Networks and Spatial Economics (2017, accepted) manuscript No.(will be inserted by the editor)

On Node Criticality in Air Transportation Networks

Xiaoqian Sun · Sebastian Wandelt∗ · Xianbin Cao

Received: date / Accepted: date

Abstract In this study, we analyze the criticality of nodes in air transportation using techniques from threedifferent domains, and thus, three essentially different perspectives of criticality. First, we examine the un-weighted structure of air transportation networks, using recent methods from control theory (maximummatching 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 betweennessand outbound traffic with transit threshold) is performed. Remarkably, all techniques identify a differentset of critical nodes; while, in general, giving preference to the selection of high-degree nodes. Our eval-uation on the international air transportation country network suggests that some countries, e.g., UnitedStates, France, and Germany, are critical from all three perspectives. Other countries, e.g., United ArabEmirates and Panama, have a very specific influence, by controlling the passenger traffic of their neighbor-hood countries. Furthermore, we assess the criticality of the country network using Multi-Criteria DecisionAnalysis (MCDA) techniques. United States, Great Britain, Germany, and United Arab Emirates are iden-tified as non-dominated countries; Sensitivity analysis shows that United Arab Emirates is most sensitiveto the preference information on the outbound traffic. Our work gears towards a better understanding ofnode criticality in air transportation networks. This study also stipulates future research possibilities oncriticality 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 componentswith different characteristics. Ambitious goals have been set for European air transportation, in termsof quality and affordability, environment, efficiency, safety, and security [15]. With rapid development ofeconomic globalization and increased demand of air traffic, congestion frequently occurs and leads to hugeeconomic 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 transporta-tion [3,38,48]. In particular, complex network theory has been widely applied to air transportation [43,41,44], where nodes are individual airports or air navigation route points. A significant number of studies havefocused on delay propagation [17], epidemic spreading [20], passenger flows [29,28], and temporal evolu-tion [27,42]. An overview of how complex network theory is applied in geography and regional science is

Xiaoqian SunSchool of Electronic and Information Engineering, Beihang University, 100191 Beijing, ChinaE-mail: sunxq@buaa.edu.cn

Sebastian WandeltSchool of Electronic and Information Engineering, Beihang University, 100191 Beijing, China;Department of Computer Science, Humboldt-University of Berlin, Unter den Linden 6, 10099 Berlin, Germany

Xianbin CaoSchool of Electronic and Information Engineering, Beihang University, 100191 Beijing, China

2 Xiaoqian Sun et al.Sheet3

Origin Destination pax Origin H1 Destination pax Origin H1 H2 Destination pax Origin H1 H2 H3 Destination pax

GB ES 12,870,059 IN AE US 355,852 AU SG AE GB 34,922 NZ AU TH AE GB 3,455

ES GB 12,828,345 US AE IN 334,656 GB AE SG AU 29,466 GB AE TH AU NZ 3,182

ES DE 10,107,228 AU AE GB 287,021 AU BN AE GB 26,911 FR GB AE AU NC 3,102

DE ES 10,071,297 GB AE AU 282,346 IN SG KR US 26,418 NC AU AE GB FR 2,928

CA US 8,571,604 IN AE GB 275,941 GB AE BN AU 24,659 FR EG YE DJ KM 2,809

US CA 8,528,398 GB AE IN 254,436 GB AE AU NZ 24,053 KM DJ YE EG FR 2,680

US MX 8,260,251 IN GB US 248,549 SA AE CN PH 22,031 MT CY AE MY AU 2,500

MX US 8,180,060 US GB IN 218,848 PH CN AE SA 21,968 VG AN KN AG GB 2,235

CN TW 7,302,766 IN SG AU 192,369 NZ AU AE GB 20,701 PH CN SA BH SA 1,724

TW CN 7,204,752 CN KR US 185,266 US CN SG IN 17,832 GB AG KN AN VG 1,493

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Tickets with direct flights (Top10) Tickets with one hop (Top 10) Tickets with two hops (Top 10) Tickets with three hops (Top 10)

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Fig. 1: Top ten tickets according to the number of passengers, grouped by number of hops. For direct flighttickets, intra-continental connections dominate: Most passengers fly between Great Britain (GB), Spain (ES), and Germany(DE), followed by flights inside North/Middle America. One-hop tickets are usually inter-continental connections. Notably,the top six tickets all go through United Arab Emirates (AE) and seven out of ten tickets have India (IN) as origin ordestination. Tickets with two or three hops have considerably less numbers of passengers and often have countries fromSouth Asia/Oceania as origin or destination.

provided in [13]. The impact of large-scale disruptions on air transportation networks has gained interestrecently [9,10], and it is one of the most important problems, besides emission reduction [47,2,33]. Further-more, Zhang et al. applied stochastic actor-based modeling to European airport network between 2003 and2009, in order to understand the effects of exogenous and endogenous factors on the network dynamics [49].Lijesen et al. developed an empirical model to explain the phenomenon of relatively high fares for flightsleaving from carriers’ hubs [30].

Radicchi et al. introduced a set of heuristic equations for the full characterization of percolation phasediagrams in finite-size interdependent networks, where the percolation transitions can be understood bythe decomposition of these systems into a) the intersection among the layers and b) the remainders ofthe layers [39]. The smoothness percolation transition depends on which decomposed parts dominate theother. Colizza et al. studied the role of airline transportation network in global epidemic spreading andevaluated the reliability of forecasts with respect to the stochastic attributes of disease transmission andtraffic flows [11]. It is shown that the complex features of airline transportation network are the origin ofheterogeneous and erratic spreading of epidemics. Barrat et al. presented statistical analysis of complexnetworks whose edges are assigned with certain weights (flow or intensity), worldwide air transportationnetwork and scientific collaboration network were used as two typical examples [4]. The correlations amongweighted quantities and topological structures were exploited. Several research on international air transporthas dealt with gravity models and trade, among which Arvis and Shepard proposed air connectivity indexto measure the importance of a country as a node in global air transport system; the value for a countryis higher if it has stronger pull on the rest of the network and the cost of moving to other countries in thenetwork is relatively low [1].

In this study, we analyze the criticality of nodes in air transportation using techniques from three differentdomains, and thus, three essentially different perspectives of criticality. First, we examine the unweightedstructure of air transportation networks, using recent methods from control theory (maximum matchingand minimum dominating set). Second, complex network metrics (betweenness and closeness) are used withpassenger traffic as weights. Third, ticket data-level analysis (origin-destination betweenness and outboundtraffic with transit threshold) is performed. As a case study, we chose the international air transportationcountry network. Moreover, we perform multi-criteria assessment of node criticality and we investigate inhow 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 trans-portation 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 trans-portation country network reveal that all techniques identify a different set of critical countries; while mosttechniques give preference to selecting high-degree nodes as critical. We also show that the criticality israther stable over time. 3) Multi-criteria assessment of the node criticality shows that four countries arenon-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 networkcriticality. 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 ourfindings from a multi-criteria assessment point of view. The paper is concluded in Section 6.

On Node Criticality in Air Transportation Networks 3

2 Background

In this section, we review relevant work on node criticality and country networks. Network controllabilityis a new research field pioneered by Liu et al. [31,32]. With the combination of tools from network scienceand control theory, Liu et al. explored the controllability of complex systems and proved that the minimumnumber of driver nodes, whose control is sufficient to fully control a system’s dynamics, is determined bythe Maximum Matching (MM), i.e., the maximum set of links that do not share start or end nodes. Thedriver nodes are those unmatched nodes which have no incoming links in the MM. Due to the existence ofmultiple 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 controlconfigurations, leading to the discovery of two distinct control modes in complex systems: Centralized anddistributed control [24]. Furthermore, a random sampling algorithm was developed to estimate the controlcapacity, which quantifies the likelihood that a node is a driver node [23]. The connection between the corepercolation and the network controllability has also been studied and it was found that the core structuredetermines the control modes [25].

While the MM technique is applied to directed networks, Nacher and Akutsu proposed a similar opti-mization procedure to identify nodes which play important roles for the control of undirected and directednetworks using the Minimum Dominating Set (MDS) technique [36]. A dominating set is a set of nodesin a network where each node itself is either an element of this set or is adjacent to an element of thisset; the MDS is the dominating set with the smallest size. Note that both the MM and MDS techniquesgenerate multiple configurations of driver nodes. Similarly with Jia et al. [24], Nacher and Akutsu classifiedthe nodes depending on the whether a node is part of all (critical), some but not all (intermittent), or doesnot participate in any MDS (redundant). The MDS technique has been used to examine the controllabilityof protein-protein networks and it was found that proteins in the MDS have a higher impact on networkresilience [46]. Moreover, Molnar et al. studied the vulnerability of dominating sets against random andtargeted node removals in complex networks [34].

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 [21]. A country-to-countrymobility network of tweet flows was constructed, where each country is a node, and the link between twocountries is weighted by the number of Twitter users exchanging tweets. Kaltenbrunner et al. analysedthe social network of sister cities: A country social network was built, where two countries are connectedif a city of one country is twinned with a city of the other country [26]. Furthermore, Deguchi et al.analyzed the world trade network, the exports and imports among the countries are the links, weightedby the annual amounts of trade [12]. The economic influences of the countries in the world trade networkwere 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 [9].

3 Construction of the International Air Transportation Country Network

We analyse the traffic patterns of passengers in international air transportation, by building a weightedand 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/destinationairports, up to three connecting airports (hops), and the number of passengers who bought this ticket duringthe whole year 2013.

Below we describe formally how we transform the airport-based ticket data into a country network. Wedenote a ticket as a tuple t = 〈[a1, ..., am], p〉, such that each ai is an airport, a1 is the origin, am is thedestination, and p is the number of passengers who bought this ticket. The ticket database consists of ntickets {t1, ..., tn}. The list of airports is unique for each ticket, i.e. for any ti = 〈Ai, pi〉 and tj = 〈Aj , pj〉,we have the constraint: i 6= j → Ai 6= Aj .

We perform pre-processing on these tickets with the following steps. First, we map each airport to thecountry of the airport and obtain for each ticket t a country ticket ct = 〈[c1, ..., cm], p〉, where each ci is acountry. Second, we replace all consecutive occurrences of the same country, since we are not interested inflights within a country, i.e. all occurrences of c, c are replaced with c. Third, we group all country ticketsbased on the country list (the aggregation from airports to countries invalidates the uniqueness constraint),

4 Xiaoqian Sun et al.

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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 CT = {ct1, ..., ctm}. The set of countries is denotedwith C.

Figure 1 shows the top ten country tickets with direct flights, one hop, two hops, and three hops inthe country network. Note that the ticket with three hops ranked ninth (PH,CN, SA,BH,SA), is a validticket, despite Saudi Arabia (SA) occurs two times as hop and as destination: This ticket is sold by CathayPacific for the city connection (Manila, Hongkong, Riad, Muharraq, Dammam) and is used frequently byaround 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 networkN = 〈N,L,w〉 consists of a set of nodes N , a set of links L ⊆ N × N , and a weight function w mappinglinks in L to real numbers. For weighted networks, we have that (n1, n2) is in the domain of w, if andonly if (n1, n2) ∈ L. The physical network Nphys = 〈N,L,w〉 is initially empty and then constructedin the following way: First, we set N = C. We iterate over all country tickets in CT . For each ticketct = 〈[c1, ..., cm], p〉, and for each 1 ≤ j < m, we check whether (cj , cj+1) ∈ L. If (cj , cj+1) ∈ L, then we setw(cj , cj+1) = w(cj , cj+1) + p. If (cj , cj+1) /∈ L, we add (cj , cj+1) to L and set w(cj , cj+1) = p.

An illustration of the physical country network for the year 2013 is shown in Figure 2, where each countryis a node, and the link between two countries is weighted by the number of travelled passengers. In total,there are 223 nodes and 5,119 links. The country network has an average degree of 23, an average clusteringcoefficient of 0.629, along with an average shortest path length of 2.254. Small average shortest path lengthbut higher clustering coefficient than its random counterpart network are two structural features for a small-world network [45]. With roughly equivalent average shortest path length, the random counterpart Erdos-Renyi 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 afew number of flight hops to travel to any other country in the world. Figure 3 shows the correlation amongselected standard network metrics in the country network. The perfect correlation between the degree (thetotal sum of all links) and in-degree (the sum of incoming links) shows that the country network is nearlysymmetric. There exist other pairs of metrics with high correlations, most of which are positively relatedto the degree (e.g., closeness).

On Node Criticality in Air Transportation Networks 5

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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 trans-portation country network. We first explain our methodology using a small network example. Secondly, wepresent the results of criticality analysis for the country network, grouped by the maximum matching andminimum dominating set, standard weighted network metrics, as well as real ticket data analysis.

6 Xiaoqian Sun et al.

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4.1 A Running Network Example

Figure 4 shows a small network example how we built the physical country network based on the ticketdata. Given a set of Origin-Destination (OD) pairs, connecting countries, and the number of passengers whobought this ticket, we construct the physical network with two different weights: Number of passengers andeffective distance. The effective distance is based on the idea that a small fraction of traffic is effectivelyequivalent to a large distance, and vice versa [7].

In our small example, two different types of tickets were sold for the OD pair (A,B). First, ten passengerschose 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 yieldstwo 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 createdfor 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: Forinstance, the link weight A→ B is rewritten from 10 to 1

10 and the link weight of A→ D is rewritten from2 to 1

2 . Intuitively, B is closer (=smaller weight) to A than D is to A, since more passengers are travelingfrom A→ B.

Below we explain each of the six criticality techniques on the small network example first, then wediscuss 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 notwo links in M share a common starting or ending node. A node is matched if it is an ending node of a linkin the matching. Otherwise, it is unmatched. A matching M with the maximum size is called a MaximumMatching (MM) [31]. The classification for the control role of each node depends on whether it acts as adriver node in all (=critical), some (=intermittent), or none (=redundant) of the control configurations [31,24].

We describe the construction of a maximum matching for an example network shown in Figure 4. First,the network is converted into an undirected bipartite graph as follows (see Supplementary of [31]): 1) For

On Node Criticality in Air Transportation Networks 7

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D is not critical E is not critical F is critical (since size increased) G is not critical

Fig. 6: Example for the construction of the minimum dominating set. The MDS for the original network is shown,together with the MDS excluding each node separately. Nodes taking part in the MDS are highlighted in yellow color. NodeA is critical, since there is no MDS without it. Node F is critical, because the size of the MDS is increased, if F cannottake part in the MDS.

each node n, we generate two new nodes nI and nO. 2) For each link (a, b) in the original network weadd a link from aO to bI in the bipartite network. 3) We compute a maximum matching in the bipartitenetwork to obtain an initial maximum matching. This matching, projected back on the original network,is visualized in Figure 5, upper left. The number of driver nodes is set to be the number of unmatchednodes, i.e. nodes which do not occur as an ending node of any link in the matching. In our example nodesA and C are unmatched. Second, we compute a maximum matching following the same methodology, forthe network with each node removed separately. The results are visualized in the remaining subfigures ofFigure 5. Since the number of driver nodes is increased from 2 to 3 when F is removed, and unchanged forall other nodes, only F is critical.

Figure 7 (a) 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/NorthAmerica 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 anydesired final state in finite time [31], by the identification of driver nodes, which (simplified) act as inputsto the network, but are not influenced by other nodes. However, in air transportation networks, each nodeserves 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 Liuet al.’s point of view cannot be applied directly to air transportation networks.

Next we discuss the 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 adjacentto an element of this set; the MDS is the dominating set with the smallest size [36]. The nodes can beclassified depending on whether a node is part of all (=critical), some but not all (=intermittent), or doesnot participate in any possible MDS (=redundant) [36]. In our running example network shown in Figure 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 successorsof F . Following the post-processing of an MDS in [36], we obtain that A and F are critical, since thereexists no MDS without either A or F . On the other hand, C is not critical, since there exists another MDSof size three (A,D,F ). Thus, only the nodes A and F are critical according to the MDS technique. Thissituation is visualized in FIgure 6.

We have computed the dominating sets using the DSP implementation of OpenOpt (https://pypi.python.org/pypi/openopt). Figure 7 (b) highlights the critical nodes (in red color) in the country networkbased on the MDS technique. 10 out of 223 countries are identified as critical: Two countries from North

8 Xiaoqian Sun et al.

(a) Maximum matching technique [31] (b) Minimum dominating set technique [36]

(c) Weighted closeness (d) Weighted betweenness

(e) Weighted OD betweenness [35] (f) Outbound traffic

Fig. 7: Network criticality for the country network. Critical countries are highlighted with red color. All six techniquesyield distinct sets of critical nodes. The maximum matching technique returns 77 countries as critical, notably almostall countries in North/South America. The minimum dominating set technique yields ten nodes; for the remaining fourtechniques we show the top ten nodes. United States is the only country selected by all six techniques.

America (United States and Canada), three countries from Europe (Great Britain, France, and Russia),three countries from Asia (Singapore, New Zealand, and Fiji), and one country from Africa (South Africa).Some countries are not surprising, e.g., United States, since this country is often associated with its strongeconomical/political power; other countries deserve more attention. Fiji, for instance, is very well connectedin the Melanesia region (Vanuatu, Solomon Islands, Papua New Guinea, and New Caledonia) and to othersmaller island countries in the South Pacific Ocean (e.g. Tuvalu and Tonga). One could say that Fiji is a hubfor all these countries, similar to the notion of hub airports [8]. Recall that the critical countries identifiedby the MDS technique are derived solely from the topological properties of the network. It is interestingto note that Germany is not in the list of critical nodes based on MDS. We think this might because twocritical European countries (Great Britain and France) already cover most of the German neighbors.

On Node Criticality in Air Transportation Networks 9

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Fig. 8: Pairs of controlling/controlled countries using the outbound traffic with a transit threshold 33%. Eachlink represents a pair of countries; pointing from the controlling country to the controlled country. The selected criticalnodes are highlighted with red color. The size of a node is proportional to the number of controlled countries. The positionof countries has been adjusted for presentation purposes.

4.3 Criticality based on Standard Weighted Network Metrics

The dynamic load on the country network corresponds to the number of passengers traveling between twocountries. It is interesting to look at the weighted network metrics with passenger traffic. Several centralitymeasures have been proposed to characterize the importance of a node in a network. We analyze twostandard 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 anetwork; while the betweenness is the number of shortest paths passing through this node [19]. We computethe weighted metrics with effective distance [7]. A node with higher value of closeness can access othernodes easily, especially in case of efficient information spreading in the network. Moreover, betweennesscan be interpreted as the extent to which a node has influence over pair-wise connections between othernodes, assuming that the importance of connections is equivalently distributed among all shortest paths foreach node pair [6]. Therefore, we think weighted closeness and weighted betweenness can be regarded asmeasures of network criticality. In our small example network, shown in Figure 4, the top three nodes withthe largest weighted closeness values are D = 1.98, A = 1.35, and C = 1.33. The top three nodes accordingto the weighted betweenness values are F = 0.33, G = 0.17, and D = 0.13.

Figure 7 (c) shows the top ten countries ranked by the weighted closeness in the country network: GreatBritain is closest to all other countries in the world, followed by United States. Recall that we use theeffective 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 strongair traffic connections inside North/Middle America. United Arab Emirates, given its central geographicposition, is close to all other countries in the network (mainly to other countries in South-West Asia).

Figure 7 (d) presents the top ten countries ranked by the weighted betweenness. Similarly with theweighted closeness, Great Britain ranks first, followed by United States. It is interesting to note that UnitedArab Emirates ranks third. With increasing economic cooperation between Asia/Europe and Asia/Africa [40],especially the boosted economic growth of China and India, United Arab Emirates serves as importantbridges for air traffic connections among these regions.

10 Xiaoqian Sun et al.

Network metrics work very well for identifying influential people in social networks, by considering short-est paths between all pairs of nodes. For transportation networks, however, one needs to consider that thereis no transportation request for all pairs of nodes, and thus this naive strategy could be misleading in somecases. For instance, even United States (the one with the most destinations, in total 180) is not connected toall countries in the world. Other countries have much fewer travel destinations, for example, passengers fromTrinidad 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 ofnodes. 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 ofnodes without any demand. To address this problem, the alternative definition of OD betweenness wasintroduced, which is computed based on shortest paths between all OD pairs only [35]. In our examplenetwork shown in Figure 4, we have eight OD pairs, three out of which are direct connections and thusnot relevant (A → B, E → G, and F → G). For the other five OD pairs, we compute the shortest pathsbetween them and identify how frequently each node occurs on these shortest paths. As a result, only nodeC and node F lie on shortest paths, according to the eight OD pairs. All other nodes do not appear on anyshortest path. This result is substantially different from the original betweenness.

Figure 7 (e) presents the top ten countries ranked by the weighted OD betweenness in the countrynetwork. Compared with the original weighted betweenness shown in Figure 7 (d), the top three countrieskeep the same; Spain, Germany, and France are still ranked in the middle position; while Australia andSouth 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. Weformalize the notion of criticality in the country network here. For each country c1, we check how muchnon-direct outbound traffic of c1 can be blocked by another country c2. We only look at traffic with c1 asorigin. In the small example network, the outbound traffic method yields, for instance, that C is controlling50% 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 ofD. In order to obtain a binary measure, the criticality of a country using the outbound traffic techniquedepends on a transit threshold, which quantifies how much a country’s outbound traffic needs to be undercontrol. In the small example network, the outbound traffic method with a transit threshold of 33% yieldsfour controlling countries (C,E, F,G).

With a transit threshold 33%, Figure 7 (f) shows the top ten countries which have the maximum numbersof controlled countries. Figure 8 presents the complete pairs of controlling/controlled countries using theoutbound traffic method, where the directed link points from a controlling country to a controlled country.The top ten critical countries are highlighted in red color. United Arab Emirates controls the outboundtraffic of a remarkable number of 33 countries, followed by Germany controlling 13 countries, and UnitedStates controlling 7 countries.

The roles of a country (being controlling or controlled) depends on the transit threshold for the outboundtraffic. Figure 9 presents the sensitivity analysis of the transit threshold, ranging from 0 to 100%. Note thata country can have double roles: A country can serve as the first hop to control the outbound traffic from theorigin countries; while the outbound traffic from this country could be controlled by its first hop countries.

4.5 Summary of the Criticality in the Country Network

We summarize the criticality of the country network using the six techniques in Figure 10. The criticalcountries using the MDS technique are listed in alphabetical order; while the critical countries using othertechniques are listed in descending order of the measures. The number of stars associated with each countryrepresents how many techniques rate this country as critical. We can observe that United States is the onlycritical country which is identified by all six network criticality techniques, followed by France and Germany(five out of six techniques); while countries from South America are least frequently chosen, followed byAfrican countries.

On Node Criticality in Air Transportation Networks 11

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Fig. 9: Number of controlling/controlled countries for variable transit threshold in the outbound traffic. Anincreasing transit threshold reduces the number of controlling/controlled countries. The number of controlled countries isalways higher than the number of controlling countries, since with a decreasing threshold, controlling countries becomecontrolled by other countries.

Sheet1

MM technique MDS technique Weighted closeness Weighted betweenness Weighted OD-betweenness Outbound traffic (33%) LIU MDS wcloseness

Antigua and Barbuda (**) Great Britain (****) Great Britain (****) Great Britain (****) United Arab Emirates (****) AG GB

Canada (***) United States (******) United States (******) United States (******) Germany (*****) CA US

Fiji (***) Spain (****) United Arab Emirates (****) United Arab Emirates (****) United States (******) FJ ES

France (*****) Germany (*****) France (*****) Spain (****) Austria (*) FR DE

Great Britain (****) Canada (***) Spain (****) Germany (*****) Panama (**) GB CA

New Zealand (**) Mexico (**) Germany (*****) France (*****) South Africa (***) NZ MX

Russia (***) France (*****) Australia (**) China (**) Ethiopia (**) RU FR

Singapore (**) Italy (*) China (**) Russia (***) Turkey (**) SG IT

United States (******) United Arab Emirates (****) South Africa (***) India (*) Fiji (***) US AE

South Africa (***) Ireland (*) Russia (***) Turkey (**) Antigua and Barbuda (**) ZA IE

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77 countries 77 countries

Page 1

Fig. 10: Tabular summary for network criticality of the country network. The critical countries using the MDStechnique are listed in alphabetical order; while the ones using other techniques are in the descending order of the measures.For each country, the number of stars shows how many techniques rate this country as critical. The top rated countries are:Unites States (6), France (5), Germany (5), Spain (4), Great Britain (4), and United Arab Emirates (4). Countries fromSouth America are least frequently chosen, followed by African countries.

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Fig. 11: Preference of the criticality techniques to select high-degree nodes. The 223 countries (x-axis) are plottedagainst their degree with increasing order (y-axis). Critical countries are marked with red squares. The critical nodes fromthe MM technique have arbitrary degrees: Some of them have rather high degree (more than 200); while most of themhave relatively low degree (less than 50). The other five techniques clearly give preference to selecting high-degree nodes,particularly betweenness and OD-betweenness.

Figure 11 shows the preference of the six criticality techniques to high-degree nodes, where criticalcountries are marked with red squares. We can observe that critical countries identified by the MM techniquehave arbitrary degrees: Some of them have rather high degree (more than 200), but most of them haverelatively low degree (less than 50); while the other five techniques give clear preference to the nodes withhigh degree.

12 Xiaoqian Sun et al.

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12). The intersection of the lines indicates there is a change in the ranking of

the countries.

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 andwith 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 leastcorrelated, followed by the outbound traffic method (ticket control). Given the twelve criteria, which countryis the most preferred one is a typical multi-criteria decision problem. Below, we apply typical Multi-CriteriaDecision Analysis (MCDA) techniques to solve this problem.

On Node Criticality in Air Transportation Networks 13

MCDA is a process to make decisions in face of multiple, potentially conflicting criteria [14], among whichELECTRE (Elimination and Choice Translation Reality) and TOPSIS (Technique for Order Preference bySimilarity to Ideal Solution) are two widely used MCDA techniques. ELECTRE methods use the concept ofoutranking relation introduced by Benayoun [5]. One alternative is preferred when this alternative is at leastas good as the other ones with respect to a majority of criteria and when it is not significantly poor regardingany 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 distanceto the positive ideal solution and the furthest Euclidean distance from the negative ideal solution [22].Mathematical description of a MCDA problem as well as the algorithms for ELECTRE and TOPSIS canbe 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 asnon-dominated: United States (US), Great Britain (GB), Germany (DE), and United Arab Emirates (AE).

Often, the preference information among multi-criteria from a decision maker is uncertain. It is inter-esting to investigate how the ranking of countries change with different preference information. Weightingfactors is one way to represent the preference information. Figure 12 shows sensitivity analysis of theweighting factors for the four non-dominated countries. For each sub-figures, x-axis is the percentage of theweighting factor for one criterion; y-axis is the TOPSIS score for all alternatives (countries); the gray lineis the current setting of the weighting factor ( 1

12 ). The intersection of the lines indicates there is a changein the ranking of the countries. Note that we compute the TOPSIS scores for all 223 countries, only theresults for the four non-dominated countries are shown.

When we slightly increase the weighting factor of the outbound traffic method (ticket control, the rightpanel in the last row), United Arab Emirates (AE) clearly outranks the other countries. Similarly, when theweighting 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 accountthe neighbor’s connections for a node (the four middle panels in the four rows), Great Britain (GB) outranksthe other countries.

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

5.2 Temporal Evolution of Network Criticality

In addition, we provide a summary on the evolution of criticality for the years 2004–2013 in Figure 13. Itcan be seen that the United States (US) shows an outstanding dominance in the country network: With twoexceptions (2004/2006) US is always critical for all methods, followed by France and Germany. Interestingly,Germany is not critical for MDS, while France is not important for MM. Moreover, our experiments revealthat, except from MM, countries are often either critical over a long period, or not at all. This supports ourhypothesis, that the method of Liu is rather inappropriate for analyzing air transportation networks.

5.3 Alternative Choices for Network Modelling

In general, there are other options to model air transportation networks. For instance, instead of usingthe number of passengers on physical links, one could also use the Origin-Destination (OD) demand forcomputing the effective distance. We report some initial results for weighted betweenness and weightedcloseness using both link weights for year 2013 in Figure 14. It can be seen, that the top ranked countriesare critical with respect to both approaches. Future research could analyze the reasons for these slightdeviations and the impact on operational air transport industry. We have performed additional experimentsfor the MDS on both network instances and found that the number of dominating nodes is slightly reducedin the OD network, which can be explained by the larger number of links in the network. The criticalcountries from the physical network (US, FR, AU, FJ) remain critical in the OD network as well.

14 Xiaoqian Sun et al.

Country Max.−Matching MDS WCloseness WBetweeness WODBetweenness Outbound traffic Country

US●

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BT

Fig. 13: Temporal evolution of the criticality for top 30 countries. We show the critical countries for the timeperiod 2004–2013 and each methodology, respectively. A dot indicates that the country is critical for that year. Countriesare encoded with their ISO 3166-1 alpha-2 code.

In addition, besides the number of passengers, which was used in our study, one could assign the numberof flights (or its effective distance equivalent) to links. However, most of such measures are highly correlatedwith the number of passengers, particularly for frequent connections. We show an example in Figure 15. Inour own experiments, the top five critical countries remained the same when weighted by the number ofdepartures, instead of the number of passengers (data not shown).

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 networkwith three different perspectives of criticality: Recent methods from control theory (maximum matchingand minimum dominating set) were used to examine the unweighted structure of the country network; twostandard network metrics with passenger traffic as weights; as well as adjusted weighted origin-destinationbetweenness and outbound traffic based on the real ticket data. We studied different roles of the countriesin 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 ArabEmirates and Panama) have a very specific influence, by controlling the passenger traffic of their neighbor-hood 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

On Node Criticality in Air Transportation Networks 15

5 10 15 20

510

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Fig. 14: Comparison of results for weighted betweenness (left) and weighted closeness (right) on the physicalnetwork and on the origin-destination network. There is a strong positive correlation between the ranking of countriesregarding both networks. The R-square is 0.988 (left) and 0.976 (right), respectively.

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2 5 10 20 50 100 200 500 1000 2000

5e+

031e

+04

2e+

045e

+04

1e+

052e

+05

Number of departures

Num

ber

of p

asse

nger

s

Fig. 15: Comparison of results for the number of passengers and the number of flights (as the effectivedistance between countries). The number of passengers and the number of flights are strongly correlated, with correlationcoefficient 0.7218. The ranking of countries could change slightly when changing from the number of passengers to the numberof flights. However, countries which are highly critical in one network are also highly critical in the other network.

as non-dominated countries using ELECTRE; Sensitivity analysis of TOPSIS showed that United ArabEmirates 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. Futurework could focus on the combination of robust control theory with network science [37]. The six criticalitytechniques also open up future research possibilities on general transportation networks. Moreover, other di-rections include the passenger’s route choice behavior for node criticality, e.g. with common constraints [47],and also look at other network instances, for instance, the networks of networks with multiple modalities [18]or other abstractions [16].

16 Xiaoqian Sun et al.

Appendix

This appendix presents the mathematical description of a Multi-Criteria Decision Analysis (MCDA) problemas well as the algorithms for ELECTRE (Elimination and Choice Translation Reality) [5] and TOPSIS(Technique for Order Preference by Similarity to Ideal Solution) [22].

Decision Matrix

Let Ai be the i-th alternative (i = 1, 2, ...,m) and xj be the j-th criterion (j = 1, 2, ..., n). Suppose xij standsfor the value of criterion xj with respect to alternative Ai. Then, a quantitative MCDA problem can berepresented using decision matrix, as shown in Table 1.

Table 1: Decision Matrix

Alternatives Criteria

A1 x11 x12 . . . x1n

A2 x21 x22 . . . x2n

......

.... . .

...Am xm1 xm2 . . . xmn

ELECTRE (Elimination and Choice Translation Reality)

The algorithm of ELECTRE I is composed of the following nine steps [5].

1. Normalize the decision matrix

R =

r11 r12 ... r1nr21 r22 ... r2n...

.... . .

...rm1 rm2 ... rmn

, rij =xij√m∑i=1

x2ij

, i = 1, 2, ...,m, j = 1, 2, ..., n (1)

2. Calculate the weighted normalized decision matrix.

V = RW =

r11 r12 ... r1nr21 r22 ... r2n...

.... . .

...rm1 rm2 ... rmn

w1

w2

. . .

wn

(2)

3. Determine the concordance and discordance sets.For each pair of alternatives Ak and Al, the set of decision criteria J = (j | j = 1, 2, ..., n) is divided intotwo disjoint subsets. The concordance set Ckl of Ak and Al is composed of all criteria which supportthat Ak is preferred to Al. The discordance set Dkl is the complementary subset of the concordance setCkl.

Ckl = {j |xkj ≥ xlj } , (k, l = 1, 2, ...,m, and k 6= l)Dkl = {j |xkj < xlj } = J − Ckl

(3)

4. Calculate the concordance matrix C.Each element of the concordance matrix C is calculated by the sum of the criteria weights which arecontained in the concordance set. For example, the element ckl between Ak and Al is calculated byEquation 4.

On Node Criticality in Air Transportation Networks 17

C =

− c12 ... c1nc21 − ... c2n...

.... . .

...cm1 cm2 ... −

ckl =∑

j∈Ckl

wj (4)

5. Calculate the discordance matrix D.Each element of the discordance matrix D reflects the degree to which one alternative is worse than theother. For instance, the element dkl between Ak and Al is calculated by Equation 5.

D =

− d12 ... d1nd21 − ... d2n

......

. . ....

dm1 dm2 ... −

dkl =

maxj∈Dkl

|vkj − vij |

maxj∈J|vkj − vij |

(5)

It should be noticed that differences among weighting factors are contained in the concordance matrixC, while differences among criteria values are reflected in the discordance matrix D.

6. Determine the concordance dominance matrix.A concordance threshold c needs to be chosen to perform the concordance test. Alternative Ak possiblydominates alternative Al, if the element ckl exceeds at least a certain threshold c, that is, ckl ≥ c.In ELECTRE I, a Boolean matrix is used to convert the concordance test into numerical values (0 or1). If the concordance test is passed (ckl ≥ c), then the element is 1. Otherwise, if the concordance testis failed (ckl < c), the element is 0.

7. Determine the discordance dominance matrix.A discordance threshold d needs to be chosen to perform the discordance test. Alternative Ak possiblydominates alternative Al, if the element dkl is smaller than a certain threshold d, that is, dkl ≤ d.As with the case of the determination of the concordance dominance matrix, the discordance test isconverted into numerical values (0 or 1) by a Boolean matrix. The element is 1 when the discordancetest is passed (dkl ≤ d), and it is 0 when the discordance test is failed (dkl > d).

8. Aggregate the dominance matrix.An outranking relation can be justified only if both the concordance test and the discordance test arepassed. That is, ckl ≥ c and dkl ≤ d. The aggregated dominance matrix is calculated by an element-to-element product of the concordance dominance matrix and the discordance dominance matrix.

9. Eliminate the dominated alternatives.The aggregated dominance matrix gives the partial preference of the alternatives. In the aggregateddominance matrix, the element 1 in the column indicates that this alternative is dominated by otheralternatives. Thus, any alternative which has at least one element of 1 in the column can be eliminated.

TOPSIS (Technique for Order Preference by Similarity to Ideal Solution)

TOPSIS requires a decision matrix and weighting factors as input data, its computational steps are sum-marized as follows [22].

1. Normalize the decision matrix.

zij =xij√m∑i=1

x2ij

, i = 1, 2, ...,m, j = 1, 2, ..., n (6)

18 Xiaoqian Sun et al.

2. Calculate the weighted normalized decision matrix.

rij = wjzij , i = 1, 2, ...,m, j = 1, 2, ..., n (7)

3. Identify the positive ideal solution A∗ and the negative ideal solution A−.

A∗ ={(

maxi

rij |j ∈ J),(

mini

rij |j ∈ J)|i = 1, 2, ...,m

}= {x∗1, x∗2, ..., x∗n} (8)

A− ={(

mini

rij |j ∈ J),(

maxi

rij |j ∈ J)|i = 1, 2, ...,m

}={x−1 , x−2 , ..., x−n

}(9)

where J is the benefit criteria set (bigger criterion value is preferred), and J is the cost criteria set(smaller criterion value is preferred). Thus, the positive ideal solution is composed of the maximumvalues of benefit criteria and the minimum values of cost criteria; while the negative ideal solution iscomposed of the minimum values of benefit criteria and the maximum values of cost criteria.

4. Calculate the distance for each alternative to the positive ideal solution and the negative ideal solution,respectively.

S∗i =

√√√√ k∑j=1

(rij − x∗j

)2, i = 1, 2, ...,m (10)

S−i =

√√√√ k∑j=1

(rij − x−j

)2, i = 1, 2, ...,m (11)

5. Calculate the relative closeness for each alternative to the ideal solutions.

C∗i =S−i

S−i + S∗i, i = 1, 2, ...,m (12)

6. Rank the alternatives according to the value of C∗i .TOPSIS suggests the best alternative which has the furthest distance from the negative ideal solution(biggest value of S−i ) and shortest distance to the positive ideal solution (smallest value of S∗i ). Thus, theincrease of numerator and the decrease of denominator will lead to a bigger value of C∗i in Equation 12.In other words, the alternative which maximizes the value of C∗i ranks first.

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