Networks and Spatial Economics

, Volume 17, Issue 3, pp 737–761 | Cite as

On Node Criticality in Air Transportation Networks

Article

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 = 〈[a1, ... , am], p〉, such that each ai is an airport, a1 is the origin, am is the destination, and p is the number of passengers who bought this ticket. The ticket database consists of n tickets {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: ijAiAj.

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 ct = 〈[c1, ... , cm], p〉, where each ci 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 CT = {ct1, ... , ctm}. The set of countries is denoted with C.

Figure 1 shows the top ten country tickets with direct flights, one hop, two hops, and three hops in the country network. Note that the ticket with three hops ranked ninth (PH, CN, SA, BH, SA), 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.
Fig. 1

Top ten tickets according to the number of passengers, grouped by number of hops. For direct flight tickets, 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 or destination. Tickets with two or three hops have considerably less numbers of passengers and often have countries from South Asia/Oceania as origin or destination

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 LN × N, and a weight function w mapping links in L to real numbers. For weighted networks, we have that (n1, n2) is in the domain of w, if and only if (n1, n2) ∈ 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 ct = 〈[c1, ... , cm], p〉, and for each 1 ≤ j < m, we check whether (cj, cj+1) ∈ L. If (cj, cj+1) ∈ L, then we set w(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 Fig. 2, where each country is 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 clustering coefficient of 0.629, along with an average shortest path length of 2.254. Small average shortest path length but higher clustering coefficient than its random counterpart network are two structural features for a 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).
Fig. 2

An illustration of the country network in the year 2013. Each node is one country, the size of a node is proportional to the weighted degree, with the number of passengers as link weight. Note that the position of countries is slightly adjusted for presentation purposes, and spherical links are not always drawn as the shortest connection, but go through the center of the projected map instead

Fig. 3

Scatterplot for selected network metrics. The country network is nearly symmetric (left). Degree and closeness are highly positively correlated (center). Betweenesss and closeness are positively correlated (right)

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

Figure 4 shows a small network example how we built the physical country network based on the ticket data. Given a set of Origin-Destination (OD) pairs, connecting countries, and the number of passengers who bought this ticket, we construct the physical network with two different weights: Number of passengers and effective distance. The effective distance is based on the idea that a small fraction of traffic is effectively equivalent to a large distance, and vice versa (Brockmann and Helbing 2013).
Fig. 4

Network criticality for a small example network. In the upper part, the transformation of an OD-ticket table (left) to a physical network (right) is shown. The lower part shows the results of our six criticality techniques. Critical nodes are highlighted with red color. For the maximum matching, minimum dominating set, and outbound traffic, critical nodes are obtained by the algorithm; for the remaining network metrics, we have selected the top three nodes according to the metric values in descending order. Notably, all six techniques yield distinct sets of critical nodes. Five out of six techniques identify node F as critical. All techniques classify B as not critical (B is the only node without outgoing links)

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 AD, two passengers for DC, and two passengers for CB. 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 AB is rewritten from 10 to \(\frac {1}{10}\) and the link weight of AD 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 AB.

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).

We describe the construction of a maximum matching for an example network shown in Fig. 4. First, the network is converted into an undirected bipartite graph as follows (see Supplementary of Liu et al. (2011)): 1) For each node n, we generate two new nodes nI and nO. 2) For each link (a, b) in the original network we add a link from aO to bI 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.
Fig. 5

Example for the construction of maximum matching. The matching of the original network is shown, together with the networks where one node is removed. Links belonging to the MM are highlighted in red color. Unmatched nodes are highlighted with yellow color. The node F is the only node for which the number of unmatched nodes is increased, compared to the original network

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.

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 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.
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. Node A is critical, since there is no MDS without it. Node F is critical, because the size of the MDS is increased, if F cannot take part in the MDS

We have computed the dominating sets using the DSP implementation of OpenOpt (https://pypi.python.org/pypi/openopt). Figure 7b highlights the critical nodes (in red color) in the country network based on the MDS technique. 10 out of 223 countries are identified as critical: Two countries from North 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 strong economical/political power; other countries deserve more attention. Fiji, for instance, is very well connected in the Melanesia region (Vanuatu, Solomon Islands, Papua New Guinea, and New Caledonia) and to other smaller island countries in the South Pacific Ocean (e.g. Tuvalu and Tonga). One could say that Fiji is a hub for all these countries, similar to the notion of hub airports (Bryan and O’Kelly 1999). Recall that the critical countries identified by the MDS technique are derived solely from the topological properties of the network. It is interesting to note that Germany is not in the list of critical nodes based on MDS. We think this might because two critical European countries (Great Britain and France) already cover most of the German neighbors.
Fig. 7

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

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 (AB, EG, and FG). 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 c1, we check how much non-direct outbound traffic of c1 can be blocked by another country c2. We only look at traffic with c1 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).

With a transit threshold 33 %, Fig. 7f shows the top ten countries which have the maximum numbers of controlled countries. Figure 8 presents the complete pairs of controlling/controlled countries using the outbound 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 outbound traffic of a remarkable number of 33 countries, followed by Germany controlling 13 countries, and United States controlling 7 countries.
Fig. 8

Pairs of controlling/controlled countries using the outbound traffic with a transit threshold 33%. Each link represents a pair of countries; pointing from the controlling country to the controlled country. The selected critical nodes are highlighted with red color. The size of a node is proportional to the number of controlled countries. The position of countries has been adjusted for presentation purposes

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

Number of controlling/controlled countries for variable transit threshold in the outbound traffic. An increasing transit threshold reduces the number of controlling/controlled countries. The number of controlled countries is always higher than the number of controlling countries, since with a decreasing threshold, controlling countries become controlled by other countries

4.5 Summary of the Criticality in the Country Network

We summarize the criticality of the country network using the six techniques in Fig. 10. The critical countries using the MDS technique are listed in alphabetical order; while the critical countries using other techniques are listed in descending order of the measures. The number of stars associated with each country represents how many techniques rate this country as critical. We can observe that United States is the only critical 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 by African countries.
Fig. 10

Tabular summary for network criticality of the country network. The critical countries using the MDS technique 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 from South America are least frequently chosen, followed by African countries

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

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

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).

Often, the preference information among multi-criteria from a decision maker is uncertain. It is interesting to investigate how the ranking of countries change with different preference information. Weighting factors is one way to represent the preference information. Figure 12 shows sensitivity analysis of the weighting factors for the four non-dominated countries. For each sub-figures, x-axis is the percentage of the weighting factor for one criterion; y-axis is the TOPSIS score for all alternatives (countries); the gray line is the current setting of the weighting factor (\(\frac {1}{12}\)). The intersection of the lines indicates there is a change in the ranking of the countries. Note that we compute the TOPSIS scores for all 223 countries, only the results for the four non-dominated countries are shown.
Fig. 12

Sensitivity analysis of the weighting factors for the four non-dominated countries. x-axis is the percentage of the weighting factor for one criterion; y-axis is the TOPSIS score for the alternatives (countries); the gray line is the current setting of the weighting factor (\(\frac {1}{12}\)). The intersection of the lines indicates there is a change in the ranking of the countries

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

In addition, we provide a summary on the evolution of criticality for the years 2004–2013 in Fig. 13. It can be seen that the United States (US) shows an outstanding dominance in the country network: With two exceptions (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 reveal that, except from MM, countries are often either critical over a long period, or not at all. This supports our hypothesis, that the method of Liu is rather inappropriate for analyzing air transportation networks.
Fig. 13

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

5.3 Alternative Choices for Network Modelling

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

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

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

Comparison of results for the number of passengers and the number of flights (as the effective distance between countries). The number of passengers and the number of flights are strongly correlated, with correlation coefficient 0.7218. The ranking of countries could change slightly when changing from the number of passengers to the number of flights. However, countries which are highly critical in one network are also highly critical in the other network

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).

References

  1. Arvis JF, Shepherd B (2011) The air connectivity index: measuring integration in the global air transport network. Policy Research Working Paper Series 5722, The World Bank. http://EconPapers.repec.org/RePEc:wbk:wbrwps:5722
  2. Aziz HMA, Ukkusuri SV, Zhan X (2016) Determining the impact of personal mobility carbon allowance schemes in transportation networks. Networks and Spatial Economics, pp 1–41 doi:10.1007/s11067-016-9334-x
  3. Barabasi AL (2013) Network science. Philosophical Transactions of the Royal Society A: Mathematical. Phys Eng Sci 371:1987. doi:10.1098/rsta.2012.0375 CrossRefGoogle Scholar
  4. Barrat A, Barthélemy M, Pastor-Satorras R, Vespignani A (2004) The architecture of complex weighted networks. PNAS 101(11):3747–3752CrossRefGoogle Scholar
  5. Benayoun R, Roy B, Sussman N (1973) Manual de reference du programme electre. Psychoemtrika 38:337–369CrossRefGoogle Scholar
  6. Brandes U (2008) On variants of shortest-path betweenness centrality and their generic computation. Soc Networks 30(2):136–145. doi:10.1016/j.socnet.2007.11.001 CrossRefGoogle Scholar
  7. Brockmann D, Helbing D (2013) The hidden geometry of complex, network-driven contagion phenomena. Science 342:1337CrossRefGoogle Scholar
  8. Bryan DL, O’Kelly ME (1999) Hub-and-spoke networks in air transportation: an analytical review. J Reg Sci 39(2):275–295CrossRefGoogle Scholar
  9. Caschili S, Medda FR, Wilson A (2015a) An interdependent multi-layer model: Resilience of international networks. Networks and Spatial Economics. doi:10.1007/s11067-014-9274-2
  10. Caschili S, Reggiani A, Medda F (2015b) Resilience and vulnerability of spatial economic networks. Networks and Spatial Economics, pp 1–6. doi:10.1007/s11067-015-9283-9
  11. Colizza V, Barrat A, Barthélemy M, Vespignani A (2006) The role of the airline transportation network in the prediction and predictability of global epidemics. Proc Natl Acad Sci U S A 103(7):2015–2020CrossRefGoogle Scholar
  12. Deguchi T, Takahashi K, Takayasu H, Takayasu M (2014) Hubs and authorities in the world trade network using a weighted hits algorithm. PLoS ONE 9(7):e100,338. doi:10.1371/journal.pone.0100338 CrossRefGoogle Scholar
  13. Ducruet C, Beauguitte L (2014) Spatial science and network science: Review and outcomes of a complex relationship. Networks and Spatial Economics 14(3-4):297–316. doi:10.1007/s11067-013-9222-6 CrossRefGoogle Scholar
  14. Ehrgott M, Figueira J, Greco S (2010) Trends in multiple criteria decision analysis springerGoogle Scholar
  15. European Commission (2001) European aeronautics: a vision for 2020. Meeting society’s needs and winning global leadership (2001). Luxembourg: Office for Official Publications of the European CommunitiesGoogle Scholar
  16. Fielbaum A, Jara-Diaz S, Gschwender A (2016) A parametric description of cities for the normative analysis of transport systems. Networks and Spatial Economics pp 1–23.Google Scholar
  17. Fleurquin P, Ramasco J, Eguiluz V (2013) Data-driven modeling of systemic delay propagation under severe meteorological conditions 10th USA/Europe ATM Seminar. http://arxiv.org/ftp/arxiv/papers/1308/1308.0438.pdf
  18. Fotuhi F, Huynh N (2016) Reliable intermodal freight network expansion with demand uncertainties and network disruptions. Networks and Spatial Economics, pp 1–29. doi:10.1007/s11067-016-9331-0
  19. Freeman LC (1978) Centrality in social networks conceptual clarification. Soc Networks 1(3):215–239. doi:10.1016/0378-8733(78)90021-7 CrossRefGoogle Scholar
  20. Gomes MFC, Pastore, Rossi L, Chao D, Longini I, Halloran ME, Vespignani A (2014) Assessing the International Spreading Risk Associated with the 2014 West African Ebola Outbreak. PLoS Currents. doi:10.1371/currents.outbreaks.cd818f63d40e24aef769dda7df9e0da5
  21. Hawelka B, Sitko I, Beinat E, Sobolevsky S, Kazakopoulos P, Ratti C (2014) Geo-located twitter as proxy for global mobility patterns. Cartogr Geogr Inf Sci 41(3):260–271. doi:10.1080/15230406.2014.890072 CrossRefGoogle Scholar
  22. Hwang CL, Yoon K (1981) Multiple attribute decision making methods and applications: a state of the art survey springerGoogle Scholar
  23. Jia T, Barabasi AL (2013) Control capacity and a random sampling method in exploring controllability of complex networks. Sci Report 3:2354. doi:10.1038/srep02354 CrossRefGoogle Scholar
  24. Jia T, Liu YY, Csóka E, Pósfai M, Slotine JJ, Barabási AL (2013) Emergence of bimodality in controlling complex networks. Nature communications, 4. doi:10.1038/ncomms3002
  25. Jia T, Pósfai M (2014) Connecting core percolation and controllability of complex networks. Sci Report 4:5379. doi:10.1038/srep05379 CrossRefGoogle Scholar
  26. Kaltenbrunner A, Aragón P, Laniado D, Volkovich Y (2014) Not all paths lead to rome: Analysing the network of sister cities. In: elmenreich W, Dressler F, Loreto V (eds) Self-Organizing Systems, Lecture Notes in Computer Science, vol 8221. Springer, Berlin Heidelberg, pp 151–156Google Scholar
  27. Kotegawa T, DeLaurentis D, Noonan K, Post J (2011) Impact of commercial airline network evolution on the u.s. air transportation system Ninth USA/europe air traffic management research and development seminar, berlin, GermanyGoogle Scholar
  28. Lehner S, Gollnick V (2014) Function-structure interdependence of passenger air transportation: Application of a systemic approach 14Th AIAA aviation technology, integration, and operations conference, atlanta, USA, 16-20 JuneGoogle Scholar
  29. Lehner S, Koelker K, Luetjens K (2014) Evaluating Temporal Integration of European Air Transport 29Th congress of the international council of the aeronautical sciences (ICAS), st. petersburg, Russia, 7-12 SeptemberGoogle Scholar
  30. Lijesen MG, Rietveld P, Nijkamp P (2004) Do european carriers charge hub premiums?. Networks and Spatial Economics 4(4):347–360. doi:10.1023/B:NETS.0000047112.68868.48 CrossRefGoogle Scholar
  31. Liu YY, Slotine JJ, Barabási AL (2011) Controllability of complex networks. Nature 473:167–173. doi:10.1038/nature10011 CrossRefGoogle Scholar
  32. Liu YY, Slotine JJ, Barabási AL (2012) Control centrality and hierarchical structure in complex networks. Plos one 7(9):e44,459CrossRefGoogle Scholar
  33. Mascia M, Hu S, Han K, North R, Van Poppel M, Theunis J, Beckx C, Litzenberger M (2016) Impact of traffic management on black carbon emissions: a microsimulation study. Networks and Spatial Economics. pp 1–23. doi:10.1007/s11067-016-9326-x
  34. Molnár Jr F, Derzsy N., Szymanski BK, Korniss G (2015) Building damage-resilient dominating sets in complex networks against random and targeted attacks. Scientific reports, 5Google Scholar
  35. Monechi B, Others (2013) Exploratory analysis of safety data and their interrelation with flight trajectories and network metrics Air transportation system conferences - interdisciplinary science for innovative air traffic managementGoogle Scholar
  36. Nacher JC, Akutsu T (2014) Analysis of critical and redundant nodes in controlling directed and undirected complex networks using dominating sets. Journal of Complex Networks, p cnu029Google Scholar
  37. Nacher JC, Akutsu T (2014) Structurally robust control of complex networks. arXiv:1410.2949
  38. Newman M (2010) Networks-An Introduction oxford university pressGoogle Scholar
  39. Radicchi F (2015) Percolation in real interdependent networks. Nat Phys 11(7):597–602CrossRefGoogle Scholar
  40. Rose C (2012) Discourses on japan and china in africa: Mutual mis-alignment and the prospects for cooperation. Japanese Studies 32(2):219–236. doi:10.1080/10371397.2012.708397 CrossRefGoogle Scholar
  41. Sun X, Wandelt S (2014) Network similarity analysis of air navigation route systems. Transportation Research Part E: Logistics and Transportation Review 70 (0):416–434. doi:10.1016/j.tre.2014.08.005 CrossRefGoogle Scholar
  42. Sun X, Wandelt S, Linke F (2015) Temporal evolution analysis of the european air transportation system: air navigation route network and airport network. Transportmetrica B: Transport Dynamics 3(2):153–168. doi:10.1080/21680566.2014.960504 Google Scholar
  43. Vitali S, Et al (2012) Statistical regularities in atm: Network properties, trajectory deviations and delays Second SESAR innovation daysGoogle Scholar
  44. Wandelt S, Sun X (2015) Efficient compression of 4D-trajectory data in air traffic management. IEEE Trans Intell Transp Syst 16(2):844–853. doi:10.1109/TITS.2014.2345055 Google Scholar
  45. Watts D, Strogatz S (1998) Collective dynamics of small-world networks. Nature (393):440–442Google Scholar
  46. Wuchty S (2014) Controllability in protein interaction networks. Proc Natl Acad Sci 111(19):7156–7160CrossRefGoogle Scholar
  47. Yang X, Ban XJ, Ma R (2016) Mixed equilibria with common constraints on transportation networks. Networks and Spatial Economics. pp 1–33. doi:10.1007/s11067-016-9335-9
  48. Zanin M, Lillo F (2013) Modelling the air transport with complex networks: A short review. The European Physical Journal Special Topics 215(1):5–21. doi:10.1140/epjst/e2013-01711-9 CrossRefGoogle Scholar
  49. Zhang S, Derudder B, Witlox F (2015) Dynamics in the european air transport network, 2003—9: An explanatory framework drawing on stochastic actor-based modeling. Networks and Spatial Economics. pp 1–21. doi:10.1007/s11067-015-9292-8

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.School of Electronic and Information EngineeringBeihang UniversityBeijingChina
  2. 2.Department of Computer ScienceHumboldt-University of BerlinBerlinGermany

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