Robustness of Air Transport Networks

CHAPTER 7

Robustness of Air Transport Networks Contents 7.1. Robustness of the world airport network 7.1.1 Effect of errors and attacks on largest connected component 7.1.2 Robustness of the weighted world airport network 7.2. Robustness of regional airport networks 7.3. Robustness of airline airport networks 7.3.1 Airline business model and network robustness 7.3.2 Emergence of robustness in airport networks as multi-layered networks 7.4. Delays and delay propagation 7.4.1 Constructing functional delay networks 7.4.2 The European delay propagation network References

181 182 188 192 197 197 199 200 201 204 208

7.1 Robustness of the world airport network In the first part of this chapter, we examine the robustness of the world airport network (WAN). Recall that the airport network is an undirected network, where nodes are airports or metropolitan areas (sometimes referred as cities), connected by an undirected edge if there is at least a direct flight connecting them in a specific time window. In Chapter 4, we have performed a topological analysis of the world airport network, including degree and betweenness distributions and structural organisation. The world airport network is the result of the aggregation of routing decisions of airlines, some of them acting in the context of airline alliances. These decisions are mediated by existing regulations and agreements, sometimes resulting in extensions of freedoms of the air for airlines operating in a specific state. Routing decisions are also made on grounds of spatial relationships: flights between distant airports have lower density (that is, lower demand) than to close airports, and will be operated by wide-body planes, which need to consolidate a higher demand. As a conclusion, we can say that the structure of the world airport network is the result of geo-political and cost-driven constraints [16], which do not take network robustness into account [26]. Robustness analysis of the world airport network can help to Air Route Networks Through Complex Networks Theory https://doi.org/10.1016/B978-0-12-812665-3.00014-1

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assess the impact of the closure of vital airports on global air traffic and to detect vital airports for network functionality.

7.1.1 Effect of errors and attacks on largest connected component The most usual analysis of robustness of complex networks is the assessment of the effect of node isolation on the size of the largest connected component, taken as critical functionality measure. To analyse tolerance to errors, the nodes to isolate are selected at random, and tolerance to attacks is analysed by selecting vital or important nodes with the objective of causing the maximal damage on critical functionality. This was the approach adopted on early analysis of robustness of airport networks [9] and our analysis of the robustness of the world airport network [26]. In Chapter 6, we presented an introduction to robustness analysis of complex networks. Here we will apply this analysis to a sample of the world airport network considering flights scheduled in August 2018, obtained from the OAG database. Our sample of the world airport network has N = 3914 nodes and E = 27429 undirected edges. The values of average path length and clustering coefficient are L = 4.011 and C = 0.627. These values are similar to those obtained for other samples of the world airport network (see Table 4.2) of previous research on airport networks. Like other airport networks, this network has a power-law degree distribution with an exponential cut-off (see Fig. 7.1). This exponential cut-off is caused by limitations of absorption of potential demand by airports with high degree (number of connections). The node betweenness of this network also follows a similar power-law distribution with exponential cut-off. The tolerance to errors of complex networks when the critical functionality measure is the size of the giant connected component can be analysed using percolation theory. As the fraction of isolated nodes reaches a critical value fc , a phase transition occurs from the existence of a giant component (a largest connected component of a size of the order of N) to a total network disconnection. For complex networks of a high number of nodes N, this transition depends on average degree k: fc = 1 −

k k2  − k

.

(7.1)

For our sample of the world airport network, we have that k = 11.871 and k2  = 797396, so the critical value of the fraction of isolated nodes

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Figure 7.1 Log-log plots of the cumulative degree (left) and betweenness (right) distributions of the world airport network (airport nodes) of August 2012.

is almost one, exactly fc = 0.985. This means that the world airport network has a high tolerance to incidents consisting of removal of nodes at random. A network of the same number of nodes and edges built following the Erdös–Rényi model, where each pair of nodes can be connected with the same probability, has a critical value of fcER = 0.914, so the world airport network is more resilient to errors than its equivalent Erdös–Rényi network. The tolerance to attacks of the world airport network was examined in Lordan et al. [26] for a sample of flights scheduled from November 2011 to November 2012. Here we will reproduce some of the analysis carried out in that paper for a sample including connections scheduled on August 2018. We proceed in this way to allow comparison of results obtained in different time windows, as significant differences on air transport can appear in summer and winter seasons [28]. The most common way to evaluate tolerance to attacks is by selecting nodes according to a centrality criterion. We have selected four centrality measures: • The degree of a node is equal to the number of edges attached to the node. In the context of airport networks, the degree represents the number of direct connections of an airport, that is, the number of different destinations that passengers can reach from each airport with a direct flight.

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• The eigenvector centrality is a measure of centrality of a node that de-

pends of the centrality of the nodes attached to it. This measure takes into account the fact that for an airport, it can be more relevant being connected to central airports than to peripheral airports. This measure is computed obtaining the eigenvector corresponding to the maximum eigenvalue of the adjacency matrix. • The betweenness measures how often a node is in the middle of shortest paths connecting any pair of nodes in the network (for a more precise definition, see Section 3.4 of Chapter 3). An airport with high betweenness is in the shortest paths of many travels with connecting flights. • The harmonic closeness is the harmonic mean of the distance of a node with the rest of nodes. We use the harmonic closeness instead of closeness, as the latter measure is undefined for networks where any pair of nodes is not connected [5]. The degree and eigenvector centrality can be considered as local measures of centrality, as they depend on the nodes connected directly to each node. The betweenness and harmonic closeness, on the contrary, can be considered global measures of centrality, as the value of a node depends on its relationship with the rest of nodes in the network. Once selected a measure of centrality, we will examine the evolution of the size of the largest connected component of the network as a function of the fraction of isolated nodes. As we intend to maximise the damage inflicted to the network, we will choose the nodes of highest centrality. When performing the attack, we can adopt two schemes or strategies [20]: • In simultaneous attacks, we isolate a specific fraction of nodes according to the values of centrality calculated before the attack starts. So we are modelling an attack when all nodes are isolated at the same time. This scheme is sometimes known in the literature as a non-adaptive strategy of node isolation. • In sequential attacks the value of centrality is recalculated each time a node is isolated. Each time, we isolate the node of highest value of the recalculated centrality. Now we are modelling a situation in which a node is isolated after the other and is sometimes called an adaptive strategy. In Fig. 7.2, we present the evolution of the size of largest connected component when the world airport network is exposed to simultaneous and sequential attacks based on the four values of centrality. We can see that the world airport network has a low tolerance to attacks: the network is disconnected for f  0.22 for simultaneous attacks and for f  0.17

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Figure 7.2 Evolution of the size of the largest connected component as a function of the fraction of removed nodes for simultaneous (left) and sequential (right) attacks selecting nodes with degree (deg), eigenvector (eig), betweenness (btw) and harmonic closeness (clo). Attacks performed on the world airport network of August 2018.

for sequential attacks. The last value is similar to that obtained by Lordan et al. [26] for a sample of the world airport network covering flights from November 2011 to November 2012. We have performed attacks for two local measures of centrality (degree and eigenvector) and two global measures of centrality (betweenness and harmonic closeness). An extensive analysis by Iyer et al. [20] of effectiveness of several simultaneous and sequential attacks on complex networks concluded that the degree was the most effective strategy on simultaneous attacks, and the betweenness for sequential attacks. Their analysis included a collection of network models and empirical (real-world) networks but not the world airport network. The results of Fig. 7.2 confirm the effectiveness of betweenness for sequential attacks but are inconclusive with simultaneous attacks: whereas for low values of f , the betweenness is more effective than the degree, for f  0.10, the simultaneous degree attack achieves a steep reduction of network functionality. To compare the effectiveness of different attacks, we can use the parameter R defined by Schneider et al. [30] as R=

N 1  Q (F ) , N i=1

(7.2)

where Q (F ) is the size of the giant component when F = fN nodes are isolated. More effective attacks will yield smaller values of R. In Table 7.1,

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Table 7.1 Value of parameter R for simultaneous (sim) and sequential (seq) attacks by degree (deg) and betweenness (btw) for the world airport network (WAN) and an Erdös– Rényi (ER) network of same nodes and edges. In bold, value of R for the most effective strategy for each attack. Network Attack btw deg

ER ER WAN WAN

seq sim seq sim

0.41649 0.46549 0.03538 0.09469

0.42046 0.46711 0.08219 0.09136

we list the values of R for several attacks on the world airport network and an Erdös–Rényi network of the same number of nodes and edges. We can obtain the following conclusions from the values presented in Table 7.1: • When the world airport network (WAN) and the Erdös–Rényi network (ER) suffer simultaneous attacks, the effectiveness of betweenness and degree attacks is similar, although betweenness is slightly better in the ER network, and degree yields slightly better results in the WAN. This behaviour of the WAN is similar to that observed by Iyer et al. [20] in other empirical networks. • When experiencing sequential attacks, the performance of betweenness is better than degree in the WAN and ER networks. This is in accordance with the behaviour observed by Iyer et al. [20]. Observing Fig. 7.2, we can conclude that a sequential attack based on betweenness is the most effective strategy to disrupt the WAN. • The ER is much more robust than the WAN (recall that the maximum value of R is 0.5). Networks with homogeneous degree distribution are more robust than networks with heterogeneous degree distribution. In those networks, an attack based on isolating nodes with high degree (which usually also have the highest betweenness) can disrupt the network effectively [1,18,20]. Knowing the first airports that have been selected to isolate in each attack can give us insight about what are the airports on which air traffic mostly depends in the world airport network. The ten airports selected in simultaneous and sequential attacks by degree and betweenness are listed in Tables 7.2 and 7.3, respectively. From Tables 7.2 and 7.3 we observe that results are different for each strategy and attack, but we can observe some common trends:

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Table 7.2 First ten airports to be isolated by degree in a sequential and simultaneous attack to the world airport network. Simultaneous Sequential

FRA CDG IST AMS PEK MUC ATL ORD DFW DXB

Frankfurt am Main Charles de Gaulle Atatürk Amsterdam Airport Schiphol Beijing Capital Munich Hartsfield Jackson Atlanta Chicago O’Hare Dallas Fort Worth Dubai

FRA CDG IST AMS PEK MUC ATL ORD DFW DXB

Frankfurt am Main Charles de Gaulle Atatürk Amsterdam Airport Schiphol Beijing Capital Munich Hartsfield Jackson Atlanta Chicago O’Hare Dallas Fort Worth Dubai

Table 7.3 First ten airports to be isolated by betweenness in a sequential and simultaneous attack to the world airport network. Simultaneous Sequential

ANC FRA DXB LAX CDG

Ted Stevens Anchorage Frankfurt am Main Dubai Los Angeles Charles de Gaulle

ANC FRA ORD SEA MSP

IST PEK ORD NRT AMS

Atatürk Beijing Capital Chicago O’Hare Narita Amsterdam Airport Schiphol

DXB IST CDG PEK LAX

Ted Stevens Anchorage Frankfurt am Main Chicago O’Hare Seattle Tacoma Minneapolis–St Paul International Dubai Atatürk Charles de Gaulle Beijing Capital Los Angeles

• The first ten airports to be isolated by degree are the same for the si-

multaneous and sequential attacks, that is, the ten airports with highest degree of the WAN. • The rankings of simultaneous and sequential attacks by betweenness are quite different from the third airport, and thus sequential attacks by betweenness provide an alternative listing of important airports. The strategy of sequential betweenness disconnects the Alaskan network by isolating Anchorage (ANC) and Seattle Tacoma (SEA). This strategy also disconnects other USA airports, like Minneapolis (MSP), before disconnecting large hubs like Dubai (DXB) or Istanbul Atatürk (IST).

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The list of top airports selected by degree and betweenness have the same biases detected in the connectivity analysis: in fact, the simultaneous attacks select the airports of highest centrality before disconnecting the network. The betweenness ranking depends on network topology (hence the presence of Alaskan hubs among the most vital airports) and tends to score low US non-Alaskan airports. The high effectiveness of a betweenness-based strategy of node isolation to disrupt the world-airport network in both simultaneous and sequential attacks can be explained by the anomalous centrality of the world airport network. In this network, there are central nodes (i.e., of high betweenness) that are not well-connected (i.e., of high degree). In Fig. 5.1, we present a scatter plot of degree and betweenness for the world-airport network. There we see that airports like Anchorage International (ANC), Dubai International Airport (DXB) or Los Angeles International (LAX) have anomalous values of centrality. These airports appear in the ranking of vital airports by betweenness of Table 7.3.

7.1.2 Robustness of the weighted world airport network The results of robustness analysis on the unweighted world airport network presented biases caused by the prevalence of network topology. The extreme case is the Alaskan network, with a large number of airports and scheduled flights but with low number of seats offered, which has a large importance in network connectivity, as the connection of these airports with the rest of the airport network depends on two Alaskan hubs, the international airports of Anchorage and Fairbanks. A possible way to reduce topological bias and obtain a most realistic picture of the effect of attacks on the world airport network is to consider the weighted world airport network. The measures of capacity and intensity of a connection available from OAG data are the frequency of flights, number of available seats and available seats by kilometre (ASKs). We can use these measures to define edge weights wij , representing the intensity or frequency of the direct connection between nodes i and j. Here we will use available seats as weight for the world-airport network. Once an edge weight is defined, we can calculate the node strength as si =



wij .

(7.3)

j ∈N

Node strength is a local measure of centrality, as it considers only direct connections. In unweighted graphs, we use the node betweenness bi as a

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Figure 7.3 Total strength of the largest connected component (left) and size of the largest connected component as functions of the fraction of isolated notes in degreebased (deg) and betweenness-based (btw) simultaneous attacks (weights are available seats).

global measure of centrality, as it depends of the number of shortest paths among all pairs of nodes that pass through node i. For weighted graphs, we can define a measure of betweenness considering shortest paths between nodes calculated using a effective distance for each edge [10]: lij =

dij . wij

(7.4)

The effective distance is directly proportional to geographical distance dij between nodes and inversely proportional to edge weight: the larger the edge weight, the more intense the relationship between the pair of airports. Once established the node strength and weighted betweenness as local and global measures of centrality, we can consider the effect of simultaneous and sequential attacks selecting nodes with these two measures. As a critical measure of functionality for weighted graphs, Dall’Asta et al. [10] considered the largest value of strength carried by a connected component. This measure is equal to twice the value of the sum of edge weights included in that connected component. In Figs. 7.3 and 7.4, we present the results of simultaneous and sequential attacks, respectively, on the world airport network. The sample of the network included all flights scheduled in August 2018, and we have used available seats as edge weight. The analysis of degree and strength distributions for this sample yields similar results to those obtained in Section 5.4 of

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Figure 7.4 Total strength of the largest connected component (left) and size of the largest connected component as functions of the fraction of isolated notes in degreebased (deg) and betweenness-based (btw) sequential attacks (weights are available seats).

Chapter 5. Here are some of the insights we can obtain from the robustness analysis of the weighted world airport network: • When we adopt the largest value of traffic in a connected component as critical functionality, the world airport network is more fragile than when we consider the size of largest component in the unweighted network. This suggests that the robustness analysis of the unweighted network may underestimate the effect of isolation of central nodes. In Figs. 7.3 and 7.4, we present the evolution of total strength and size of largest connected component: we can appreciate that the network is disconnected according to the weighted criticality measure, even if the size of largest connected component is relatively large (compare values of both measures for values of f between 0.05 and 0.15). • Although both node isolation strategies have a similar performance when disrupting the network, the local measure (strength) is more effective in simultaneous attacks, and the global measure (weighted betweenness) is more effective in sequential attacks. We can conclude that the prevalence of global measures to assess node importance observed for unweighted centralities does not appear when evaluating weighted measures. In Table 7.4, we list the first ten nodes selected by node strength, measured by available seats, and in Table 7.5 the first ten airports to be removed by weighted betweenness measured by effective distance, with available seats

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Table 7.4 First ten airports to be selected by strength (measured by available seats) in simultaneous and sequential attacks. Simultaneous Sequential

ATL PEK DXB LAX HND ORD LHR FRA CDG HKG

Hartsfield Jackson Atlanta Beijing Capital Dubai Los Angeles Tokyo Haneda Chicago O’Hare London Heathrow Frankfurt am Main Charles de Gaulle Chek Lap Kok

ATL PEK DXB HND LAX ORD LHR FRA CGK CDG

Hartsfield Jackson Atlanta Beijing Capital Dubai Tokyo Haneda Los Angeles Chicago O’Hare London Heathrow Frankfurt am Main Soekarno–Hatta Charles de Gaulle

Table 7.5 First ten airports to be selected by weighted betweenness (measured by effective distance with seats as weights) in simultaneous and sequential attacks. Simultaneous Sequential

LHR DXB JFK BKK LAX SIN HKG LGA MIA BOS

London Heathrow Dubai John F Kennedy Suvarnabhumi Los Angeles Singapore Changi Chek Lap Kok La Guardia Miami General Edward Lawrence Logan

LHR CDG LAX DXB HKG SIN KUL BKK ICN IST

London Heathrow Charles de Gaulle Los Angeles Dubai Chek Lap Kok Singapore Changi Kuala Lumpur Suvarnabhumi Incheon Atatürk

as weights. As we found in Section 5.4 of Chapter 5, selecting available seats as edge weight leads to a classification where airports with high volumes of domestic traffic are prevalent. The airports to isolate when evaluating the robustness of the weighted airport network arguably offer a more significant perspective of node importance. The weighted ranking presents airports belonging to the US, European and Asian networks, whereas the unweighted lists (Tables 7.2 and 7.3) present more European and Asian airports. The weighted listing is free of topological biases, as the importance of the Alaskan network is largely reduced as differences in intensity of connections are considered.

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The analysis of robustness of the weighted world airport network complements the analysis of weighted connectivity carried out in Section 5.4 of Chapter 5. Considering connectivity and robustness analysis, the use of weighted measures offers a more balanced perspective of the world airport network. The use of total available seats in a direct connection as weight gives more importance to domestic traffic, whereas using available seats per kilometre gives more relevance to intercontinental traffic.

7.2 Robustness of regional airport networks In the previous section, we have performed an extensive robustness analysis of the world airport network. Considering that this network has a multicommunity structure revealed by the existence of airports of anomalous centrality [17], the robustness analysis of regions of the world airport network can give additional insight into the structure of airport networks. Previous research on robustness analysis of the world airport network has defined subsets in terms of political and geographical criteria. In an early study, Chi and Cai [9] studied the robustness of the airport network of the United States. Later, Boccaletti et al. [4] performed a multiscale vulnerability analysis of the Italian airport network, Hossain et al. [19] a robustness analysis of the Australian airport network and [33] a multilayered robustness analysis of the Greek airport network. Recently, Du et al. [12] studied the robustness of the Chinese airport network to edge removal. We performed two studies on subsets of the airport network: in Lordan and Sallan [22], we analysed the European airport network, and in Lordan and Sallan [23], we compared the robustness of regional airport networks, considering as regions the divisions of the world airport network into regions established in the OAG database. In this section, we compare the robustness of regional airport networks defined as communities of the short-haul airport network defined in Section 4.4.3 of Chapter 4. We build a short-haul world airport network by removing routes of more than 4,800 km (considered long-haul). We consider that regional airport networks are connected by short-haul flights and that long-haul flights are connected mainly by airports belonging to different regions. To identify communities, we used the Louvain clustering algorithm [3] using seats offered in August 2018 as edge weight. In Table 7.6, we present a brief description of each of the ten defined regional airport networks.

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Table 7.6 The regions defined by the ten largest communities of the short-haul airport network (see Section 4.4.3 of Chapter 4 for details of region definition). Region Description

North America (NA) Europe (EU) India and Middle East (ME) Oceania (OC) East and South Asia (EA) China (CH) South America (SA) Russia and Eastern Europe (RU) West Africa (WA) Indonesia (IN)

North and Central America, Marshall Islands Europe, Turkey and North Africa (Morocco and Argelia). Eastern Africa, Middle East and India. Australia, New Zealand, Papua New Guinea and Pacific Islands. Japan, Taiwan, Philippines and Southeast Asia (except Indonesia). Mainland China. South America, except Colombia and Venezuela. Russian Federation, Central Asia and Eastern Europe. West and South Africa. Indonesia.

Table 7.7 R parameter for sequential and simultaneous attacks based on node degree and node betweenness for regional airport networks. The most effective sequential and simultaneous attack for each region is presented in bold. Region sim deg sim btw seq deg seq btw

North America (NA) Europe (EU) India and Middle East (ME) Oceania (OC) East and South Asia (EA) China (CN) South America (SA) Russia and Eastern Europe West Africa Indonesia

0.07053 0.20039 0.08409 0.03685 0.08373 0.18747 0.07948 0.10362 0.06811 0.11229

0.06685 0.18190 0.10003 0.05010 0.09360 0.16773 0.07548 0.10373 0.06755 0.10965

0.05268 0.18058 0.08457 0.03259 0.08018 0.16150 0.07145 0.09167 0.06276 0.10882

0.03399 0.14783 0.06273 0.03386 0.06444 0.14510 0.06237 0.08312 0.05044 0.08542

To evaluate the robustness of regional airport networks, we obtained the parameter R resulting from attacking each region with simultaneous and sequential attacks based on node degree and betweenness. The results are listed in Table 7.7. We can make several observations from Table 7.7: • China and Europe have values of R higher than the other regional networks.

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Figure 7.5 Values of parameter R for sequential attacks with node betweenness strategy versus average degree for each regional airport network (see Table 7.6 for abbreviations).

• For all regions, sequential attacks are more effective than simultaneous

attacks. • Unlike models of complex networks and other real-world complex

networks, the degree does not always outperform the betweenness in simultaneous attacks: in five regions, the degree is more effective, and in the other five, the betweenness is more effective. In the most robust regions, the betweenness is the best strategy in simultaneous and sequential attacks. Oceania is the only regional network where the degree is more effective in sequential attacks than the betweenness. In Chapter 6, we have learned that networks with higher average degree tend to be more robust. In Fig. 7.5, we compare the average degree with parameter R obtained in sequential attacks with node betweenness strategy. There we can observe that regional airport networks with higher average degree tend to be more robust, although China and Indonesia regions tend to be more robust than expected given their average degree, and North America less robust. As scale-free networks are less robust to intentional attacks than random networks, a first attempt to explain differences in network robustness would look for differences in degree distribution. In Fig. 7.6, we present the degree distribution and robustness behaviour of North America and China

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Figure 7.6 Log-log plot of degree distribution (left) and evolution of largest connected component with a sequential attack (right) of North America (above) and China (below) regional airport networks.

regional airport networks. We can see that both networks have a degree distribution following a power law with an exponential cut-off, similar to the world airport network. Differences on network robustness can be explained from network structural characteristics different from degree distribution. In Chapter 6, we have introduced smart rewiring techniques that increased network robustness maintaining degree distribution [30]. The result of smart rewiring are networks with an onion-line structure [27], where nodes of high degree are interconnected forming a core, so that paths between nodes of high degree tend to have nodes of equal degree [32]. In real-world complex networks, these cores can appear spontaneously and can be detected

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Figure 7.7 Values of parameter R for sequential attacks with node betweenness strategy versus average degree for each regional airport network (see Table 7.6 for abbreviations).

with k-core decomposition analysis [11]. The k-core of a graph is a set of nodes whose degree is equal to or larger than k, so that cores of low values of k are embedded into cores of higher k. Du et al. [14] used the k-core decomposition to elaborate a multilevel decomposition of airport networks where a core is equal to the k-core of maximum k. In [22] and [23], we have used the fraction of nodes in the core as a predictor of network robustness, as a high value of this parameter suggests a naturally occurring onion-like structure. We have computed this parameter for each of the ten regional airport networks, and in Fig. 7.7, we have related this value with the same value of R of Fig. 7.5. We can see that China and North America have high and low fractions of nodes in the k-core of maximum degree, respectively. The high value of robustness of China network can be explained by the existence of a large core of interconnected nodes of high degree, whereas the opposite occurs in the North America network. In Fig. 7.8, we present the airports belonging to the core of North America and China airport networks. We can appreciate that the fraction of airports of North America network belonging to the core is much smaller than China’s network. We can argue that China’s air transport has grown around a smaller number of airports than that of North America, facilitating the spontaneous building of an onion-like structure.

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Figure 7.8 The airports of the North America and China regional airport networks. Airports belonging to the core of the network (k-core of the highest degree) are represented as large circles.

7.3 Robustness of airline airport networks Most research on robustness of airport networks has considered complete airport networks, being regional or local. These networks include all city pairs covered by air travel in the considered regions and time windows, irrespective of the airline or airlines that cover each route. Deregulation of air transport has given airlines considerable degrees of freedom to define their offer of routes, so most route decisions are made at airline level. Taking this into account, Lordan et al. [25] called for examining topology and robustness of airline and alliance airport networks, and Zanin [35] pointed out the need of considering the multi-layered nature of airport networks, as these can be considered as the aggregation of complementary layers of airline airport networks. In this section, we examine the robustness properties of a sample of airline airport networks, and we study when network robustness emerges from the aggregation of layers of airline airport networks.

7.3.1 Airline business model and network robustness Lordan et al. [24] reported the results of the topological analysis of airline airport networks, examining differences and similarities between different airline business models. They found that hubs of full-service carriers were usually more globally and locally central than operational bases of pure low-cost carriers. As a result, the robustness of airline networks of low-cost carriers was higher than that of full-service carriers. In Section 4.5 of Chapter 4, we have examined the topological properties of airlines adopting different business models: low-cost, hybrid, full-

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Table 7.8 Parameter R for sequential and simultaneous attacks based on node degree and node betweenness for selected airline airport networks. The most effective strategy for simultaneous attacks for each airline is printed in bold. Airline sim deg sim btw seq deg seq btw

Ryanair Volotea Southwest Airlines Transavia Vueling Norwegian American Airlines Lufthansa China Eastern Airlines Emirates

0.18766 0.18250 0.15192 0.02367 0.04126 0.03391 0.01685 0.00995 0.05869 0.00805

0.17691 0.17529 0.14590 0.02256 0.03957 0.03916 0.01560 0.01069 0.06292 0.00984

0.17297 0.15062 0.14284 0.02173 0.03904 0.03148 0.01591 0.00817 0.05556 0.00786

0.16848 0.14437 0.13856 0.02325 0.04073 0.03258 0.01626 0.01067 0.05590 0.01071

service and Arabian Gulf. The list of selected airlines can be checked in Table 4.9. In Table 7.8, we present the value of the parameter R of network robustness for a sample of ten airline airport networks, similarly as presented in Table 7.7 for regional airport networks. Airline airport networks seem to have a robustness pattern different from most real-world networks, as for the majority of selected airlines, the most effective strategy for simultaneous attacks is selecting the nodes of highest betweenness, and the most effective for sequential attacks is selecting nodes of highest degree. The sample included three airlines adopting the full-cost business model: Ryanair, Volotea and Southwest Airlines. These airlines are the most robust of the sample, all of them with values of R above 0.1. Low-cost airlines offer point-to-point flights, and their central airports play the role of operational bases, so they have lower degree heterogeneity [24]. Airlines with hybrid business models are Transavia and Vueling, both subsidiaries of full-service airlines offering flights with low-cost traits, like fare unbundling [15] together with Norwegian, a low-cost airline, which is entering the long-haul market. These airlines have values of R lower than low-costs: the main airports of hybrid airlines play a role closer to the hub of a full-service carrier than operational bases of low cost. The lowest values of robustness in the sample are for fullservice airlines (American Airlines, Lufthansa and China Eastern) and for Emirates, which follows a business model consisting in offering long-haul flights from its Dubai hub [29]. There is a sharp difference in robustness between these airlines and hybrid and low-cost airlines: for the former, the

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Figure 7.9 Evolution of the size of the largest connected component as a function of the fraction of removed nodes of airport networks of American Airlines (left) and Southwest Airlines (right).

removal of their hubs, which are the nodes of highest betweenness and centrality, leads to a severe disconnection of the airline airport network. Differences in network robustness are illustrated in Fig. 7.9, which presents the evolution of the largest connected components of Southwest Airlines and Lufthansa when experiencing sequential attacks.

7.3.2 Emergence of robustness in airport networks as multi-layered networks In addition to the interest of airline airport networks and its relationship with airline business model, these networks can be considered the layers of regional and global airport networks, as the later are the result of aggregation of routes offered by airlines. In Section 4.5.3 of Chapter 4, we have presented the evolution of some properties of airport networks as a function of the number of layers considered, following the methodology outlined in [8]. There we have observed that properties like the size of the largest connected component or the number of triangles depend on the number of layers, so these properties can be considered emergent by the aggregation of layers to the network, rather than being properties present in layers themselves (see Figs. 4.13 and 4.14). Following the call for examination of the influence of the multi-layered structure on properties of airport networks [35], we have examined the evolution of network robustness on regional airport networks, replicating

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Figure 7.10 Value of parameter R of network robustness for samples of m layers of the North America (left) and Europe (right) regional short-haul airport networks.

for robustness R the analysis performed on Section 4.5.3 of Chapter 4 for the Europe and North American short-haul networks. A motivation for this analysis is the comparison of variability of values of R in Tables 7.7 and 7.8: the robustness of airline networks is much more heterogeneous than that of regional networks. The evolution of robustness as we increase the number m of airlines included to build the airport network is presented in Fig. 7.10. North America and Europe airport networks present very different values of robustness, the former being less robust, although individual airlines of North America and Europe have similar values of robustness (see box-plots for m = 1 in each picture of Fig. 7.10). The high robustness of the Europe network emerges as a result of the aggregation of airline airport networks: we need around 35 out of 50 layers to achieve values of robustness similar to the total network. The patterns of aggregation of network of European airlines have lead to an airport network with higher average degree and relative size of core than in North America (see Figs. 7.5 and 7.7). So we can conclude that network robustness in airport networks emerges with aggregation of airline airport networks when this aggregation increases the average degree and relative size of core.

7.4 Delays and delay propagation In the previous sections of this chapter, we analysed the impact of major damages to the airport network, like closure of a subset of airports. Other

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malfunction of air transport network, not so salient but much more pervasive, is delays of scheduled air services. They are a source of consumer dissatisfaction and can cause serious disturbances in connecting flights. As most air traffic management authorities keep track of planned and real flights, extensive records exist of delays in air travel. Using these records, Britto et al. [6] have analysed the impact of delays on air transport, concluding that they produce a significant decrease in consumers and airlines welfare, airline’s reduction being three times larger than customer’s. Zou and Hansen [37] observed that airlines tend to pass the cost of delays to passengers, as they found a relationship between rise of fares and delays at arrival. A delay-related phenomenon that network theory can contribute to understand is the delay propagation, the apparition of delays in downstream flights originated by upstream flights [21]. The delay propagation can be analysed through examination of causal relationships in delays of pairs of airports, but this pairwise analysis can be extended defining a delay functional network. Functional networks have been defined to analyse brain [7], gene expression [36] or the immune system [31]. The development and analysis of delay functional networks is at its beginnings: Du et al. [13] have analysed the Chinese delay network, and Zanin [35] have studied the multi-layered nature of these networks, considering airline delay networks as layers. In this section, we introduce this research and convey our approach to delay functional network construction and analysis.

7.4.1 Constructing functional delay networks The nodes of a delay functional network are airports affected by delay propagation. A directional link from airport i to airport j will be established if delays in airport i cause delays in airport j. As the relationship does not need to be reciprocal, the resulting network is unweighted and directed. Building a delay propagation network helps to go beyond pairwise relationships and get a global picture of propagation of delays. Delay propagation occurs because of the existence of connected resources in air travel operations: for a flight to start, it may need aircraft and crew coming from previous flights, and it may occur that a flight needs to wait for passengers coming from a connecting flight. These connected resources are the causal link between delays of two different airports. Zanin [35] uses several measures to define links in the delay network: Pearson’s correlation, mutual information, Granger causality and transfer

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entropy. Like Du et al. [13], we have chosen Granger causality for link definition, as this measure takes into account temporal precedence to define delay relationships. The delay of a flight is defined as the difference between real time and planned time on departure and arrival. So a positive delay means that arrival or departure is later than expected. The delay of an airport is the average value of flights arriving at or departing from that airport. Analysis of several sources of delay data reveal that arrival and departure delays are strongly correlated, as delays at arrival are caused mainly by delays at departure. For this reason, we use departure delays only to define the functional network. To be able to evaluate temporal precedence, it is usual to compute the average value of delay for flights planned to depart at each hour of operation. This variable will be set to zero if no flights depart at a specific hour. So we denote as dit the average departure delay on airport i at time t. The Granger causality criterion establishes that delays at j are caused by delays at i if past values of delays at i contain information that helps predicting delays at j above and beyond the information contained in past values of delays at j alone. The existence of a causality relationship with the Granger criterion is tested with a hierarchical regression, where past values of dj are the control variable, and past values of di are the criterion. The first regression equation will be uc

djt

=

ij 

t−p

αp dj

+ εt ,

(7.5)

p=lijc

where lijc and ucij are the lower and upper bounds of lag in the control variable for evaluating the relationship between i and j. The second regression equation will be p

uc

djt =

ij 

p=lijc

u

t−p

αp dj

+

ij 

βr dit−r + ε t ,

(7.6)

p r =lij

where lijp and upij are the lower and upper bounds of lag in the predictor variables for evaluating the relationship between i and j. We can say that delays in j are caused by delays in i if the model of Eq. (7.6) has more explanatory power than the model of Eq. (7.5). This is determined with an F-test of the null hypothesis if equality of explanatory power is rejected. Note that we need to perform a hypothesis testing for each relationship.

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Usually, the lags used for the control and predictor variables are equal and related to the average flight time from i to j and start at the next time period, so lijc = lijp = 1 and ucij = upij = uij . There are two research design decisions to make when building a delay network: time window and relationships to evaluate. The time window will determine the number of observations available for the regression equations: if we take one single day, then we will have 24 − uij observations. This number will increase if we take a larger time window. Here we need to consider a trade-off: a short time window (e.g., a single day) will yield regression equations of low statistical power, meaning that the likelihood of type II errors (not detecting an existing relationship) will increase. A model built with data from a wider time window (e.g., a month or a year) will have higher statistical power, but it may detect causal relationships in different moments of the time window (e.g., different weeks or months within a year). As for the number of relationships to evaluate, the most common approach, adopted in [35,13], is to consider all pairs of airports. As delay propagation comes from connected resources, we have chosen to evaluate only relationships between pairs of airports with at least one flight in the considered time window. In this approach the delay network is a subset of the directed airport network. When using the Granger causality test to determine the delay functional network, we have to evaluate a multiple test of hypothesis. This requires being more exigent than when evaluating a single hypothesis if we want to limit the probability of type I errors (considering as existent a non-existing relationship). For instance, if we evaluate m = 100 tests of hypothesis and we fix an alpha level of α = 0.05, then the probability of committing at least one type I error is 1 − (1 − α)m = 1 − (1 − 0.05)100 = 0.994.

(7.7)

If we want to control type I error rates in a context of multiple tests of hypothesis, we need to lower the value of α for each individual test. The most drastic solution is rejecting null hypothesis with p-values lower than α/m. This is the Bonferroni correction (in fact, credited to Oliver Jean Dunn). The Bonferroni correction is a procedure of controlling the familywise error rate (probability of making at least one type I error), which can be too conservative for large values of m. The development of research requiring testing a high number of hypothesis has led to the development of methods of control of the false discovery rate. The aim of these methods

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is to limit the rate of false discoveries when performing multiple test of hypothesis. Among the most common methods of control of false discovery rate, there are the Benjamini–Hochberg method [2] for independent tests and the Benjamini–Yekutieli method [34] for arbitrary dependence between tests. Adopting a method of false discovery rate control leads to a more precise delay network than that without adopting any method or that obtained by fixing an arbitrary value of density for the delay network. Finally, as pointed out by [35], delay networks are multi-layered in nature. Although some delays can be attributed to weather of air traffic control, some causes of delay propagation can be found within the domain of the airline, as aircraft, crew or passengers are connected resources that can be behind delay propagation. Then, it can be interesting to study airline delay propagation networks.

7.4.2 The European delay propagation network For illustrative purposes, we have built several delay networks from delay data obtained from the DDR2 (Demand Data Repository) of Eurocontrol for two different time windows: • The third week of August 2018 (from 13 to 19 August). • The second week of November 2018 (from 12 to 18 November). The selected weeks represent summer and winter seasons, which may have significant differences in terms of flights scheduled [28]. We have considered flights arriving at and departing from airports of countries members of the European Civil Aviation Conference, including Switzerland. For each airport, we have computed the hourly average depart delay for both time windows. We have adopted the following additional criteria for building the network: • We have considered only relationships corresponding to the routes with at least one flight. • The Granger causality tests are dependent, so we have chosen the p-values that allow a failure discovery rate of 0.05 adopting the Benjamini–Yekutieli criterion. Prior to analysing the weekly delay networks, we have examined the size of the delay networks for the four weeks of August and November 2018: the results are presented in Table 7.9. Delay networks seem to be larger in seasons of high traffic (as is August in Europe) than in seasons of low traffic (like November in Europe). This seems a logical consequence of the airport system being closer to maximum capacity in the summer season.

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Table 7.9 Nodes and edges of the European delay networks for four weeks of August and November 2018. In bold, the delay networks selected for further analysis. August November N E N E

236 168 192 167

748 204 287 196

82 140 77 61

60 129 51 48

Table 7.10 Correlation about outdegree, in-degree and traffic for August (upper diagonal) and November (lower diagonal) time windows.

s kin kout

s 1.000 −0.329 0.414

kin −0.008 1.000 −0.459

kout 0.455 0.072 1.000

As the delay network is unweighted and directed, two relevant node magnitudes are worth analysing: out-degree and in-degree. The out-degree kout is the number of edges coming out from node i, so it is the number i of airports affected by delays at airport i. The in-degree kin i is the number of edges incident to node i and represents the number of airports that are causing delays at airport i. It may be worth considering the relationship between these two measures of the delay network with total traffic s, which is the total number of flights arriving at or departing from the airport. Airports with high traffic are more connected with other airports, so we may expect high values of out- and in-degrees for these airports. To confirm this, we have computed the correlation matrix between the three measures for the two one-week time windows considered. In both cases, we see that correlation of out-degree with traffic is stronger than with indegree. This means that larger airports tend to amplify the effect of delays (see Table 7.10). To examine the role of airports in the delay network in more detail, we have constructed two plots for each time window: a picture of the delay network with the geographical position of nodes and a second map presenting information of traffic volume of each airport and indicating if it amplifies (kout > kin ) or mitigates (kin > kout ) processes of delay propagation.

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Figure 7.11 Left, the delay functional network for European flights for the third week of August 2018 (from 13 to 19). Right, the nodes (airports) of the delay network with size proportional to traffic. In grey, airports amplifying delay propagation; in black, airports mitigating delay propagation.

Figure 7.12 Left, the delay functional network for European flights for the second week of November 2018 (from 12 to 18). Right, the nodes (airports) of the delay network with size proportional to traffic. In grey, airports amplifying delay propagation; in black, airports mitigating delay propagation.

In Figs. 7.11 and 7.12, we show the results for the third week of August and the second week of November 2018, respectively. As advanced in Table 7.9, the delay network in August (the European summer season) is denser than in November (the European winter season). In both time windows, we observe that the larger European airports mitigate delay networks, with the exception of London Heathrow in the summer season. This behaviour is different from that obtained by Du et al. [13] for the Chinese delay network, where larger airports amplify delay propagation. Delay propagation comes from connected resources, many of them coming from airline operations. Then delay functional networks can be

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Figure 7.13 The Ryanair delay functional network for the third week of August 2018. Plots have analogous meaning as in Figs. 7.11 and 7.12. To determine plot size, we have considered traffic coming from Ryanair only.

Figure 7.14 The Vueling delay functional network for the third week of August 2018. Plots have analogous meaning as in Figs. 7.11 and 7.12. To determine plot size, we have considered traffic coming from Vueling only.

considered multilayered networks [35], each layer being constructed examining hourly average delays for each airline in each airport. Constructing a multi-layered delay functional network as union of airline delay networks can be problematic if we want to control failure discovery rate at a reasonable level, but we can examine some individual airline delay networks. We have chosen two airlines that operate mainly in Europe, Vueling and Ryanair. Ryanair has a large, relatively decentralised airport network, whereas Vueling’s airport network is more centred around its hub in Barcelona-El Prat. The plots for the Ryanair network are presented in Fig. 7.13, and Fig. 7.14 shows analogous plots for the Vueling network. The delay network of Ryanair is denser than that of Vueling, as the former has a larger, more decentralised airport network. In the case of

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Ryanair, we observe that its bases in Ireland, United Kingdom and Spain amplify delay propagation. This can be attributed to the intensive use of Ryanair’s aircraft and crew in point-to-point flights, sometimes with tight schedules that may be prone to delay propagation. Vueling delay network is less dense and centred in Barcelona. In this case, we observe that the effect of Barcelona-El Prat airport is to mitigate delay propagation, as in the European delay network. The differences of delay propagation networks of both airlines point out differences in business model, as Ryanair seems to be more focused on efficiency in aircraft and crew than Ryanair. Delay propagation networks can give insights about the effect of connected resources in airline and airport operations on air traffic delays, a source of inefficiencies and dissatisfaction with air travel, which leads to reduce customer satisfaction and the rise of air fares. As the construction of these networks relies on hypothesis testing about relationships between time series of delays, the obtention of significant results calls for extension of time windows to gain statistical power (reduction of type II errors) and to selecting significant p-values using techniques that control false discovery rate (type I errors). The application of the later techniques may help to build delay propagation networks with a (relatively) high value of nodes and edges, controlling for type I and type II statistical errors in hypothesis testing.

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