# On Node Criticality in Air Transportation Networks

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## Abstract

In this study, we analyze the criticality of nodes in air transportation using techniques from three different domains, and thus, three essentially different perspectives of criticality. First, we examine the unweighted structure of air transportation networks, using recent methods from control theory (maximum matching and minimum dominating set). Second, complex network metrics (betweenness and closeness) are used with passenger traffic as weights. Third, ticket data-level analysis (origin-destination betweenness and outbound traffic with transit threshold) is performed. Remarkably, all techniques identify a different set of critical nodes; while, in general, giving preference to the selection of high-degree nodes. Our evaluation on the international air transportation country network suggests that some countries, e.g., United States, France, and Germany, are critical from all three perspectives. Other countries, e.g., United Arab Emirates and Panama, have a very specific influence, by controlling the passenger traffic of their neighborhood countries. Furthermore, we assess the criticality of the country network using Multi-Criteria Decision Analysis (MCDA) techniques. United States, Great Britain, Germany, and United Arab Emirates are identified as non-dominated countries; Sensitivity analysis shows that United Arab Emirates is most sensitive to the preference information on the outbound traffic. Our work gears towards a better understanding of node criticality in air transportation networks. This study also stipulates future research possibilities on criticality in general transportation networks.

### Keywords

Network criticality Air transportation systems Complex network## 1 Introduction

Air transportation is a complex socio-technical system, since there are large numbers of different components with different characteristics. Ambitious goals have been set for European air transportation, in terms of quality and affordability, environment, efficiency, safety, and security (European Commission 2001). With rapid development of economic globalization and increased demand of air traffic, congestion frequently occurs and leads to huge economic losses. It is critical to understand the complexity and emergent behavior of air transportation.

Network science provides powerful tools for understanding the structures and dynamics of air transportation (Barabasi 2013; Newman 2010; Zanin and Lillo 2013). In particular, complex network theory has been widely applied to air transportation (Vitali and Et al 2012; Sun and Wandelt 2014; Wandelt and Sun 2015), where nodes are individual airports or air navigation route points. A significant number of studies have focused on delay propagation (Fleurquin et al. 2013), epidemic spreading (Gomes et al. 2014), passenger flows (Lehner et al. 2014; Lehner and Gollnick 2014), and temporal evolution (Kotegawa et al. 2011; Sun et al. 2015). An overview of how complex network theory is applied in geography and regional science is provided in (Ducruet and Beauguitte 2014). The impact of large-scale disruptions on air transportation networks has gained interest recently (Caschili 2015a, b), and it is one of the most important problems, besides emission reduction (Yang et al. 2016; Aziz et al. 2016; Mascia et al. 2016). Furthermore, Zhang et al. applied stochastic actor-based modeling to European airport network between 2003 and 2009, in order to understand the effects of exogenous and endogenous factors on the network dynamics (Zhang et al. 2015). Lijesen et al. developed an empirical model to explain the phenomenon of relatively high fares for flights leaving from carriers’ hubs (Lijesen et al. 2004).

Radicchi et al. introduced a set of heuristic equations for the full characterization of percolation phase diagrams in finite-size interdependent networks, where the percolation transitions can be understood by the decomposition of these systems into a) the intersection among the layers and b) the remainders of the layers (Radicchi 2015). The smoothness percolation transition depends on which decomposed parts dominate the other. Colizza et al. studied the role of airline transportation network in global epidemic spreading and evaluated the reliability of forecasts with respect to the stochastic attributes of disease transmission and traffic flows (Colizza et al. 2006). It is shown that the complex features of airline transportation network are the origin of heterogeneous and erratic spreading of epidemics. Barrat et al. presented statistical analysis of complex networks whose edges are assigned with certain weights (flow or intensity), worldwide air transportation network and scientific collaboration network were used as two typical examples (Barrat et al. 2004). The correlations among weighted quantities and topological structures were exploited. Several research on international air transport has dealt with gravity models and trade, among which Arvis and Shepard proposed air connectivity index to measure the importance of a country as a node in global air transport system; the value for a country is higher if it has stronger pull on the rest of the network and the cost of moving to other countries in the network is relatively low (Arvis and Shepherd 2011).

In this study, we analyze the criticality of nodes in air transportation using techniques from three different domains, and thus, three essentially different perspectives of criticality. First, we examine the unweighted structure of air transportation networks, using recent methods from control theory (maximum matching and minimum dominating set). Second, complex network metrics (betweenness and closeness) are used with passenger traffic as weights. Third, ticket data-level analysis (origin-destination betweenness and outbound traffic with transit threshold) is performed. As a case study, we chose the international air transportation country network. Moreover, we perform multi-criteria assessment of node criticality and we investigate in how far our results are evolutionary stable, by analyzing data for multiple years.

The three main contributions of our paper are: 1) We investigate the criticality of nodes in air transportation networks by using techniques from three different domains: Recent methods from control theory, standard weighted network metrics, and ticket data-level analysis. 2) Our experiments on the air transportation country network reveal that all techniques identify a different set of critical countries; while most techniques give preference to selecting high-degree nodes as critical. We also show that the criticality is rather stable over time. 3) Multi-criteria assessment of the node criticality shows that four countries are non-dominated among all 223 countries and the outbound traffic is the most sensitive criterion.

This paper is organized as follows. In Section 2, we review the state-of-the-art literature on network criticality. Section 3 presents how to construct the country network based on real ticket data. In Section 4, we present the results of network criticality for international air transportation. Section 5 discusses our findings from a multi-criteria assessment point of view. The paper is concluded in Section 6.

## 2 Background

In this section, we review relevant work on node criticality and country networks. Network controllability is a new research field pioneered by Liu et al. (2011, 2012). With the combination of tools from network science and control theory, Liu et al. explored the controllability of complex systems and proved that the minimum number of driver nodes, whose control is sufficient to fully control a system’s dynamics, is determined by the Maximum Matching (MM), i.e., the maximum set of links that do not share start or end nodes. The driver nodes are those unmatched nodes which have no incoming links in the MM. Due to the existence of multiple control configurations, Jia et al. classified the role of each node in control: Critical, intermittent, or redundant, depending on whether the node acts as a driver node in all, some, or none of the control configurations, leading to the discovery of two distinct control modes in complex systems: Centralized and distributed control (Jia et al. 2013). Furthermore, a random sampling algorithm was developed to estimate the control capacity, which quantifies the likelihood that a node is a driver node (Jia and Barabasi 2013). The connection between the core percolation and the network controllability has also been studied and it was found that the core structure determines the control modes (Jia and Pósfai 2014).

While the MM technique is applied to directed networks, Nacher and Akutsu proposed a similar optimization procedure to identify nodes which play important roles for the control of undirected and directed networks using the Minimum Dominating Set (MDS) technique (Nacher and Akutsu 2014a). A dominating set is a set of nodes in a network where each node itself is either an element of this set or is adjacent to an element of this set; the MDS is the dominating set with the smallest size. Note that both the MM and MDS techniques generate multiple configurations of driver nodes. Similarly with Jia et al. (2013), Nacher and Akutsu classified the nodes depending on the whether a node is part of all (critical), some but not all (intermittent), or does not participate in any MDS (redundant). The MDS technique has been used to examine the controllability of protein-protein networks and it was found that proteins in the MDS have a higher impact on network resilience (Wuchty 2014). Moreover, Molnar et al. studied the vulnerability of dominating sets against random and targeted node removals in complex networks (Molnár Jr et al. 2015).

Recently, country networks have also been studied in social network fields. Hawelka et al. analysed geo-located Twitter messages in order to uncover global patterns of human mobility (Hawelka et al. 2014). A country-to-country mobility network of tweet flows was constructed, where each country is a node, and the link between two countries is weighted by the number of Twitter users exchanging tweets. Kaltenbrunner et al. analysed the social network of sister cities: A country social network was built, where two countries are connected if a city of one country is twinned with a city of the other country (Kaltenbrunner et al. 2014). Furthermore, Deguchi et al. analyzed the world trade network, the exports and imports among the countries are the links, weighted by the annual amounts of trade (Deguchi et al. 2014). The economic influences of the countries in the world trade network were investigated using two network metrics: Hubs (exporting) and authorities (importing). Cashili et al. constructed an interdependent multi-layer international trade model for 40 countries, including economic, socio-cultural and physical layers, and examined the influence of shocks on the interdependent networks (Caschili et al. 2015a).

## 3 Construction of the International Air Transportation Country Network

We analyse the traffic patterns of passengers in international air transportation, by building a weighted and directed country network based on the ticket data from the Sabre Airport Data Intelligence (ADI, http://www.airdi.net) in the year 2013. Each flight ticket provides the following information: Origin/destination airports, up to three connecting airports (hops), and the number of passengers who bought this ticket during the whole year 2013.

Below we describe formally how we transform the airport-based ticket data into a country network. We denote a ticket as a tuple *t* = 〈[*a*_{1}, ... , *a*_{m}], *p*〉, such that each *a*_{i} is an airport, *a*_{1} is the origin, *a*_{m} is the destination, and *p* is the number of passengers who bought this ticket. The ticket database consists of *n* tickets {*t*_{1}, ... , *t*_{n}}. The list of airports is unique for each ticket, i.e. for any *t*_{i} = 〈*A*_{i}, *p*_{i}〉 and *t*_{j} = 〈*A*_{j}, *p*_{j}〉, we have the constraint: *i* ≠ *j* → *A*_{i} ≠ *A*_{j}.

We perform pre-processing on these tickets with the following steps. First, we map each airport to the country of the airport and obtain for each ticket *t* a country ticket *c**t* = 〈[*c*_{1}, ... , *c*_{m}], *p*〉, where each *c*_{i} is a country. Second, we replace all consecutive occurrences of the same country, since we are not interested in flights within a country, i.e. all occurrences of *c*, *c* are replaced with *c*. Third, we group all country tickets based on the country list (the aggregation from airports to countries invalidates the uniqueness constraint), such that the number of passengers is summed up over all country tickets with the identical country list. After the pre-processing, we have a set of country tickets *C**T* = {*c**t*_{1}, ... , *c**t*_{m}}. The set of countries is denoted with *C*.

*P*

*H*,

*C*

*N*,

*S*

*A*,

*B*

*H*,

*S*

*A*), is a valid ticket, despite Saudi Arabia (SA) occurs two times as hop and as destination: This ticket is sold by Cathay Pacific for the city connection (Manila, Hongkong, Riad, Muharraq, Dammam) and is used frequently by around 100 passengers per month. Moreover, note that since demographics of passengers are not available, origin/destination airports of passengers are not necessarily the same as their home countries.

Next, we describe how to obtain the physical network from the country tickets. In general, a network \(\mathcal {N}=\langle N,L,w\rangle \) consists of a set of nodes *N*, a set of links *L* ⊆ *N* × *N*, and a weight function *w* mapping links in *L* to real numbers. For weighted networks, we have that (*n*_{1}, *n*_{2}) is in the domain of *w*, if and only if (*n*_{1}, *n*_{2}) ∈ *L*. The physical network \(\mathcal {N}_{phys}=\langle N,L,w\rangle \) is initially empty and then constructed in the following way: First, we set *N* = *C*. We iterate over all country tickets in *CT*. For each ticket *c**t* = 〈[*c*_{1}, ... , *c*_{m}], *p*〉, and for each 1 ≤ *j* < *m*, we check whether (*c*_{j}, *c*_{j+1}) ∈ *L*. If (*c*_{j}, *c*_{j+1}) ∈ *L*, then we set *w*(*c*_{j}, *c*_{j+1}) = *w*(*c*_{j}, *c*_{j+1}) + *p*. If (*c*_{j}, *c*_{j+1})∉*L*, we add (*c*_{j}, *c*_{j+1}) to *L* and set *w*(*c*_{j}, *c*_{j+1}) = *p*.

*small-world*network (Watts and Strogatz 1998). With roughly equivalent average shortest path length, the random counterpart Erdös-Rényi networks have much lower clustering coefficients (around 0.15) than the country network. Therefore, we can say that the country network has a small-world property. In other words, it takes passengers only a few number of flight hops to travel to any other country in the world. Figure 3 shows the correlation among selected standard network metrics in the country network. The perfect correlation between the degree (the total sum of all links) and in-degree (the sum of incoming links) shows that the country network is nearly symmetric. There exist other pairs of metrics with high correlations, most of which are positively related to the degree (e.g., closeness).

## 4 Empirical Analysis of Criticality in the Country Network

In this section, we present the methodology to assess the criticality of nodes in the international air transportation country network. We first explain our methodology using a small network example. Secondly, we present the results of criticality analysis for the country network, grouped by the maximum matching and minimum dominating set, standard weighted network metrics, as well as real ticket data analysis.

### 4.1 A Running Network Example

In our small example, two different types of tickets were sold for the OD pair (*A*, *B*). First, ten passengers chose a direct flight from *A* to *B*, which is reflected with link weight 10 in the physical network. Second, two passengers chose to go from *A* to *B* using the two hops *D* and *C*. In the physical network, this yields two passengers for the weight of link *A* → *D*, two passengers for *D* → *C*, and two passengers for *C* → *B*. The latter two links have a weight of eight passengers in the physical network; six of which are created for the OD pair (*D*, *B*) using *C* as a hop. Similarly, all other links in the physical network are populated. Finally, the effective distance network is obtained by replacing each link weight with its reciprocal value: For instance, the link weight *A* → *B* is rewritten from 10 to \(\frac {1}{10}\) and the link weight of *A* → *D* is rewritten from 2 to \(\frac {1}{2}\). Intuitively, *B* is closer (=smaller weight) to *A* than *D* is to *A*, since more passengers are traveling from *A* → *B*.

Below we explain each of the six criticality techniques on the small network example first, then we discuss the results of the criticality analysis for the country network.

### 4.2 Criticality based on Maximum Matching and Minimum Dominating Set

We recall the rationale of *maximum matching* first. For a directed graph, a link subset *M* is a matching if no two links in *M* share a common starting or ending node. A node is matched if it is an ending node of a link in the matching. Otherwise, it is unmatched. A matching *M* with the maximum size is called a Maximum Matching (MM) (Liu et al. 2011). The classification for the control role of each node depends on whether it acts as a driver node in all (=critical), some (=intermittent), or none (=redundant) of the control configurations (Liu et al. 2011; Jia et al. 2013).

*n*, we generate two new nodes

*n*

_{I}and

*n*

_{O}. 2) For each link (

*a*,

*b*) in the original network we add a link from

*a*

_{O}to

*b*

_{I}in the bipartite network. 3) We compute a maximum matching in the bipartite network to obtain an initial maximum matching. This matching, projected back on the original network, is visualized in Fig. 5, upper left. The number of driver nodes is set to be the number of unmatched nodes, i.e. nodes which do not occur as an ending node of any link in the matching. In our example nodes

*A*and

*C*are unmatched. Second, we compute a maximum matching following the same methodology, for the network with each node removed separately. The results are visualized in the remaining subfigures of Fig. 5. Since the number of driver nodes is increased from 2 to 3 when

*F*is removed, and unchanged for all other nodes, only

*F*is critical.

Figure 7a shows the critical nodes (in red color) in the country network based on the MM technique. Among the 223 countries, 77 countries are identified as critical, including most countries from South/North America and several African countries. These results seem to be counter-intuitive and deserve explanation. According to Liu et al., a system is said to be controllable if it can be driven from any initial state to any desired final state in finite time (Liu et al. 2011), by the identification of driver nodes, which (simplified) act as inputs to the network, but are not influenced by other nodes. However, in air transportation networks, each node serves as an input and output, i.e., passengers can travel from every country to any other country in general, as long as there is a connecting set of flights. Therefore, we think the concept of driver nodes regarding Liu et al.’s point of view cannot be applied directly to air transportation networks.

*Minimum Dominating Set*(MDS) technique. In an undirected or directed graph, a dominating set is a set of nodes where each node itself is either an element of this set or is adjacent to an element of this set; the MDS is the dominating set with the smallest size (Nacher and Akutsu 2014a). The nodes can be classified depending on whether a node is part of all (=critical), some but not all (=intermittent), or does not participate in any possible MDS (=redundant) (Nacher and Akutsu 2014a). In our running example network shown in Fig. 4, one MDS is {

*A*,

*C*,

*F*}, since node

*B*is a successor of

*C*, node

*D*is a successor of

*A*, and

*E*/

*G*are successors of

*F*. Following the post-processing of an MDS in Nacher and Akutsu (2014a), we obtain that

*A*and

*F*are critical, since there exists no MDS without either

*A*or

*F*. On the other hand,

*C*is not critical, since there exists another MDS of size three (

*A*,

*D*,

*F*). Thus, only the nodes

*A*and

*F*are critical according to the MDS technique. This situation is visualized in Fig. 6.

### 4.3 Criticality based on Standard Weighted Network Metrics

The dynamic load on the country network corresponds to the number of passengers traveling between two countries. It is interesting to look at the weighted network metrics with passenger traffic. Several centrality measures have been proposed to characterize the importance of a node in a network. We analyze two standard weighted network metrics for the country network: Weighted closeness and weighted betweenness. Below we explain why we select these two metrics.

Recall that the closeness of a node is the reciprocal of the sum of its distance to all other nodes in a network; while the betweenness is the number of shortest paths passing through this node (Freeman 1978). We compute the weighted metrics with effective distance (Brockmann and Helbing 2013). A node with higher value of closeness can access other nodes easily, especially in case of efficient information spreading in the network. Moreover, betweenness can be interpreted as the extent to which a node has influence over pair-wise connections between other nodes, assuming that the importance of connections is equivalently distributed among all shortest paths for each node pair (Brandes 2008). Therefore, we think weighted closeness and weighted betweenness can be regarded as measures of network criticality. In our small example network, shown in Fig. 4, the top three nodes with the largest weighted closeness values are *D* = 1.98, *A* = 1.35, and *C* = 1.33. The top three nodes according to the weighted betweenness values are *F* = 0.33, *G* = 0.17, and *D* = 0.13.

Figure 7c shows the top ten countries ranked by the weighted closeness in the country network: Great Britain is closest to all other countries in the world, followed by United States. Recall that we use the *effective distance* as link weight and a large fraction of traffic is effectively equivalent to a short distance. Six out of the top ten countries are from Europe (Great Britain, Spain, Germany, France, Italy, and Ireland), which indicates strong intra-European traffic. Canada and Mexico are also top ranked because of strong air traffic connections inside North/Middle America. United Arab Emirates, given its central geographic position, is close to all other countries in the network (mainly to other countries in South-West Asia).

Figure 7d presents the top ten countries ranked by the weighted betweenness. Similarly with the weighted closeness, Great Britain ranks first, followed by United States. It is interesting to note that United Arab Emirates ranks third. With increasing economic cooperation between Asia/Europe and Asia/Africa (Rose 2012), especially the boosted economic growth of China and India, United Arab Emirates serves as important bridges for air traffic connections among these regions.

Network metrics work very well for identifying influential people in social networks, by considering shortest paths between all pairs of nodes. For transportation networks, however, one needs to consider that there is no transportation request for all pairs of nodes, and thus this naive strategy could be misleading in some cases. For instance, even United States (the one with the most destinations, in total 180) is not connected to all countries in the world. Other countries have much fewer travel destinations, for example, passengers from Trinidad and Tobago only travel to 34 distinct destinations (mostly countries in North/South America). Taking into account all other destinations for Trinidad and Tobago would distort the betweenness scores of nodes. We will further discuss this issue in the next section.

### 4.4 Criticality based on Real Ticket Data

In air transportation networks, computing all shortest paths is often impractical since there are pairs of nodes without any demand. To address this problem, the alternative definition of OD betweenness was introduced, which is computed based on shortest paths between all OD pairs only (Monechi and Others 2013). In our example network shown in Fig. 4, we have eight OD pairs, three out of which are direct connections and thus not relevant (*A* → *B*, *E* → *G*, and *F* → *G*). For the other five OD pairs, we compute the shortest paths between them and identify how frequently each node occurs on these shortest paths. As a result, only node *C* and node *F* lie on shortest paths, according to the eight OD pairs. All other nodes do not appear on any shortest path. This result is substantially different from the original betweenness.

Figure 7e presents the top ten countries ranked by the weighted OD betweenness in the country network. Compared with the original weighted betweenness shown in Fig. 7d, the top three countries keep the same; Spain, Germany, and France are still ranked in the middle position; while Australia and South Africa are replaced by another two new countries (India and Turkey) in the list.

The last method we use for criticality analysis is the outbound traffic based on the real ticket data. We formalize the notion of criticality in the country network here. For each country *c*_{1}, we check how much non-direct outbound traffic of *c*_{1} can be blocked by another country *c*_{2}. We only look at traffic with *c*_{1} as origin. In the small example network, the outbound traffic method yields, for instance, that *C* is controlling 50 % of the outbound passengers from node *D* (all these passengers have destination *B*). The other 50 % of the outbound passengers travel via *G*. Thus, *C* and *G* have a high control over the outbound traffic of *D*. In order to obtain a binary measure, the criticality of a country using the outbound traffic technique depends on a transit threshold, which quantifies how much a country’s outbound traffic needs to be under control. In the small example network, the outbound traffic method with a transit threshold of 33 % yields four controlling countries (*C*, *E*, *F*, *G*).

### 4.5 Summary of the Criticality in the Country Network

## 5 Discussion

### 5.1 Multi-Criteria Assessment for the Network Criticality

In this section, we are interested how the six network criticality techniques correlate with each other and with additional network metrics, from the perspective of multi-criteria assessment. In our experiments, we observed that degree is highly correlated with most measures; while the MM proposed by Liu is least correlated, followed by the outbound traffic method (ticket control). Given the twelve criteria, which country is the most preferred one is a typical multi-criteria decision problem. Below, we apply typical Multi-Criteria Decision Analysis (MCDA) techniques to solve this problem.

MCDA is a process to make decisions in face of multiple, potentially conflicting criteria (Ehrgott et al. 2010), among which ELECTRE (Elimination and Choice Translation Reality) and TOPSIS (Technique for Order Preference by Similarity to Ideal Solution) are two widely used MCDA techniques. ELECTRE methods use the concept of outranking relation introduced by Benayoun et al. (1973). One alternative is preferred when this alternative is at least as good as the other ones with respect to a majority of criteria and when it is not significantly poor regarding any other criteria. ELECTRE classifies a set of alternatives into dominated and non-dominated solutions. TOPSIS is based on the idea that the chosen alternative should have the shortest Euclidean distance to the positive ideal solution and the furthest Euclidean distance from the negative ideal solution (Hwang and Yoon 1981). Mathematical description of a MCDA problem as well as the algorithms for ELECTRE and TOPSIS can be found in Appendix.

In this section, we apply ELECTRE I to the country network, taking into account all twelve criteria, with equal weighting factors for the initial evaluation. Among the 223 countries, four are identified as non-dominated: United States (US), Great Britain (GB), Germany (DE), and United Arab Emirates (AE).

When we slightly increase the weighting factor of the outbound traffic method (ticket control, the right panel in the last row), United Arab Emirates (AE) clearly outranks the other countries. Similarly, when the weighting factor of betweenness is slightly increased (the left panel in the second row), United States (US) ranks first immediately. When the network is weighted with the traffic information or taking into account the neighbor’s connections for a node (the four middle panels in the four rows), Great Britain (GB) outranks the other countries.

It should be kept in mind that different techniques often generate different sets of critical nodes. Therefore, one cannot distinguish, in general, which result is right or wrong; it depends on what network features are considered to define the importance of nodes. For instance, if the user wants to evaluate topological roles of nodes in a network, then the maximum matching and minimum dominating set techniques should be applied; if the user is interested in their roles with real traffic information, network metrics with passenger traffic as weights and ticket data-level analysis are recommended.

### 5.2 Temporal Evolution of Network Criticality

### 5.3 Alternative Choices for Network Modelling

## 6 Conclusion

In this research, we investigated the criticality of nodes in air transportation networks. As a case study, we chose the international air transportation country network. We identified critical nodes in the network with three different perspectives of criticality: Recent methods from control theory (maximum matching and minimum dominating set) were used to examine the unweighted structure of the country network; two standard network metrics with passenger traffic as weights; as well as adjusted weighted origin-destination betweenness and outbound traffic based on the real ticket data. We studied different roles of the countries in international air transportation. Our results suggest that some countries (such as United States, France, and Germany) are critical from all three perspectives of criticality. Other countries (such as United Arab Emirates and Panama) have a very specific influence, by controlling the passenger traffic of their neighborhood countries. Furthermore, the node criticality was also assessed using Multi-Criteria Decision Analysis (MCDA) techniques. United States, Great Britain, Germany, and United Arab Emirates were identified as non-dominated countries using ELECTRE; Sensitivity analysis of TOPSIS showed that United Arab Emirates was most sensitive to the preference information of the outbound traffic method (ticket control).

Our work gears towards a better understanding of node criticality in air transportation networks. Future work could focus on the combination of robust control theory with network science (Nacher and Akutsu 2014b). The six criticality techniques also open up future research possibilities on general transportation networks. Moreover, other directions include the passenger’s route choice behavior for node criticality, e.g. with common constraints (Yang et al. 2016), and also look at other network instances, for instance, the networks of networks with multiple modalities (Fotuhi and Huynh 2016) or other abstractions (Fielbaum et al. 2016).

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