node to study the centrality values in a circular area or by setting the topological radius. This would help us to investigate the manner in which the metrics would change at the local levels. From an emergency perspective, this approach can evaluate the performances of this approach in areas around places of interest such as hospitals or public facilities.

(7Z)

The authors thank Monika Niederhuber for her support in processing the network data.