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Data Mining and Knowledge Discovery

, Volume 32, Issue 3, pp 830–847 | Cite as

Structural robustness and service reachability in urban settings

  • Sofiane Abbar
  • Tahar Zanouda
  • Javier Borge-Holthoefer
Article
Part of the following topical collections:
  1. Special Issue on Data Mining for Smart Cities

Abstract

The concept of city or urban resilience has emerged as one of the key challenges for the next decades. As a consequence, institutions like the United Nations or Rockefeller Foundation have embraced initiatives that increase or improve it. These efforts translate into funded programs both for action “on the ground” and to develop quantification of resilience, under the for of an index. Ironically, on the academic side there is no clear consensus regarding how resilience should be quantified, or what it exactly refers to in the urban context. Here we attempt to link both extremes providing an example of how to exploit large, publicly available, worldwide urban datasets, to produce objective insight into one of the possible dimensions of urban resilience. We do so via well-established methods in complexity science, such as percolation theory—which has a long tradition at providing valuable information on the vulnerability in complex systems. Our findings uncover large differences among studied cities, both regarding their infrastructural fragility and the imbalances in the distribution of critical services.

Keywords

Complex networks City resilience City robustness Percolation LBSNs 

1 Introduction

Over the past years the concept of urban or city resilience has repeatedly appeared in many contexts. While there is a general agreement that it is related to the capacity of a urban area to confront uncertainty and/or risk, it is not clear how it should be operationalized or quantified. Meerow et al. (2016) identify as many as 25 different definitions of urban resilience in the literature, related more or less strictly to Engineering, Environmental Sciences, Business and Finance, or Social Sciences. Indeed, it seems common sense that “generic resilience” (capacity of a system to recover its initial state after a shock) can be mapped onto a myriad of particular definitions which differ only in the number of affected subsystems (infrastructural, social, ecological) and the time scale of the disturbance itself: a shock may arrive under the form of a natural disaster (fast, system-wide), an economic crisis (slow, specific) or the long-term urban footprint on the surrounding environment (very slow and hardly visible in the city itself).

These semantic difficulties go well beyond pure academia, and can affect actual efforts towards increasing urban resilience. Public and private institutions, like the United Nations or the Rockefeller Foundation, are currently supporting initiatives which focus, explicitly or implicitly, on this topic. If, however, its definition is unclear, these programs encounter difficulties at making decisions—whether a given investment makes sense or not. In this direction, the “100 Resilient Cities Program” (100RC) at the Rockefeller Foundation has recently released some guidelines with regard to what makes a city resilient (City Resilience Index, CRI).1 This effort to build an agreed rationale is interesting, because it breaks down the problem into four dimensions (see Fig. 1), which is already a significant advance toward operationalization.

In this paper we focus on the infrastructural side of the 100RC, which breaks this dimension into three goals: reliable mobility and communications; reduced exposure and fragility; and effective provision of critical services. We relate each of these goals to different aspects of network robustness, which has a longstanding tradition in complex systems, and percolation theory in particular (Albert et al. 2000; Callaway et al. 2000; Cohen et al. 2000) Our claim is that the current availability of large digital datasets, and of solid foundations in complex networks can offer a suitable quantitative complement to the evaluation of city resilience, which—as in the example of CRI—typically relies on general traits (e.g. exposure of a zone to flooding hazards).
Fig. 1

Guidelines to the CRI. Resilience is mapped onto 4 key dimensions (outmost ring), which are in turn subdivided into 12 specific goals

Our study relates the first aforementioned goal with the robustness of the road network against random failure of its segments, inasmuch “reliable mobility” relates to daily connectivity between places, people and services. In this sense, the network is prone to fail anywhere, anytime (accidents, congestion, system-wide shocks), as opposed to targeted attacks on critical parts of the infrastructure which is the second goal (“reduced exposure and fragility”). Finally, we combine this network approach with large datasets on service location extracted from Foursquare2—a location-based social network (LBSN)—to understand to what extent a broken infrastructure can affect the reachability of critical services (“effective provision of critical services”), i.e. whether a city offers a balanced and redundant supply of resources. Such services include but are not limited to “medical center” and “transportation infrastructure”.

In order to place the paper in the right context, we have chosen to build on the previous work of Wang (2015) and expand it to several cities included in 100RC for which we could collected data. We also introduce a new perspective into service reachability offering a study at scale which allows for a cross-city robustness comparison that has—to the best of our knowledge—not been addressed before. The remaining of the paper is organized as follows. In Sect. 2 we present different data sources used in this study as well as the various pre-processing steps we undertook to prepare it. Section 3 describes methods. Section 4 discusses our main findings and results. Section 5 concludes the paper with further remarks.

2 Data

To conduct our study, we focus on cities participating in the 100RC program to which we deliberately added 5 cities from the Gulf Cooperation Council (GCC) in the Middle East. These cities are Abu-Dhabi and Dubai (UAE), Doha (Qatar), Manama (Bahrain), and Riyadh (Saudi Arabia). Figure 2 gives an overview of the geographical distribution of the 54 cities considered. We can easily see that the cities span all continents (except Antarctic) with 4 cities from Africa, 14 from Asia, 10 from Europe, 14 from North America, 6 from South America, 1 from Central America, and 4 from Oceania. The geographic diversity and cultural heterogeneity of these cities provide two advantages. First, it diminishes the biases one could observe in studies that focus on cities from a particular country or continent. Furthermore, it allows a more robust generalization of the findings.

For each city, we requested two types of data: the road network and the geographic distribution of different services available in the city. We describe in the following section the data used and the processes by which it was prepared. We also report some initial observations.
Fig. 2

The geography of cities covered in this study. As we see, cities are from around the world and span six continents

2.1 From open street map to road networks

OpenStreetMap (OSM)3 is a popular open source mapping service built by a community of volunteer mappers who contribute to the tagging of cities by maintaining data about buildings, shops, roads, metro stations and any other service that has a value to be on a map. As a crowd-sourced platform, OSM relies on its active community of contributors to create accurate maps by leveraging their local knowledge on neighborhoods. Because OSM databases are mainly contributed by volunteers, it is fair to highlight that its accuracy and completeness vary from a city to another. Generally speaking, large touristic cities like Paris, New-York, and Singapore tend to expose data with high resolution and quality as compared to smaller and isolated cities. However, recent research by Barrington-Leigh and Millard-Ball (2017) has found that globally, OSM is 83% complete, and more than 40% of countries—including several ones in the developing world—have fully mapped street networks. Thus, it is safe to rely on this unique crowd-sourced data to conduct a study like this one.

While OSM provides APIs to query its databases, we decided to use a more comprehensive online service called Mapzen4 to download “extracts” of relevant cities. The extracts we requested come in the form of shape files that are easy to process. Mapzen updates on a daily basis its city extracts by querying OSM.

2.2 Creating road networks

The shape file of cities come with different files defining different services in the city such as transport points, building, places, administrative boundaries, and roads. All these objects are made in OSM with nodes. A node is defined with two mandatory attributes that are its ID (unique identifier), its geographic position provided as WGS84 pair of coordinates (longitude, latitude), and a list of complementary information such as names, addresses, and types of locations. Complex objects such as buildings are defined as polygons of nodes whereas paths (segments of roads) are defined as lists of nodes.

Given the road shape file (i.e., list of paths) of a city, our objective is to create a real road network in which nodes are intersections and edges are road segments between them. The major challenge in this task is to know how to merge different paths (streets) into the same transportation unit (road). Different techniques have been proposed in the literature to deal with this problem. Some of them rely on the street names (SN) to group them (Jiang and Claramunt 2004) whereas other methods are based on the geometry of streets and look at the intersection continuity between streets (ICN) (Porta et al. 2006). An extensive discussion of the pros and cons of the different approaches can be found in Wang (2015). Given the lack of consensus, we apply a three-steps heuristic widely adopted in the computer science community (Chen et al. 2016; Stanojevic et al. 2018), which renders the simplest object among all the options. Note that heuristics yielding a larger set of nodes and edges (which are irrelevant in terms of traffic dynamics) would artificially modify the behaviour of percolation processes (see Sect. 3). Figure 3 provides an illustrative example of the method.
Fig. 3

Building road network from OSM paths. Given three roads, we first identify the set of relevant nodes. Nodes number 1, 3, 4, 6, 7 and 9 are relevant because they start/end roads. Nodes 4 and 6 are also intersections as both belong to more than one path. But node 4 is removed in the third step at it holds a degree \(k=2\)

Step 1 We identify the list of relevant nodes. A node is considered relevant to the road network if and only if it verifies one of the following two conditions: (i.) The node is the beginning or the end of a path. (ii.) The node belongs to more than one path. The first condition captures all nodes necessary to the definition of roads whereas the second condition captures intersections between roads.

Step 2 We iteratively scan through the list of all paths available in the city shape file. For each path, we create an edge between every successive pair of relevant nodes identified in step one. This process discards intermediate nodes that OSM mappers use primarily to shape road segments.

Step 3 We recursively prune from the resulting graph all nodes with degree \(k=2\) and replace them with an edge connecting the two nodes to which they are linked. The intuition here is that nodes with degree \(k=2\) do not add any semantic to the road network as it is envisioned in this study.

Examples of real road networks of four cities generated with our heuristic are shown in Fig. 4.
Fig. 4

Road networks of some selected cities as extracted from OpenStreetMap. We see that cities show different types of networks in terms of size, density, and topology a New-York, b London, c, Dakar d Dubai

In Table 1 we summarize some key structural traits of the extracted road networks. For consistency reason, we reduce the road network of each city to its first giant connected component (GCC) for which we highlight V(city) (number of nodes), E(city) (number of edges), Length (total kilometers), average degree (k). We include also Meshness \(M \in [0, 1]\), defined as the ratio of the existing facets divided by the total number of all possible facets in a graph, where \(M \sim 0\) (for tree-like graphs) and \(M \sim 1\) (for complete graphs) (Wang 2015):
$$\begin{aligned} M = (E - V + 1) / (2V - 5) \end{aligned}$$
(1)
Finally, we report the “organic” score to assess whether a road network has been planned or not (Wang 2015), computed as the proportion of dead ends (nodes with degree \(k=1\)) and unfinished intersections (nodes with degree \(k=3\)) to the total number of nodes:
$$\begin{aligned} Org = (V(k=1) + V(k=3)) / V \end{aligned}$$
(2)
Table 1

Statistics about road networks of different cities

City

\(||\hbox {V}||\)

\(||\hbox {E}||\)

Length (KM)

K

Mesh

Org

Los-Angeles

525,246

678,652

113,504.437

2.584

0.146

0.924

Paris

369,244

503,320

54,713.641

2.726

0.182

0.915

London

343,081

451,855

37,954.920

2.634

0.159

0.900

Dallas

313,147

398,649

80,613.803

2.546

0.137

0.950

Chicago

300,479

403,366

70,911.445

2.685

0.171

0.938

Melbourne

285,671

382,855

48,425.705

2.680

0.170

0.936

New-York

280,797

378,542

60,445.977

2.696

0.174

0.925

Denver-Boulder

224,441

296232

44,696.132

2.640

0.160

0.920

bangkok

205,062

251,387

64,109.417

2.452

0.113

0.966

Rotterdam

197,785

280,381

24,081.916

2.835

0.209

0.867

Mexico-city

184,568

246,154

36,659.346

2.667

0.167

0.944

Milan

177,746

242,894

22,686.829

2.733

0.183

0.911

Pittsburgh

169,723

211,419

42,600.967

2.491

0.123

0.950

Saint-Louis

167,058

203,376

44,989.002

2.435

0.109

0.953

Rio-de-Janeiro

145,383

193,222

28,436.101

2.658

0.165

0.943

Lisbon

136,740

184,803

24,091.563

2.703

0.176

0.944

Athens

136,338

188,655

25,389.023

2.767

0.192

0.907

Sydney

131,581

174,639

22,508.013

2.654

0.164

0.948

Montreal

124,373

172,043

39,126.493

2.767

0.192

0.908

Rome

111,259

147,794

17,878.176

2.657

0.164

0.920

Riyadh

107,445

156,448

24,023.621

2.912

0.228

0.980

Santiago

107,027

146,604

17,690.681

2.740

0.185

0.946

San-Francisco

99,491

134,451

17,644.981

2.703

0.176

0.898

Glasgow

93,235

118,877

11,350.264

2.550

0.138

0.953

Bengaluru

88,037

114,748

14,457.036

2.607

0.152

0.959

Porto-Alegre

79,148

108,282

23,416.511

2.736

0.184

0.918

New-Orleans

76,438

94,395

22,442.663

2.470

0.117

0.948

Chennai

72,277

92,720

11,950.411

2.566

0.141

0.958

Singapore

65,333

87,039

13,641.570

2.664

0.166

0.931

Dubai

64,462

92,849

12,127.150

2.881

0.220

0.907

San-Juan

61,883

72,247

9027.872

2.335

0.084

0.963

Barcelona

59,165

84,178

10,104.779

2.846

0.211

0.899

Amman

57,123

76,775

10,675.922

2.688

0.172

0.954

El-paso

44,326

60,167

11,000.284

2.715

0.179

0.949

Quito

43,051

56,301

10,406.483

2.616

0.154

0.944

Durban

42,171

53,345

8424.425

2.530

0.133

0.942

Boston

40,375

55,573

5656.712

2.753

0.188

0.899

Dakar

38,640

54,164

4208.776

2.804

0.201

0.949

Abudhabi

38,401

55,724

8006.597

2.902

0.226

0.880

Doha

34,743

48,544

7106.840

2.794

0.199

0.968

Juarez

33,663

46,313

7401.811

2.752

0.188

0.949

Belgrade

32,936

41,780

5840.179

2.537

0.134

0.937

Wellington

27,135

32,651

7811.626

2.407

0.102

0.967

Manama

26,781

37,143

3891.066

2.774

0.193

0.957

Medellin

24,410

33,397

5049.379

2.736

0.184

0.902

Christchurch

20,368

26,220

5667.041

2.575

0.144

0.939

Cali

20,001

28,930

3549.279

2.893

0.223

0.923

Accra

19,838

26,079

4070.770

2.629

0.157

0.974

Phnom-penh

17,283

22,296

4075.526

2.580

0.145

0.959

Mandalay

7190

9584

1706.192

2.666

0.167

0.944

Da-nang

6872

9777

1565.853

2.845

0.212

0.930

Kigali

6622

9006

2511.172

2.720

0.180

0.945

Surat

2639

3569

1040.696

2.705

0.177

0.958

Smallest and largest values for each column are highlighted in bold

We found that the meshness scores of cities in this study vary between 0.084 for San-Juan and 0.228 for Riyadh which are consistent with the scores reported in Cardillo et al. (2006) on analyzing planar graphs of twenty cities. The low scores are mainly due to the absence of squares and triangles in real-world cities. Non surprisingly, organic scores are high for almost all the cities, with a noticeable advantage for new and fast growing cities such as Accra (0.97) and Doha (0.96) as compared to old cities such as Barcelona and San-Francisco (0.89).

2.3 Service location data

Foursquare is a location-based social media network that allows users to share their location with friends through what is known as check-in. Data from such check-ins contain information about the user, the venue she is reporting on, the specific location of that venue (coordinates) and a timestamp.

In this work we exploit data that were collected by Yang et al. (2015, 2016) from April 2012 to September 2013. This rich dataset contains 33,278,683 check-ins posted by 266,909 users on 3,680,126 unique venues, that are geographically spread in 415 cities across 77 countries. For our purposes, we only pay attention to the type of service the venue belongs to according to the Foursquare category hierarchy5; and its geographic position in the format of WGS84 pairs of latitude and longitude—disregarding any other information. We have deliberately promoted the “Medical Centers” sub-category into a parent category to be able to assess how such a critical service is affected by failures in the city. At the end we obtained a list of 10 different categories of different criticality, including: Medical Center, Travel & Transport, Food, College & University, Residence, Arts & Entertainment, Shops & service, Nightlife Spot, Professional & Other Places, and Outdoors & Recreation.

Figure 5 plots the frequency distribution of different Foursquare service categories for a sample of cities for which we had access to Foursquare data. We observe that Food related services (e.g. restaurants, coffee shops, etc.) are the most frequent in most cities, followed by Shops & Services.
Fig. 5

Distribution of Foursquare service categories in different cities

To integrate the network structure with services locations, we assign each location to its closest node. This is tantamount to mapping service locations to their nearest intersection, allowing an efficient computation of the amount of reachable services (see Sect. 3). Services that are farther than 2 km for all intersections in a city are omitted from the study.

3 Methods

We review in this section the methods used in our work in order to assess the robustness of cities by studying their underlying road networks. As stated earlier, the road network is viewed as a directed graph \(G=(V, E)\), where V is the set of vertices and E is the set of edges. In any city, the road network graph is found to be connected, which means one could start navigating (traversal) from any node and reach any other node in the graph. One intuitive way of measuring the quality of the connectivity is to estimate the number of edges that need to be removed in order to create a disconnected graph. The process of deteriorating edges is called percolation in complex systems. Indeed, network percolation theory has already been exploited in the urban context for other purposes than the ones in this work (Arcaute et al. 2016; Li et al. 2015; Wang 2015).

A related formulation in data mining—and graph theory in particular—is the so called minimum-cut problem (MCP) (Stoer and Wagner 1997; Boykov and Kolmogorov 2004). In this context, a cut of a graph \(G=(V, E)\) is a partition of V into two non empty sets S and T. The crossing edges of the cut (ST) are the ones with one endpoint in each of the two sets (ST). The capacity of the cut is the sum of the weights of its crossing edges. The minimum cut (MC) is then defined as the cut with the minimum capacity. In case of unweighted graphs (which is equivalent to the case where all edges have similar weight), the minimum cut is the cut with a minimum number of crossing edges. The most popular version of MCP is the \(s-t\) minimum cut in which a pair of source and target nodes as assumed given, and seek to find the MC that disconnects them. A generalization of this problem to directed graphs where no such pair is given has been proposed by Hao and Orlin (1994) and coined as the unrestricted minimum cut problem. The idea for solving MC for the entire graph is to solve a sequence of \(2n-2\) minimum \(s-t\) cut problems, and then select the best one of them (where n is the number of nodes). Hao and Orlin (1994) proposed a fast algorithm in which the sequence of subsequent minimum \(s-t\) cut problems are carefully rearranged to allow for some remarkable optimizations. Although the idea of using MCP might sound relevant to study resilience of networks, it does suffer several shortcomings due to the planar and geometric nature of road networks in particular. Indeed, in most of the cases, the minimum \(s-t\) cut problem results in a set of singular edge that splits the whole set of vertices V into \(V-\{j\}\) and \(\{j\}\), where j is either the source or the target node in the \(s-t\) cut problem (see Table 1 for the low average degrees found in road networks.) This makes MCP trivial adaptation unfit to the problem we want to study. Therefor, we stick to the percolation process which has more interpretable and meaningful results in studying connectivity of road networks represented as planar directed graphs.

With the road networks for dozens of cities at hand, we can now proceed with the percolation dynamics in two different ways. Both of them share the idea of progressive structural deterioration (Albert et al. 2000; Callaway et al. 2000; Cohen et al. 2000), understood either as error or failure (removal of randomly chosen edges); or attack (removal of important edges, where “importance” can be quantified by some descriptor, such as high betweenness of edges, high centrality of nodes, etc.) Note that in this work we focus on bond percolation (edges are removed) as opposed to site percolation (nodes are removed).

Here we apply both Error and Attack procedures on the available networks. For the Error process, at each step an edge chosen uniformly at random is removed, until the size of the GCC (giant connected component) is equal to the size of the SLCC (second larger connected component). Connected components (and their sizes) are checked periodically (each time 1% of edges have been deleted). Because of the stochastic nature of this process, it is repeated 50 times so as to report the average expected deterioration of the network. In the Attack modality, the edge with highest betweenness centrality is removed at each step. As before, the process continues until the size of the GCC is equal to the size of the SLCC.

Following these progressive schemes, as the fraction p of removed edges increases the size of the GCC (captured in number of nodes) decreases, until the network undergoes a second-order phase transition at the critical point \(p_{c}\) (percolation threshold). Such transition can be spotted precisely—it occurs when the size of SLCC becomes maximal (Bollobás 1985). For instance, assume we have a connected network with 100 nodes. The percolation procedure starts removing edges one at a time. For each edge removal, we compute and store the sizes in nodes of GCC and SLCC. At the end of the process, we have 100 different pairs of (GCC-size, SLCC-size) that correspond to each percentage of edge removal. The percolation threshold is nothing but the percentage of removed edges that led to either the largest SLCC, or an SLCC that is equal in size to the GCC.

For each network, at \(p_{c}\) we compute the proportion of available services (gathered under ten general categories) in the remaining GCC as a proxy of the capacity of a city to deliver those services in an Error/Attack scenario.

4 Results

We first take a look at the cities’ response to Error. Figure 6 shows some examples of how random edge removal affects the size of the GCC. Typically, the size of the GCC is relatively stable, which indicates that the road structure is not suffering a large damage regarding overall reachability. At a given point (\(p_{c}\)), a transition takes place (often quite abruptly, e.g. Dakar or Abu-Dhabi), after which the network is shattered into many disconnected components. Note that Error results are only reported for a subset of cities to ease the comparison across different experiments.
Fig. 6

Random failure behavior. Evolution of the size of the GCC (y-axis) as a function of the percentage of removed edges (x-axis). Results are reported for 50 randomizations. Notably, cities react differently to edge failure due to their different \(p_{c}\)—see Fig. 8 (top)

Along this process, \(p_{c}\) can be seen as an indicator of the robustness of the system to failure. A low critical threshold can be interpreted as a limited capacity of the road infrastructure to support and recover from random disruption—accidents or congestion happening at any location, or system-wide disturbance. In Fig. 8 (top), cities are sorted according to their \(p_{c}\) against random error (averaged over 50 realizations). We see there that there is a remarkably wide range of cities, with the weakest road network corresponding to Wellington (\(p_{c}=0.17\)), and the more robust ones to Mandalay, Cali and Boston (\(p_{c}=0.35\)). Such variability suggests deep structural differences beyond the obvious features reported in Table 1 (compare for instance the two mentioned cities). We also see that most North American have made it to the top cities in terms of robustness to random error.
Fig. 7

Intentional attack behavior: evolution of the size of the GCC as a function of the percentage of removed edges. Again, cities react differently to edge failure—see Fig. 8 (bottom)

Figure 7 shows as well the progressive deterioration of the GCC as a function of the fraction of removed edges, but in this case links are targeted with respect to their betweenness centrality. The differences with respect to Fig. 6 are obvious—transitions take place much earlier and in a much more abrupt manner. In this case, the values of \(p_{c}\) are below 10% for any city under study, evidencing the large dependence (and, in turn, exposure and vulnerability) of the road network on a few central segments. Figure 8 (bottom) shows cities sorted by their percolation threshold under the targeted Attack scheme.

Interestingly, we see that the ranking of cities has drastically changed in Attack. In fact, cities from Oceania such as Wellington and Christchurch are turned to be among the most robust cities in these scheme while they were at the bottom of the ranking in the Error scheme. We also observe that North American cities are no longer grouped toward the head of the ranking as they were in the Error case. This suggest that cities, regardless of their location, behave differently the two failure schemes considered in our study. We further computed the overall Pearson correlation between the vectors of \(p_{c}\) scores and we found it as low at 0.23, indicating that there is a very low correlation between the percolation thresholds observed in Error and Attack schemes. In other words, a city with a high robustness to Error can be very vulnerable for Attack and vice versa.
Fig. 8

Percolation point \(p_{c}\) of different cities. The percolation point is computed as fraction of removed edges from the network when the size of the second larger connected component maximizes. a reports the results for the random failure in which percolation thresholds are between 17% (Wellington) and 35% (Boston, Cali, and Mandalay). b reports the results for the targeted attack. As expected, the percolation happens now much earlier, after removing only 1% of edges in New Orleans, or 2% in Phnom Penh and Mandalay), to 9% in San Francisco, Rome and Christchurch. a Random failure, b targeted attack

The third focus of our work is related to the question of service distribution. The effects of the failure of a part of the road network are not important per se, but in terms of the possibility to reach—or not—any other part of the city (and the services it delivers). Thus, besides questioning how robust the infrastructure is (i.e. how early it breaks), we wonder now about the consequences of surpassing \(p_{c}\) in terms of health, food or residential resources. Figure 9a summarizes the situation for the 27 cities for which we could find Foursquare data. In panel (a) of the same figure, we compute for each city the average proportion of available services (across the ten categories) reachable by the GCC observed at percolation threshold (\(p_c\).) Here again, we found that North and South American cities such as New Orleans, New York, Montreal, Quito, and Porto Alegre are in the top of the ranking. In the city of New Orleans for instance, only 23% of the services became unaccessible (availability around 77%.) The bottom of the ranking is fairly dominated by European cities (Rotterdam and London) in the bottom 3 cities. Indeed, a city like Rotterdam could preserve access to only 26% of all its services. Also, there seem to be a negative correlation between the percolation threshold of a city and the fraction of services it retains. That is, the higher the percolation threshold the lower the available services.

In panel (b) of Fig. 9 we report the the availability of service per city per category. We represent in red the cities that lost more than 50% of services in at least on category whereas we use green to report cities that preserve at least 50% of their services in all categories. We also plot in black the average availability across cities for each category. As one would expect, cities are showing different patterns regarding different categories. However, a close analysis reveals that on average category availabilities are somewhat equilibrated around 55% with the following breakdown: Nightlife Spot (\(60.96\%\)), Food (\(58.19\%\)), Professional & Other Places (\(55.18\%\)), Shop & Service (\(54.89\%\)), Medical Center(\(54.76\%\)), Arts & Entertainment (\(53.50\%\)), Residence (\(49.51\%\)), Outdoors & Recreation (\(49.36\%\)), College & University(\(49.19\%\)), Travel & Transport (\(49.00\%\)). This reveals that the most affected category of services across cities is Travel and Transport that includes places such as bus and metro stations, airports, rail stations, etc. In the third and last panel (c) we report examples of cities with completely different reactions to failures in terms of service providing. London (UK) as a city that severely suffers the failure by losing access to almost 70% of its services with a particular emphasis on medical centers and transport; Amman (Jordan) as a mid affected city losing access to 55% of its services; and New York (US)—as a good example—loses only 30% of its services with a high preservation of its medical centers. Of course, one could claim that the amount of reachable services at percolation threshold depends on the size of remaining giant component itself. Thus, we plot in Fig. 10 for each city the fraction of unreachable services at percolation threshold (\(y-axis\)) function of the fraction of disconnected nodes (\(x-axis\)). The intuition here is that more disconnected nodes imply more unreachable services. However, the figure shows that this is not the case all the time. For instance, while Manama (Jordan) and New York (US) lose about the same fraction of their road network at percolation threshold (74% of nodes), New York retains more than 70% of its services versus 45% only for Manama. New Orleans turned to perform best as it retains access to over 80% of its services even after losing more than 60% of its road network (nodes). Other cities such as Cali (India), El Paso (US), and Rotterdam (Holland) show a more balancer behavior as their points are close by the identity line (\(x=y\)).
Fig. 9

Service availability in the giant connected component at percolation threshold \(p_{c}\). a Reports the proportion of available services in different cities sorted from the most affected to the less affected. b In green, cities that keep at least 50% of their services in the GCC for all categories. In red, cities that lose more than 50% services for at least one category. c Focus on three different cities with three different behaviors: London in red, Amman in yellow and New York in green (Color figure online)

Fig. 10

Scatter plot of the fraction of disconnected nodes from the giant connected component at percolation threshold (\(x-axis\)) and the fraction of unreachable services for the same giant connected component (\(y-axis\)). the line in black is \(x=y\). We observe that the fraction of disconnected nodes is higher than that of unreachable for all cities

5 Discussion and perspectives

In this work we have focused on various aspects of road network robustness. To do so, we have exploited large, publicly available datasets, and the rich tradition in percolation theory. Our results indicate that phase transitions occur at different levels of deterioration (early vs. late), but also in qualitatively different manners (abrupt vs. smooth). While these differences may result from different design principles (Wang 2015), we suggest that these heterogeneous “percolation profiles” can be interpreted as a fingerprint of the city, in the sense that these profiles are unique to each of them—while at the same time incomplete (a city is much more than its road structure).

We complement the structural degradation approach with a reachability perspective, coupling the infrastructural level with the services it connects. Indeed, a fractured road network is not critical per se, but in relation to the broken flow of people and resources. The results show again heterogeneous profiles regarding service allocation. These insights may be helpful in assessing strategies for a more balanced distribution of such services that guarantee reachability a balanced coverage even in a stressed infrastructure.

Despite the simplifications that our method entails, a systematic study of these “robustness fingerprints” may lead to a classification of cities with regard to their fragility and service distribution imbalances—along the lines of Louf and Barthelemy (2014), but taking a urban resilience angle. Beyond its academic interest, such classification could lead to the design of common preventive strategies for cities whose structural features resemble each other (at least from a percolation and service allocation perspective). Furthermore, it could be enriched with other critical urban networks (e.g. water supply or power grid) expanding the scope of the study to multilayer percolation (Buldyrev et al. 2010; Gao et al. 2012), as well as other theories in the urban geography tradition (e.g. Central Place Theory Berry and Pred 1965; Christaller 1966).

All of the previous falls under an overarching vision, which is related to the existing gaps between policy actors and scientific activity: the pipeline between both extremes is often too narrow—if not inexistent. There seems to be a general agreement that resilience is a desirable feature in urban settlements—as the world population increasingly flows into cities–, but the concept becomes fuzzy when it comes to the definition of objectives and strategies. In an attempt to alleviate this, we have taken a specific angle of resilience (that of 100RC) and mapped some of their targets onto quantifiable, concrete measurements. There is a large body of work waiting ahead before we can come up with an objective set of methods to measure urban resilience, taking into account the wide range of temporal and spatial scales a city encloses.

Footnotes

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Copyright information

© The Author(s) 2018

Authors and Affiliations

  1. 1.Qatar Computing Research Institute - HBKUDohaQatar
  2. 2.Internet Interdisciplinary Institute (IN3-UOC)Universitat Oberta de CatalunyaBarcelonaSpain

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