Vulnerability of cities to toxic airborne releases is written in their topology
As a starting point, we compare the vulnerability of four districts in Lyon, Paris (France), Firenze (Italy) and New York (US). These cities were chosen as emblematic of different topologies, resulting from different historical urban layering. The historic center of Firenze (panel b in Fig. 1) is mainly characterized by a dense urban fabric with a medieval signature of narrow and winding streets24. In Paris (panel c), Haussman’s renovation plan at the end of the 19th century supplemented the North–South and East–West ancient crossroad by a second network of concentric large avenues25. The rectilinear grid of Manhattan, New York, originates from 181126,27 and extends along the spine of Manhattan island (panel d). Despite the significant difference in size, a similar regular pattern is found in the modern urban area of Lyon (panel a), developed in the second half of the 19th century. In the insets of Fig. 1, we report for each city a polar histogram of the orientation of the streets. Although greater variability is observable for the orientation of the streets in the urban areas of Firenze (panel b) and Paris (panel c), two main orthogonal axes are found in the spatial structure of each city.The urban networks analysed in this work were delimited in order to be large enough to include the distinctive patterns of these four cities. The edges of the areas were traced along physical boundaries (e.g., rivers, parks, railways, large avenues) which act as elements of discontinuity in the dispersion process. Where not possible, the break was forced along wide streets.We promptly computed vulnerability maps for the selected urban areas by means of the centrality metric we derived in20 and recall in the “Methods” section. The nodes with the highest centrality values (V) are the most vulnerable as they correspond to the best spreading locations in the urban fabric. The spreading potential of a node is evaluated based on the extent of the area that is contaminated when the release takes place in this same node.We report in Fig. 1 the vulnerability maps of the four urban areas for the indicative scenario of a wind blowing at an angle (phi =45^circ). In the insets of Fig. 1, the wind direction is indicated with a red arrow. Given the different orientation and structure of the street networks, (phi) is defined as a clockwise angle with respect to the main axis of the city, which is identified as the longest bar in the polar histogram of street orientation.To extend the analysis to multiple meteorological scenarios, we estimated the vulnerability of each node (seen as a spreading source) for eight different wind directions ((phi =0^circ), (45^circ), (90^circ), (135^circ), (180^circ), (225^circ), (270^circ), (315^circ)). In this way, for each city, we obtained an extended dataset of vulnerability values that we represent in a compact way by means of a cumulative distribution function, as shown in Fig. 2a. The intercept of the cdf represents the nodes with null vulnerability. These are mostly located along the physical edges of the domain where the pollutant gas is blown away by the wind without affecting other streets. Where the delimitation of the network is forced (for example on the sides of central park as regards Manhattan), the interruption of the propagation, in the vulnerability model, is also constrained. This does not result in any artificial effect when the boundary is located upwind with respect to the network (propagation carries on from the boundary towards the considered urban area). On the other hand, when the boundary is downwind, vulnerability can be there underestimated. Considering the multiple wind directions simulated and the small number of nodes belonging to these edges (1% of the total number of network nodes), this effect has been calculated negligible to the purposes of this work.According to the mean values (vertical dashed lines) of the distributions reported in Fig. 2, New York is the most vulnerable city on average, while Firenze is the most protected. The vulnerability of New York and Lyon are the most sensitive to changes in wind direction, as shown by Fig. 2.b, where a polar histogram reports the mean vulnerability for each city for the eight directions of the approaching wind. In general, the spreading potential is more effective when the wind is oblique ((phi =45^circ ,) (135^circ), (225^circ), and (315^circ)) to the main orthogonal axes of the street network, as evidenced by the higher vulnerability observed for the dark gray sectors of Fig. 2b. We also notice that vulnerability for parallel ((phi =0^circ ,) (phi =180^circ)) and perpendicular ((phi =90^circ ,) (phi =270^circ)) wind directions is quite similar. This seems counterintuitive as previous studies (e.g.,28,29) have reported that a perpendicular wind is much more unfavorable for the dispersion of pollutants in a street. In this regard, we underline that (phi) is here defined with respect to the main axis of the city, so for (phi =90^circ) not all streets will be perpendicular to the wind direction. For example, in the regular network of Manhattan we expect the number of perpendicular streets to be similar to that of parallel streets, when (phi =90^circ).Figure 1Vulnerability maps for (a) Lyon, (b) Firenze, (c) Paris, and (d) New York for a wind direction of (45^circ) with respect to the main axis of the urban fabric. The polar histograms in the insets report the distribution of street orientation, while the red arrows represent the wind direction with respect to the street network. Panels a1–d1 show the urban pattern in a rectangular area of 0.5 km(^2) (reported in panels a–d) for the cities of Lyon, Firenze, Paris, and New York, respectively. Background images made with QGIS 2.18 (https://qgis.org).Full size imageFigure 2Vulnerability distribution for different cities and wind directions. (a) Cdf of node vulnerability for the different cities under eight different wind directions. The mean vulnerability is shown as a dashed line and reported numerically together with the standard deviation (in parentheses). (b) Mean vulnerability of city networks for each wind direction. Colors blue, yellow, green and magenta correspond to the urban networks of Lyon, Firenze, Paris and New York, respectively.Full size imageThe reasons for the different resilience of cities (and their patterning) to gas propagation are embedded in the centrality metric adopted to compute urban vulnerability. The key factors for node vulnerability can then be analytically recognized in the metric definition (Eqs. 4–5 in “Methods”): the highest vulnerabilities are achieved when the set of reachable nodes ((mathcal {V})) from the source node is large, and the paths connecting the source and the reachable nodes ((d_{sr})) are short, i.e. the propagation cost ((omega)) along the paths is minimal. In other words, the spots in a city (i.e. nodes in a network) with the highest spreading potential are those from which a toxic plume can reach many other locations with significant concentration. Going beyond the vulnerability results, we aim here to decompose the aforementioned elaborate and meaningful quantities (the set of reachable nodes, the shortest paths, the propagation cost) in elementary properties of the urban area in order to link the vulnerability of a city to its tangible characteristics.We start by disassembling the propagation cost associated to each street. Given a source node, a pollutant plume will propagate along the streets downwind the node. The propagation cost of each street (Eq. 4) describes the decay of concentration that the plume undergoes when it propagates along the street. Neglecting physico-chemical transformations, this cost depends on the transport processes within the streets and is a function of two dimensionless quantities: a geometrical ratio between the length (l) and height (h) of the street canyon, and a dynamic ratio between the exchange rate of pollutants towards the atmosphere above roof level (v) and the advective velocity along the longitudinal (u) axis of the street. According to30 and31, these two velocities can be parametrized as a function of the external wind intensity, the cosine ((theta)) of the angle between the wind direction and the orientation of the street, the geometry of the street canyon (its length l, height h and width w) and the aerodynamic roughness of building walls. As detailed in the Methods, the dependence of the propagation cost on the external wind intensity disappears as both velocities u and v scale linearly with it. Assuming constant aerodynamic resistance of the surfaces, the parameters l, h, w, (theta), remain the relevant building blocks for the propagation cost along a street.We underline that the parametrizations adopted here for the transport mechanisms in a street are based on the up-to-date literature and are currently employed in operational models (see the Methods section for mode details). Any refinements to this transport model may be included in the future. In this case, the cost associated to each street may depend on additional parameters that, however, we expect to be of second-order importance to those listed above.While pollutant transport in a single street canyon (i.e. the propagation cost) has been easily broken down into its basic elements, the information enclosed in the shortest paths ((d_{sr})) and in the set of reachable nodes from the source ((mathcal {V})) is much more challenging to trace back to evident properties of the city. These quantities depend on the sequence of streets that must be traveled to connect a source node to the surrounding nodes, i.e. on the way the streets are interconnected. The information is thus primarily topological. However, we point out that the interconnectivity of the network is not frozen, but dynamic, as it is given by the reaction of the urban structure to the direction of the external wind. In fact, the links of the street network are directed according to the orientation of the approaching wind. Moreover, the connectivity between the nodes is limited by the decay of the concentration along the streets. Although a target node may be reached from the source node by means of a path across the network, the two nodes may not actually be connected by a propagation path as the pollutant concentration may vanish along the path. For these reasons, traditional descriptors of network topology cannot be applied directly to describe the topological component of the vulnerability. Instead, we have to look for tailored and simple indicators that can express the wind-driven interconnectivity of the street network and the reachability potential between the nodes.Focusing on a node as spreading source, we infer that the number of links in its downwind area gives a first estimate of the potential for a release in the node to affect many other locations in the network. To delimit this downwind area, we adopt the concept of n-hop neighborhood32,33. Two nodes are n hops apart if it is possible to reach the target node from the source node by traveling n links. We identify the downwind area of the source node as the subnetwork composed by the nodes that are reachable from the source via at most n hops along the directed links. We propose the number of links in this neighborhood (k) as a suitable measure of reachability from the node. This reachability depends upon three features: (i) the local structure of the street network, (ii) the direction of the wind, and (iii) the topological distance n. This latter parameter is intuitively correlated to the intensity of the release. More precisely, it depends on the ratio between the magnitude of the toxic release at the source and the threshold value for pollutant concentration to be significant. In this work, n is taken as constant and its value is obtained from an optimization analysis detailed in the “Methods” section (Fig. 6) .Once the (n-hop) neighborhood of a node is delimited, the number of links k is not exhaustive in giving information about the properties of the paths connecting the source to the other nodes of the neighborhood. For the same k, different structures of the neighborhood can take place (see Fig. 6b), with consequent different outcomes for the propagation process that we are breaking down to basic components. The higher the number of links outgoing each node of the neighborhood, the higher the potential concentration for the k links, as they are topologically closer to the source. This feature can be accounted for by means of a simple branching index (b) for the node neighborhood, defined in Eq. 8 as the average outdegree for the nodes belonging to the neighborhood34.The disassembling analysis presented above suggests that the spreading potential of a node, and thus its vulnerability, mainly depends on the topological parameters k and b and on the geometrical characteristics of its neighborhood, i.e. L, H, W, (Theta), where the capital letters are used to indicate the local average (over the n-hop neighborhood) for the length (l), height (h), width (w), and orientation ((theta)) of the street canyons.In adopting averaged geometrical properties, we are assuming that these characteristics are rather homogeneous in the surroundings of a node. While the height, width and length of the street canyons are actually quite uniform on a local scale, especially in European city centers, the same does not apply to the orientation of the streets. The streets of a neighborhood intersect each other at different angles (e.g., at (90^circ) in grid plans), and the intensity of the wind in the streets changes strongly with their orientation. Low wind streets act as bottlenecks in the propagation paths, thus strongly influencing the spreading dynamics. For this reason, the standard deviation of street orientation in the neighborhood ((sigma)) is expected to be an additional topological index of node vulnerability.To assess whether the identified parameters are valuable basic elements of node vulnerability, we perform a regression analysis adopting a simple (but versatile) non linear model of the form:$$begin{aligned} V_{pred}=alpha L^beta H^gamma W^delta Theta ^epsilon k^zeta b^eta (1-sigma )^lambda . end{aligned}$$
(1)
We estimate the coefficients (alpha) to (lambda) by means of a nonlinear least square technique (namely the fitnlm function in Matlab) that minimizes the sum of the squares of the residuals between the predicted vulnerability (V_{pred}) and the vulnerability V obtained from the centrality metric (Eq. 5 in “Methods”). The regression is performed considering all the scenarios presented in this study: four different urban networks and eight different wind directions. The p-values for the coefficients (alpha) to (lambda) tend to zero, indicating that the relationships between the independent variables and the observations (V) are statistically significant. Note that in Eq. (1) we adopt (1-sigma) as predictor, instead of (sigma), to avoid null entries, as (sigma) takes value in [0 1). To explain the reason for this range for (sigma), we point out that the angle between the wind direction and the street axis is defined in [(-90^circ) (90^circ)]. As a consequence, the cosine ((theta)) of the angle varies in [0 1] and the standard deviation of (theta) (i.e. (sigma)) varies in [0 1).The scatter plot in Fig. 3 compares (V_{pred}) against V. Points correspond to the nodes of the four urban networks in the eight wind scenarios. The figure suggests that 80% of the spreading capacity (V) of a spot in a city can be grasped from the basic geometrical and topological characteristics of its neighborhood. To identify the most influential parameters in the regression, we evaluate the gain in the coefficient of determination (R^2) as they are progressively included in the model (red circles in the inset). The quantities are entered in order to optimize (R^2) at each addition. Alternatively, the role of each parameter can be evaluated adopting the concept of unique contribution (triangles in the inset), i.e. the loss in the coefficient of determination induced by the exclusion of the parameter from the model35. Both analyses reveal k and (sigma) as the main indicators for the vulnerability of a node. Actually, more than 60% of the total variance (inset in Fig. 3) is explained by these two parameters, unveiling the crucial role of topology in governing the dynamics of pollutants in urban areas. The effect of the geometrical properties (L, H, W, (Theta)) of the street canyons is secondary. Among these, the contribution of the building height (H) is the most remarkable as its contribution, combined with that of the two topological parameters k and (sigma), brings the correlation to almost its maximum value.Given these results, it is enlightening to show some tangible examples of how the three simple indicators k, (sigma) and H dominate urban vulnerability. We wonder which of these properties determine the distinct vulnerability of neighboring areas belonging to the same district, and which ones differentiate the resilience of cities with a different urban history.Figure 3Correlation of node vulnerability with basic geometrical and topological parameters of the street network. Color (blue to red) is associated to point density. Left y-axis of inset: trend of the coefficient of determination (R^2) as the urban indicators are progressively included in the model. Right y-axis of inset: unique contribution of the indicators.Full size imageFigure 4a shows the spatial distribution of the key parameters k, (sigma) and H and of node vulnerability, for Manhattan and a wind direction (phi =45^circ). In panel b, high street reachability (k) is observed in the central part of Midtown, in the heart of Downtown, and near Wall Street. An homogeneous distribution in the orientation of the streets with respect to the incident wind (low values for (sigma) and thus high (1-sigma)) is especially found in Midtown (panel c). Finally, in panel d, high-buildings (H) distinguish the Financial District and East Midtown. A perfect match between the four layers is not expected as vulnerability is given by the synergistic contribution of the different parameters. However, in line with the results of the regression model shown in Fig. 3, a positive correlation is observable between the most vulnerable areas (circled areas comprising the nodes with highest V in panel a of Fig. 4) and those with the highest values for the three indicators. In these areas, high buildings inhibit the vertical exchange of pollutants between the streets and the atmosphere above, as largely discussed in literature (see e.g.36,37). This inhibition limits the concentration decay along the propagation paths and facilitates large-scale contamination. Moreover, the great number (k) of streets topologically close to the node increase the impact of the release. The effect of (sigma) is significant especially for the vulnerability of Midtown. Here, since (phi =45^circ) and the street network is regular, (theta) (the cosine of the wind-street angle) is almost the same for all the streets. Therefore, the standard deviation of (theta) ((sigma)) is low and the predictor (1-sigma) is high. Physically, this means that the external wind approaches all the streets with almost the same angle. As a consequence, the intensity of the longitudinal wind in the streets is similar (the street aspect ratio is also similar) and the propagation takes place equally along both the dominant and lateral segments of the street network38, thus favoring the spread over large areas. Although high values of H and (1-sigma) can be detected in the North-East corner of Midtown too, here the vulnerability is mitigated by a higher discontinuity in the urban pattern (low k). This feature, together with the great overlapping of red areas in panel a with those in panel b, evidences the key role of street reachability (k) in the heterogeneity of vulnerability between areas of the same urban district.Figure 4Street network of Midtown and Downtown Manhattan. Node color is associated to node vulnerability (V), and to its key indicators: street reachability (k), inhomogeneity in street orientation ((sigma)), and average height of buildings in the node neighborhood (H).Full size imageFrom these observations, we move to a broader view and investigate the structural fragility of a city as a whole. In Fig. 5a–c, we report the probability density function (pdf) of the key parameters k, (1-sigma) and H. For each city, the pdf is calculated over all the network nodes and for the different wind directions. So, each pdf is representative of eight different networks for the same urban area. In panel a, the distributions for the four cities are quite similar but the tails of the pdfs highlight that the highest values for street reachability (k) occur in the street networks of Lyon and New York. The homogeneity in street orientation with respect to the wind (panel b), expressed by (1-sigma), exhibits a bimodal distribution and a slightly higher mean for the regular street network of New York. The two peaks are associated to distinctive wind scenarios, as will be discussed below. Also in this case, the observation of the tails of the pdfs reveals that high values of the vulnerability indicator are more probable in Lyon and New York. Finally, the distribution of building height (panel c) presents the most marked difference between the considered architectures, with high-rise buildings contributing to the heavy pdf tail of Manhattan. Comparing these results with those in Fig. 2a, Manhattan’s greatest vulnerability appears to be due to the greater depth of the urban canyons (high H) and the greater homogeneity, on average, in wind-street orientation (high (1-sigma)). Conversely, the medieval structure of Firenze, with higher heterogeneity in street orientation (low (1-sigma)) and low buildings (low H), enhances street ventilation and hinders propagation over long distances. Moreover, the tails of the pdfs for k and (1-sigma) reveal the role of topology in the higher variability of vulnerability values (given by the standard deviation of the pdfs in Fig. 2) for the street networks of Lyon and Manhattan.After discussing the behavior of the single parameters, we assess the synergistic contribution of the three quantities. To this aim, we define a simple correlation index (rho =widehat{k} cdot (1-widehat{sigma }) cdot widehat{H}), where the hat denotes a min-max normalization of the parameters, i.e. the range of values of each parameter is rescaled in [0, 1]. For the urban areas of Manhattan, Lyon, Paris, and Firenze, (rho) gives 0.039, 0.017, 0.012, and 0.011, respectively. This ranking complies with the ranking inferable in Fig. 2 for the average vulnerability of the cities. This result confirms that vulnerability occurs when the three parameters are correlated, as already evidenced in Fig. 4.To make the picture even more fascinating, it is worth noting that the role of topology, shown above as key, is dynamic as it varies according to the direction of the wind impacting the urban fabric. In panels d to f of Fig. 5, the pdfs of k, (1-sigma) and H are distinguished for four wind directions ((phi =0^circ), (45^circ), (90^circ) and (135^circ)). For each angle, the statistics are calculated over the examined cities, together. Although wind orientation alters the direction of the network links, and thus the delimitation of the n-hop neighboring area of each node, street reachability (panel d) and building height (panel f) remain statistically invariant for the different wind directions, suggesting a rather isotropic structure of the urban fabric. On the other hand, the variability in street orientation with respect to the wind (panel e) presents two distinctive trends for wind directions aligned with or oblique to the main axes of the street network. To explain this behavior, we refer to the simple case of a grid-like urban plan, like Manhattan’s plan. When (phi =0^circ) or (90^circ), (theta) (the cosine of the angle between the street and the wind direction) mostly switches between 0 (for the streets aligned with the wind) and 1 (for the orthogonal streets), resulting in a high standard deviation over the neighborhood (low (1-sigma)). When (phi =45^circ) or (135^circ), instead, the incident angle (theta) mainly takes intermediate values, leading to higher values for (1-sigma). This distinctive behavior is clearly detectable in the two peaks that we have observed in panel b for the regular grid of Lyon and New York. The left peak of the bimodal distribution corresponds to the scenarios with aligned wind directions, while the right peak occurs for oblique wind directions over the city. A more irregular street pattern in Firenze and Paris adds random contributions to the way the wind approaches the street, thus altering this bimodal shape. Going back to panel e, the greater homogeneity in wind-street orientation (higher (1-sigma)) for (phi =45^circ) or (135^circ) gives insights into the higher vulnerability found for the scenarios with these wind directions in almost all cities (dark gray sectors in Fig. 2b). This result is confirmed by the correlation ((rho)) between the three rescaled parameters ((widehat{k}), (1-widehat{sigma }), (widehat{H})). The correlation (rho) is estimated separately for the different wind directions, but considering the nodes from the four urban areas together. For oblique wind directions, (rho) is about twice ((rho =0.035)) the value found for the aligned wind directions ((rho =0.018)).Figure 5Probability density function of the key parameters k, (1-sigma), H. In the first row, each curve refers to a city and includes vulnerability data from eight different wind directions. In the second row, each curve corresponds to a specific wind direction and includes vulnerability data from the four cities, together.Full size image More
