A global assessment of national road network vulnerability

Every country relies on a well-functioning road system. However, we do not have a clear understanding yet of the vulnerability of each of these road networks to different forms of disruption. In this study, we aim to better understand how road networks are affected by different disruptive events, to identify hotspots of road network vulnerabilities, and to better target where and what type of future investments can be made to develop more resilient networks. To do so, we developed a fully open-source modelling framework to expose over 200 country road systems across the world to random, local, and targeted disruption schemes. For each country, we assessed the impact of such disruptions on intra-country travel activities and regional accessibility. The results highlight the vulnerability of road systems in mountainous and small-island countries owing to the limited availability of alternative routes. Additionally, we find that, on average, low-income countries experience a collapse of road-system services with much fewer disruptions, relative to high-income countries, due to the lack of redundancy in their systems. While the value of goods and services disrupted may be higher in wealthier countries, the results highlight that from an equity perspective, transport infrastructure investments are more desired in low-income country networks.


Introduction
Robust transport networks form the backbone of a well-functioning society. Resilient and reliable infrastructure is therefore an essential part of the Sustainable Development Goals . Target 9.1 in particular, calls for 'reliable, sustainable and resilient infrastructure, including regional and trans-border infrastructure, to support economic development and human well-being, with a focus on affordable and equitable access for all' . However, this goal is currently far from being achieved. In many countries, firms and households still suffer from a lack of reliable and resilient infrastructure systems . Recent estimates find that infrastructure disruptions impose costs between $391 billion and $647 billion per year on households and firms in low-and middle-income countries . Given the limited budgets available to invest in infrastructure, building resilient and reliable services requires a sound understanding of transport network characteristics to ensure that every dollar spent contributes to the goal of improving the reliability of services. While previous studies have shown that strengthening individual infrastructure assets exposed to natural hazards can already be cost-effective based on the direct physical damage to these specific assets , accounting for the costs of cascading failures and potential nation-wide disruption may further help to target adaptation efforts towards the right places (He et al 2022).
Infrastructure systems are organized in networks of links, which establish connectivity between a set of nodes, to facilitate a flow-of people, goods, material, and services-between them. Historically, transportation infrastructure networks have not been primarily designed with resilience in mind. Rather, their typologies were constrained by natural obstacles, such as rivers and mountains, and resulted from historical settlement and job localization choices (Strano et al 2012, Wang et al 2015. For example, in the United States, many former portage sites (i.e. sites that required overland hauling due to obstacles to water navigation) are still important nodes for the transport network, long after their historical advantage has become obsolete (Bleakley and Lin 2012). As such, transport planners often must work with what they have, keeping in mind that changing network shapes can take decades.
Network analyses have been used by planners for prioritising transport investments, helping transport authorities allocate their budget to the assets that yield the maximum benefits for users (Espinet et al 2018). The ranking of transport assets based on their impact on network functionality is often referred to as a criticality analysis (Jafino et al 2020). One category of criticality analysis involves systematically removing each asset (node or link) in a network and assessing the resulting cost for users. Other metrics are based on, for example, network topologies (Jafino et al 2020). More recently, network models have been used for impact assessment of natural hazards and prioritisation of interventions to reduce risk (Rozenberg et al 2017, He et al 2022. Typically, hazard maps representing individual events are overlaid with network models, and the resulting loss of network functionality and cost for users are assessed. For climate adaptation investment prioritisation, the criticality of transport assets is combined with their exposure and vulnerability to natural hazards risk (Rozenberg et al 2017, Espinet et al 2018.
Natural hazards typically disrupt more than one road segment at a time, and disruptions are often spatially correlated (e.g. when the network is affected by a flood). Segments that do not appear critical in a single-node assessment can become critical when removed simultaneously with other segments. There is thus a need to systematically understand the network loss resulting from the simultaneous destruction of road segments, rather than focussing on the criticality of individual segments. The consequences of simultaneous link disruptions in a transport network depend on many network characteristics (i.e. its length, density and connectivity). Since an infrastructure network shape is static in practice (e.g. a new road cannot be built instantaneously), the network topological attributes are viable indications of its coping capacity against disruptive events (Jenelius 2009, Wang et al 2020, Dong et al 2022, van Ginkel et al 2022. In this study, we aim to investigate the role of country and network characteristics on the resilience of road networks. We aim to better understand which country characteristics influence whether a road network will or will not fail after the loss of some of its components; characteristics investigated include economic characteristics, such as income level and the degree of urbanisation, and geographic characteristics, such as how mountainous the country is or whether or not it is landlocked. To do so, we expose 208 national road networks (see supplementary material) to random, local, and targeted disruption schemes to determine how a country's road system 'responds' to different forms of network disruption. These disruption schemes mimic different ways in which various natural hazards may disrupt road networks (e.g. a flash-flood can be described as a local or targeted attack, whereas a winter storm may be better described as a random attack). Random attacks are applied through systematically removing random combinations of links from connected road networks. Local attacks are applied by removing all edges within a specific grid cell (0.5, 0.1 and 0.01 degrees). Targeted attacks are applied by removing single road edges. The outcome of this analysis is threefold: (i) understanding how road networks are affected by different disruptive events, (ii) identification of the type of disruption to which each country is most vulnerable to, (iii) pin-pointing hotspots of road network vulnerabilities. Figure 1 outlines the steps taken to assess the vulnerability of country-level road networks to different disruption schemes (Oliveira et al 2016). The first step is to prepare each country's network for our analysis, ensuring that all networks are standardised and comparable (Panel 1 A, section 2.1). The second step is to prepare an origin-destination (O-D) matrix containing potential travel flows within each country. To ensure a standardised approach, we assign 100 unique O-D points to each country, representing the start and end points of travel (Panel 1B and 1D, section 2.2). Third, the O-D trips and prepared networks allow us to run random, local, and targeted attacks on each country network (Panel 1 C, section 2.3). Finally, we analyse the outcomes by identifying (i) which country and network characteristics explain the level of disruption caused by each attack; (ii) what part of each country's network causes the largest disruptions; and (iii) what the associated societal costs are of these disruptions (Panel 1E, sections 2.4, and 2.5). The entire approach is written in Python 3.9 and publicly available .

Network preparation
To perform this analysis, we make use of the available geospatial information of roads through OpenStreetMap (OSM). As mentioned in previous studies (Meijer et al 2018, Nirandjan et al 2022, OSM provides good coverage of country road networks, and can be considered one of the most consistent geospatial datasets to be used for a global analysis on road networks . In 2017, Barrington-Leigh and Millard-Ball (2017) estimated that approximately 80% of the OSM road network was complete. Despite the notion of previous work to integrate different road databases to have a more complete coverage of all roads (Meijer et al 2018, Wenz et al 2020, we only use OSM. This decision is twofold: (i) we perform a network analysis, in which a fully connected network is essential. While combining two different geospatial road databases may be feasible on a regional level, this is not realistic within a global analysis; (ii) we only include higher-tier roads (see next paragraph), whereas the largest gaps within OSM are mostly on lower-tier roads.
To use OSM's road network, we start from the position of extracting all the 'way' relations in OSM with a 'highway' tag in them-the typical notation in OSM for roads. Next, we only select roads that are tagged as trunk, motorway, primary, secondary and tertiary. We consider these roads to be the core network of a country. Additionally, the completeness of the road network reduces substantially for lower-tier roads (i.e. unpaved tracks or residential roads). Therefore, focusing on a higher-tier road network increases the fairness of comparisons across countries. We note that ferries are not included in the analysis. This means that, particularly for some countries (e.g. Mozambique or DR Congo), our results can be considered pessimistic estimates, as alternative routes by ferry are not integrated. Moreover, in the case of Archipelagos and other island groups, we consider only the island with the densest road network (i.e. where the country's capital is situated) in our analysis.
As illustrated in Panel (A) of figure 1, several steps are taken to simplify the network, reducing the total number of nodes and edges within each network. Simplification is necessary to perform a percolation on the largest networks (e.g. Brazil, the United States, or China). The first simplification step is to remove all roundabouts and replace them with a single node. The second step is to add nodes where roads connect but where a node is missing in OSM (e.g. a T-junction that is incorrectly mapped). This ensures a fully connected base network. The third step is at the core of the simplification process. We merge all edges that are connected through nodes with a connectivity degree of two and drop these nodes. In other words, we only consider intersections where at least three road segments are connected. This reduces the number of edges by approximately 30-40%, on average. Within this merging process, we keep the road characteristics of the first road segment in the list of segments to be merged. Finally, we add the necessary road characteristics to perform a Dijkstra shortest path analysis weighted by travel time. To achieve this, we require the maximum (allowed) travel speed, length, and resulting travel time for each edge. The completeness of the maximum travel speed attribute in OSM varies substantially between countries. To fill in the missing values for an edge, we consider the mode of maximum travel speed over all edges within a country that have the same road type. For example, if the travel speed for a primary road is missing, we consider the modal travel speed for all primary roads within a country. This modal travel speed for primary roads will then be used to fill the missing values for the respective primary road edges. This network preparation results in road network graph G for each country in this analysis.

O-D preparation
To identify the vulnerability of each road network, we model movement along the road network from a collection of origins and destinations that describe human travel patterns in a country. The most straightforward approach would be to use subnational administrative units as the origin and destination of trips (e.g. Database of Global Administrative Areas, GADM). Unfortunately, the number of subnational administrative units is very inconsistent between countries, even among countries with similar geographical sizes. For example, Romania has 42 first-level subdivisions within the GADM classification (one level below the national level), whereas the United Kingdom has only four subdivisions (the countries are roughly of equal size). Such inconsistency makes it difficult to consistently compare countries and to select one specific level of subdivision (i.e. first or second-level) to be used for all countries. To overcome this inconsistency, we create a grid for each country. This grid ranges from 100 cells (a grid of 10 × 10, the smallest country) up to 10 000 cells (a grid of 100 × 100, the largest country). All countries in between are assigned a value between 100 and 10 000, based on the size of their country (Panel (B) in figure 1). This results in varying grid cell sizes across countries: for the United States, a grid cell measures 0.326 • ; for Kenya, 0.149 • ; and for Bahrain, 0.021 • (see supplementary material for an overview of the grid cell amount and size for each country).
This means that for the United States, we are mostly measuring cross-country journeys across the interstate system and traffic between large urban agglomerations; however, for a small island developing state (SIDS) like Aruba, we are more likely to measure all intra-urban journeys, where network redundancy may be higher. Although we acknowledge that this simplification does not fully represent the actual network usage in each country, it allows us to consistently compare between countries the relation between the physical topology of the national network and its vulnerability to disruptions.
We populate the O-D matrix with 100 O-D points per country, resulting in 9900 unique trips, but 4950 routes (100 × 99 × 0.5) as we do not account for one-way roads. To select these 100 O-D points, we use the population share of each grid cell as its probability of being chosen. Grid cells with a large population have a higher probability of being included in the analysis. WorldPop is chosen as the base data to assign population values to each grid cell, given its wide geographic availability and robust methodology (Tatem 2017). The WorldPop raster data are summarised at the chosen grid level size for each country (supplementary material). The sum of the population within each grid unit is calculated and mapped to the geographic centroid of that polygon. A simple demand model is constructed to describe the demand for journeys between any two points in the O-D set using the population values and the prepared road network for travel times. The demand for the journey between points A and B is described by: where Distance A,B is the travel time between A and B, expressed as a fraction of the longest travel time between any two points in the O-D set. Demand is then normalised to 100 trips, with the minimum demand set to 1 trip, to ensure all points in the O-D set had at least some demand for the journey between them. This process generates a demand matrix of shape (100 100), with values ranging from 1 to 100, which we call the 'baseline demand matrix' (Panel (C) in figure 1). The travel time for a hypothetical trip between A and B is also calculated for the O-D set and generates a matrix of shape (100 100). This will be further referred to as the 'baseline O-D matrix' . The O-D matrix is connected to a country's road network by connecting the geographic centroid of each O-D point to the nearest node on the road network.

Percolation approach
Originally, percolation theory dealt with the removal of nodes or edges in a graph to demonstrate phase transitions, such as finding the giant component (Callaway et al 2000). It is now commonly used in complex network modelling to assess the robustness of a network to permutations (Buldyrev et al 2010). The basic concept is to incrementally remove nodes or edges from a network and record how this affects the overall connectivity, or another defined function. Edge removal simulates a road being unable to facilitate the flow of people and goods. By measuring the effect of this on journeys, we can see how failure percolates through the system. Following the removal of a road, some trips will become impossible, some will be delayed, while others will remain unaffected. By recording the impact of incremental failure, this approach allows us to quantify the resilience of a transport network to permutations. Networks that have well-placed redundancy (i.e. alternative paths between important nodes) will experience less travel and supply chain disruptions following edge removal. We will measure the connectivity (i.e. the existence of a path between) and shortest travel times for demand-weighted journeys. As illustrated in Panel (D) of figure 1, we use three different percolation approaches to assess the vulnerability of each country's road network: (i) random attack, (ii) local attack, and (iii) targeted attack (Yuan et al 2016).
The three disruption schemes differ in their methodology for selecting edges for deletion and the number of edges deleted in any given iteration. The general structure of the experiment remains the same for each attack method. We start with the road graph G, the baseline O-D, and the demand matrix, and then remove a set of edges (this set could also contain only a single edge), giving us the modified graph G' . For each G', a revised matrix of travel times is prepared for each point in the O-D set, travelling to every other point. As we consider 100 unique points in each country's O-D set, this generates a new travel time matrix of shape (100 100), further referred to as the 'revised O-D matrix' . The difference between the revised O-D matrix and the baseline O-D matrix describes the increase in travel time for making any trip between a pair of points A and B in the O-D set. These are then weighted by the baseline demand matrix to calculate several summary statistics for each G'. The most important metrics are the percentage of trips disrupted, which can be described as the fraction of total trips that can no longer be completed via any route in G' (i.e. no routes between some pair of points (A, B) in the O-D set can be designed on G'), and the percentage of trips delayed, which can be described as the fraction of total trips that will take longer compared to the baseline travel time.

Targeted attack
The targeted attack approach can be considered to be the simplest disruption scheme. Here we estimate the impact of removing a single edge once at the time within a country's road network (representative for a local pluvial flood event or landslide). More specifically, we do not incrementally remove more edges but focus on the impact of the removal of a single edge from the road graph. This approach will help us identify the importance of certain road segments in the functioning of the network.

Local attack
To estimate the impact of local attacks (representative of an earthquake or flood event), we again start from the road graph G, the O-D matrix, the baseline demand matrix and the baseline O-D matrix. However instead of removing a single road edge, we now remove a cluster of road edges simultaneously. We carry out this analysis at three different cluster sizes: 0.5 • (±50 km at the equator), 0.1 • (±10 km at the equator) and 0.05 • (±5 km at the equator). Hence, we divide each country into a grid of 0.5, 0.1 or 0.05 degrees. We remove all roads within one specific grid cell. We do this for all grid cells but do not incrementally remove multiple grid cells at the same time.

Random attack
Random attacks have been previously used to assess the vulnerability of road networks to disruptions (Wang et al 2019, Dong et al 2022, van Ginkel et al 2022). The choice of roads to be removed is randomised. The removed road segments belong to a set 'λ' , which generates an altered road network graph, G'. This process is repeated with increments 'β' of 1% until 30% of the network is removed. Between 30 and 50%, we randomly remove 2% of the network at each increment. After 50%, we take incremental steps of 5%, as most of the road networks have already been fully disconnected. This process is repeated until there are no longer any possible journeys. For each incremental change, 200 iterations are calculated-up to 200 different sets of roads λ are removed from the network, generating up to 200 different versions of G' for each value of β. Moreover, we reselect the 100 OD-points from the grid (see section 2.2) within each iteration to ensure we cover as many possible trips as possible.

Travel cost impacts
Furthermore, we estimate the travel time on a given route between two points (i.e. the origin and destination) in the O-D set. If an alternative route has to be taken due to the inaccessibility of the baseline route, we can derive the additional travel time. This additional travel time can be considered to be the travel time loss. The sum of these losses across all routes is then calculated to derive an estimate of the total loss arising from disruptions to G built into G'. This value is expressed as a percentage reduction in total surplus, as calculated against the baseline G. For each pair of points in the O-D set, there are three known pieces of information-the baseline demand for the trip, without disruptions; the originally predicted travel time; and the adjusted travel time for the changes made to G', which in the event of no disruption was equal to the travel time under G. To calculate surplus, the duration of trips needed to be converted into a cost or price. We priced the time of travellers at the hourly per capita GDP of the country, expressed in USD. By assuming travel cost elasticity µ, we are able to calculate the surplus loss arising from the disruptions to the network (figure 2) underneath the surplus demand curve. The surplus demand curve can be described as a generic linear function, where trip cost is a function of trip demand. The cost elasticity µ describes the slope of the surplus demand curve, whereas a describes the maximum trip cost in a given country and is commonly defined as the y-intercept.
In figure 2, the original demand and cost combination (equilibrium E 1 ) for a given trip is expressed as (D 1 , C 1 ), as modelled on G. If we use the adjusted network G' , the equilibrium is E 2 , with reduced demand D 2 owing to a higher trip price C 2 (as travel time increases). The original surplus is represented by a triangle (C 1 E 1 ). Now at equilibrium 2, the area (C 1 C 2 E 2 E 1 ) represents lost surplus. This region is calculated for all trips and summed. This is compared to the sum of area C 1 E 1 for all trips to identify the percentage reduction in surplus arising from G' . There is a large uncertainty in the elasticity µ; thus, we follow Litman (2011) and use a range between −0.36 and −0.12. As the number of unique trips in our approach is capped at 9900 (100 × 99), we are not producing travel time loss values that mimic reality. As such, we only consider the percentage increase in travel time loss when trips are disrupted and delayed, assuming that our sample of trips is representative of all trips in the country. The added value of this travel time loss metric, relative to the percentage of disrupted and delayed trips, is that both are combined into a single metric. Thus, it can be interpreted as a proxy for the relative social welfare losses attributable to reduced network connectivity.

National road vulnerability index (NRVI)
By combining the results of the targeted, local and random attacks for each country, we develop a normalised index which allows us to create a ranking of countries to identify the least and most vulnerable national road networks. For local and targeted attacks, we include the maximum percentage of disrupted trips and the percentage of grids (local attacks) and road segments (targeted attack) that cause a disruption. For the random attack, we include the mean percentage of disrupted trips when 20% of the network is removed. For each of these indicators, we normalise them across all countries between 0 and 1. The final NRVI is an equally weighted average of the different percolation approaches.

Targeted attack on individual road segments
The targeted attack results allow us to identify the components of the road system that play a critical role in at least some parts of the population. This shows which individual road segments in the network can cause severe travel disruptions, thereby mimicking small-scale natural hazards, such as local flood events or landslides. It should be noted that in most road networks, many individual road segments can be removed without significant consequences. However, approximately 2.2% of the road segments in an average country are so critical that their removal blocks access to a part of the country (i.e. at least one O-D point loses access to the rest of the country). Removal of the most critical road segment in a country may fully disrupt (no detour possible) 13.7% of all trips (global average). Low-income countries have approximately twice as many edges which can cause inaccessibility for at least one part of the country, compared to high-income countries (3.0% vs. 1.5%).
Interestingly, even if the percentage of road segments that can cause full disruption of travel between at least two parts of the country is low, the impact can be locally severe. This is highlighted in figure 3(B), countries such as Canada, Russia, and the Philippines have relatively large road networks, in which very few road segments can cause a full disruption of at least one connection between two O-D pairs (figure 3(A)). When these few road segments are unavailable, their impact can be substantial. For example, in Canada, several road segments along the Trans-Canada Highway (particularly north of Deception Bay) could potentially prevent trips between the eastern and western parts of Canada. An alternative route would only be possible when travelling through the United States.
In Guyana, Bhutan, Papua New Guinea and The Gambia, we find that the failure of a single road segment could cause a complete disruption of approximately 50% of all trips. These road segments are generally located centrally in these countries, causing a disconnect in up to half of the country. This can be directly attributed to the fact that they have one critical road crossing the country with few (or in some cases, zero) alternative routes. Alternative routes are often not possible owing to geographical characteristics, including rivers with limited crossings (e.g. The Gambia), the physical topography of the country (e.g. the mountains in Bhutan), or dense rainforests (for example, Guyana). Most other countries that experience such high levels of disruption (figure 3) can be explained by comparable reasons (e.g. Mozambique or Tajikistan). Next to the importance of the physical geographical characteristics of a country, we observe that on average, countries with lower income levels experience larger network disruptions than those with higher income levels.

Local attack on road clusters
Local attack analysis highlights the vulnerability of a country's road network to the removal of a cluster of road segments. These local attacks are a proxy for the degree of disruption caused by spatially correlated hazards, such as volcanic eruptions, earthquakes, medium-to large-scale floods, or tropical cyclones that cause flooding, extreme rainfall, and high-gust winds across a large area. The results highlight that the number of disrupted trips as a result of removing clusters grows substantially based on the size of the disruption scheme. Across all countries, the average number of trips that are fully disrupted between at least two parts of the country is ∼9.8% as a result of a local attack cluster of 5 × 5 km, ∼13.2% for a cluster of 10 × 10 km, and ∼27.3% for a cluster of 50 × 50 km. For 5 × 5 km clusters, the maximum number of trips that are fully disrupted as a result of these edge failures is 26.8%, which increases to ∼30.9% for 10 × 10 km clusters and ∼45.5% for 50 × 50 km clusters. The maximum number of trips that would be delayed decreases with an increase in cluster size (e.g. ∼34% for a 5 × 5 km and ∼7.5% for a 50 × 50 km cluster). This shows that, as we will also observe in the results for the random attack analysis (section 3.3), when a critical number of edges is removed, re-routing is no longer possible, and we experience a steep increase in fully disrupted travel trips. Figure 4 presents the results of removing 50 × 50 km clusters across various regions of the world. The results, perhaps not fully surprising, show that larger countries (e.g. France, Poland, and South Africa) or countries with very dense road networks (e.g. The Netherlands) accommodate fewer clusters of edges that can cause a full disruption of travel trips, relative to the smaller countries. Within Europe, for example, both the amount of fully disrupted and delayed trips are most prominent in eastern European countries.  The results for the local attack method show, on a global scale, a less pronounced difference between high-income and low-country countries relative to the targeted (section 3.1) and random attack (section 3.3). This is largely driven by the various microstates in Europe such as Andorra and San Marino (as highlighted in figure 4(B) as grid cells with high levels of disruption), and the various high-income small islands, such as the British Virgin Islands and Saint Kitts and Nevis (as highlighted in figure 4(A) as grid cells with high levels of disruption). To reduce the influence of such outliers, we make a comparison based on the median value. The median of the average number of trips that are fully disrupted as a result of a 10 × 10 km cluster is not substantially different for Sub-Saharan Africa (median value of 3.4%) versus the rest of the world (3.1%). However, the maximum number of trips that can be fully disrupted due to a 10 × 10 km cluster increases towards 19.5% for Sub-Saharan Africa, relative to 15.9% for the rest of the world. For the least developed countries in the world, the maximum number of fully disrupted trips further increases towards 29.8% (relative to 13.3% for the rest of the world). Excluding microstates and small islands, we find the largest average number of trips fully disrupted due to a 10 × 10 km cluster in The Gambia (∼17.8%) Brunei (∼17.3%), followed by and Guyana (∼14.6%).
Many of the SIDS and other small countries experience large disruptions with the removal of all roads within a 5x5km cluster. On average ∼25% of all trips become disrupted due to the removal of all roads within a 5 × 5 km cluster on SIDS, relative to a global mean of ∼5% for all other countries. This is because, for small countries, these disruptions are relatively large compared to their country size, which is consistent with the extreme vulnerability of small countries to natural disasters, as disasters can affect their entire economy at the same time. For example, on Mahé, the largest island of Seychelles, we find a maximum possible disruption of approximately 75% of the trips due to a single 5 × 5 km cluster. Interestingly, for the Seychelles we find only a maximum disruption of approximately 5% of all the trips due to the failure of a single road segment (i.e. a targeted attack, section 3.1). Similar differences between targeted and local attacks can be found for the British Virgin Islands (5%-75%) and Malta (2%-28%).
These results highlight the importance of certain road clusters in a country's network. In other words, many countries have clusters of roads that (1) are spatially close to each other and can therefore easily be disrupted by a single natural hazard and (2) are critical for the performance of the national road network. In the case of smaller countries, this is predominantly a road network within the country's capital.

Random attack on country-scale networks
The random attack analysis highlights how the disruption of a randomised selection of road segments can influence the nationwide flow of movements. These random attacks can be interpreted as a proxy for national-scale wind or snowstorm events because they randomly disrupt some percentage of road segments in a large spatial area. The results show that 50% of all trips are fully disrupted within a country when approximately 19% (global average) of the segments are randomly removed from the road network. After the random removal of 39% (global average) of the segments within a country's road network, we find that, on average, 95% of the trips are fully disrupted. Figure 5 provides a summarised overview of the random attack analysis, aggregated through various characteristics (i.e. income groups, density of built-up area), and split out based on the percentage of disrupted trips (figure 5(I)), delayed trips (figure 5(II)), and travel-time loss (figure 5(III)). The results show that low-income countries experience approximately 30% more disrupted trips at a 20% network loss relative to high-income countries (Panel I.A). Moreover, the results show that the lower bound of fully disrupted trips as a result of network loss in low-income countries is, on average, almost always higher than the upper bound of fully disrupted trips in high-income countries. In other words, we find that the road networks of the least vulnerable low-income countries are in almost all cases still more vulnerable than those of the most vulnerable high-income countries.
The higher levels of network vulnerability for SIDS found in some of the previous results are not shown in the random attack results. Panel I.B. shows that at 20% network-level loss, SIDS experience approximately 15% less disrupted trips. This can be explained through Panel I.D, which describes the disrupted trips in relation to the average percentage of built-up area within a country. The countries with the highest percentages of built-up area experience almost 40% less disrupted trips compared to countries that have the lowest percentages of built-up area. Several SIDS are among the countries with the highest percentages of built-up area, such as Aruba, Barbados and Puerto Rico. Moreover, this also relates to the way we subdivided each country into a grid of 100 cells. For small islands, this mostly means that we look at intra-urban journeys where network redundancy may be higher (see section 2.2). Whether a country is mountainous makes a large difference to the percentage of disrupted trips. The most mountainous countries experience almost 35% more disrupted trips compared to the least mountainous countries. This is in line with the results found for the targeted and local attacks, as most mountainous countries are dependent on specific tunnels or mountain passes between valleys, and often there are only a few routes that can reach certain areas.
Panel II in figure 5 presents the results for the delayed trips. For several country characteristics (i.e. income level and average percentage of built-up area), the percentage of delayed trips shows a contrasting pattern relative to the percentage of isolated trips. More specifically, countries with relatively few isolated trips experience a rapid increase in delayed trips. For example, 80% of the trips are delayed in high-income countries when roughly 5% of the network is lost, while at that stage only less than 10% of the trips are not possible anymore. Interestingly, whether a country is landlocked or mountainous (Panels II.D and II.F, respectively) also affects delayed trips more negatively. In other words, both the percentage of isolated trips and the percentage of delayed trips are higher in these countries. This indicates that such countries can be considered to be the most vulnerable to random disruptions across the network. This is also in line with the results found in our targeted (section 3.1) and local attack analyses (section 3.2). Mountainous countries have clear geographical obstacles to building a more redundant road network. Landlocked countries may have less developed road networks around the borders, because under normal conditions traffic may use highways just across the border (e.g. Austria or Luxembourg).
Panel III in figure 5 presents the results for the travel time losses. The travel time loss combines both the isolated and delayed trips into one metric. Interestingly, now the difference between high and low-income countries is much less pronounced, whereas the percentage of built-up area and whether a mountainous country is much more persistent. The travel time losses combine both the disrupted and delayed trips and can be considered as a proxy for societal disruption. As such, these results highlight that despite the fact that high-income countries are much less vulnerable to fully disrupted trips (i.e. parts of the country are no

National Road Vulnerability Index (NRVI)
We create a NRVI (see Methods) to compare the overall road network vulnerability of each country. In our results, the United States, China and India show the lowest combined network vulnerability, whereas The Comoros, Bhutan and Trinidad and Tobago experience the highest overall road network vulnerability (figure 6(C), supplementary material). In general, countries with the largest road networks are the least vulnerable to disruptions. However, some of the smallest networks are the most vulnerable ( figure 6(A)). Among the most vulnerable countries (supplementary material), we primarily observe SIDS (e.g. Nauru, Saint Lucia), highly mountainous countries (e.g. Bhutan, Andorra) and some of the lowest-income countries (e.g. Somalia and Madagascar). Figure 6(B) highlights the vulnerability of the least flat countries, which have a median country ranking of 132, compared with 89 for the most flat countries. While China is also part of the >66th percentile of the least flat countries (hence the leftmost whisker towards almost 0), this category includes countries such as Chile (ranked 128th), Papua New Guinea (ranked 203th), and Bhutan (ranked 206th). Figure 6(D) shows the variation across the continents. North America, Europe, and South America show an overall prioritisation towards strengthening the already existing road network to lower its vulnerability, while countries in Asia and Africa vary substantially more. These countries require a more detailed approach to identify their needs. For example, Zimbabwe's road network scores relatively high in our analysis (ranked 27th), indicating that the redundancy in the network is relatively high. This would suggest that reducing its vulnerability would mostly benefit from ensuring good quality of the existing road system. Senegal (ranked 117th), on the other hand, would most likely benefit more from increasing the number of all-weather roads between key locations.
It is important to note that we do not include the quality of roads in this analysis, and all results are purely based on how the network topology is configured through a country's set of nodes and road edges. This does provide us with some interesting results. When we compare our index to global rankings of road quality (Schwab 2019) or mean speed score (Moszoro and Soto 2022), we find some substantial differences. While the road networks of countries such as Brazil and Nigeria are identified as relatively least vulnerable in our results, they score poorly on road quality and travel speeds. Our interpretation here is that for such countries, future investments should not necessarily be prioritised towards more road infrastructure, but rather be focused on strengthening the already existing infrastructure (figure 6(C)). On the other hand, most of the countries that we identify as the most vulnerable (e.g. Papua New Guinea, South Sudan), also happen to be the countries that often have very low road qualities (Schwab K 2019, Moszoro andSoto 2022). For these countries, a more holistic approach is required, where a balanced approach should be taken between strengthening crucial road segments, and investing in new segments where network redundancy is most lacking.

Discussion and conclusion
In this study we have analysed country-level road network vulnerabilities through the use of a globally applicable open-access modelling framework. The interoperability of our approach across all countries allows for a consistent comparison of vulnerabilities between countries. This enables us to suggest a prioritisation for future investments to strengthen the resilience of the road network. Our analysis finds that low-income countries suffer disproportionately from single-link disruptions and random attacks compared to high-income countries; they have twice as many individual edges on average, which may cause a full disruption of at least one route, and the disruption of one edge causes the full disruption of more trips compared to high-income countries. Low-income countries are always more vulnerable to random attacks and experience 30% more disrupted trips at 20% network loss than high-income countries. These differences are caused by lower network redundancy and call for increasing the number of all-weather roads between key locations, while immediately strengthening the critical road segments that are constrained by geography and can disrupt half of the countries' trips (e.g. Guyana, Bhutan, Papua New Guinea, and The Gambia).
As highlighted by He et al (2022), it is difficult to find a clear relationship between 'traditional' network topology characteristics (e.g. network density or connectivity) and the level of network vulnerability across different income groups and disruption schemes. Therefore, we refrained from presenting these (weak) correlations within this analysis. However, it should be noted that previous studies have identified them. For example, Kasmalkar et al (2020) found a clear relationship between the general characteristics of the road network and its level of resilience. However, they used a much more detailed traffic-simulation model. A future analysis in which our approach is enriched by a more detailed network flow model may provide clearer relationships between general network characteristics and network vulnerability to disruptions across countries. Yet, the complexity and diversity of real-world country networks may remain to complex and diverse to identify such clear correlations (He et al 2022).
Moreover, our analysis confirms the extreme vulnerability of SIDS road networks to disruptions caused, for example, by natural hazards (Rozenberg et al 2021). This can be directly attributed to the relative size of the overall network compared to the size of the potential attack (disruptions of all roads within 5 × 5 km can cause very high levels of disruption in a small country). Given the geographical constraints on increasing the redundancy of the road network in SIDS, the priority should be to strengthen the most critical road segments while increasing inter-modal redundancy around the most critical clusters of roads (for example with water or air transport) to ensure that evacuation and access to health centres is possible in the case of a natural hazard (Rentschler et al 2021, Petricola et al 2022. Similar results and recommendations are valid for mountainous countries that are constrained in the kilometres of all-weather roads that can be built at an affordable cost (Oh et al 2019). However, as raised by Wenz et al (2020) and Laurance and Arrea (2017), expanding infrastructure should be carefully considered in conjunction with its environmental impacts. While our analysis supports SDG9, it is important to note that expanding road networks may conflict with other SDGs such as climate change mitigation (SDG13) or biodiversity conservation (SDG15). As such, a holistic approach, taking into account all societal and environmental impacts is essential for future infrastructure investment decisions (Laurance and Arrea 2017).
While our analysis finds that high-income countries have more resilient road networks overall, we still find individual network weaknesses in various countries. The results also demonstrate the dependence of some countries on foreign road networks. For example, we find that the road network of Canada contains road segments which can disrupt half of the simulated travel trips across the country (i.e. segments along the Trans-Canada Highway, north of Deception Bay). This highlights the dependency of some countries on the road networks of neighbouring countries and the importance of open borders to allow for multiple routes and a more redundant transport system. For example, van Ginkel et al (2022) showed that the vulnerability of Austria's road network decreases when one also includes the road networks of neighbouring countries (e.g. Germany).
Summarising these results, a map of global priorities based on our analysis shows that investments in new roads could be prioritised in Central America, the Caribbean, Sub-Saharan Africa, Southeast Asia, and the Pacific, focusing on low-income countries and SIDS, provided that the geography of these countries allows for more redundancy in networks. In other countries, the priority is to strengthen critical road segments, particularly if they are exposed to natural hazards or other threats.

Data availability statement
The data that support the findings of this study are openly available at the following URL/DOI: 10.5281/ zenodo.7234361.