Exploring the Regional Structure of the Worldwide Air Traffic and Route Networks

Te topological structure of the world air transportation network has been the subject of much research. However, to better understand the reality of air networks, one can consider the trafc, the number of passengers, or the distance between fights. Tis paper studies the weighted world air transportation network through the component structure, recently introduced in the network literature, by using the number of fights. Te component structure is based on the community or multiple core-periphery structures and splits the network into local and global components. Te local components capture the regional fights of these two mesoscopic structures (dense parts). Te global components capture the inter-regional fights (links between the dense parts). We perform a comparative analysis of the world air transportation network and its components with their weighted counterparts. Moreover, we explore the strength and the s-core of these networks. Results display fewer local components well delimited and more global components covering the world than the unweighted world air transportation network. Centrality analysis reveals the diference between the top airports with high trafc and the top airports with high degrees. Tis diference is more pronounced in the global air network and the largest global component. Core analysis shows similitude between the s-core and the k-core for the local and global components, even though the latter includes more airports. For the world air network, the North and Central America-Caribbean airports dominate the s-core, whereas the European airports dominate the k-core.


Introduction
Te concept of a complex network paradigm provides a more profound comprehension of diverse, interconnected systems, including social networks, economics, epidemics, and infrastructure [1].Infrastructure networks such as power grid networks [2], road networks [3][4][5], and airport networks [6] receive a lot of attention.Te air transport system connects all countries in the world.Tis infrastructure has a direct impact on society and the global economy.Indeed, millions of people and goods transit through the air every day.Te COVID-19 pandemic has signifcantly impacted the global air transportation network.Te virus spread rapidly through air travel, and research has shown the consequences of political shutdowns on national and international economies.Reference [7] provides further insights into these repercussions.Tus, understanding the air transportation system can help policymakers make decisions that can improve or afect this system.Network science provides a simple way to represent and understand this infrastructure.Tus, several studies are devoted to the air transportation network, including structure, dynamics, and robustness.In a previous study, we analyzed the world air transportation network through a new mesoscopic structure called the component structure [8].A network contains two types of components.Te dense parts of the network form the local components.Te interactions between the local components are called the global components.Terefore, one must extract the dense areas to build the component structure.To do so, the community structure and the core-periphery structure are good candidates.Indeed, the communities constitute cohesive groups of nodes sparsely connected [9][10][11].Te core-periphery [12][13][14] structure contains two groups of nodes (core and periphery) with three types of connections.Te core nodes are tightly connected.Te periphery nodes are almost not connected.Te links between the core and periphery nodes are relatively dense.Networks can exhibit a multi-core-periphery structure [15].One can extract the dense parts using any algorithm to uncover the communities or the various cores to form the local components.In the world air transportation network, the local components capture the regional destinations, while the global components represent the interregional fights.In the previous study, we analyzed the unweighted world transportation network.Terefore, the results of our investigations are related to the network infrastructure.Indeed, it uses no information on the dynamics of the infrastructure.Tis simplifcation can lead to centrality anomalies [15,16] and hide critical information about the fow of fights and passengers in the infrastructure.Indeed, the trafc in the various routes can be pretty different, and failing to integrate these diferences can lead to misleading conclusions.
Tis paper presents a comparative analysis of both weighted and unweighted world transportation networks.Te unweighted network focuses on the infrastructure, representing various routes connecting airports, while the weighted network considers the passenger trafc on these routes.Te analysis is carried out at three levels: macroscopic, mesoscopic, and microscopic.
(1) Macroscopic Level.Tis level allows us to identify and highlight the global characteristics that diferentiate the two types of networks.
(2) Mesoscopic Level.At this level, we adopt a component structure representation to explore and distinguish the infuences of geography and economy.Five geographical areas (local components) of the weighted world air transportation network are investigated.Furthermore, we also study the network formed by inter-regional fights.To facilitate our analysis, we examine and compare the k-core and score of the network.
(3) Microscopic Level.In this level of analysis, we evaluate and contrast the degree and strength of airports, aiming to compare highly connected airports to those experiencing heavy trafc.
By conducting a comprehensive analysis across these three levels, this paper aims to understand better the differences and interactions between the weighted and unweighted world transportation networks.
Te rest of the paper is organized as follows.Section 2 reports a review of related studies of the air transportation network.Section 3 describes the data, and Section 4 examines the network community structure.Section 5 reports the analysis of the component structure.Section 6 explores the topological properties of the local and global components.Section 7 presents the topological properties of the world air transportation network.Section 8 presents the results of a comparative analysis of the strength centrality of the components and the world air transportation network.Section 9 discusses the results of the core structure analysis.Finally, we conclude in Section 10.

Literature Review
Te literature reports numerous studies of unweighted air transportation networks [17][18][19][20].Tey cover national, regional, and worldwide networks and include investigations at the macroscopic, mesoscopic, and microscopic scales.All these networks share some common characteristics.Tey are generally small-world and scale-free.Moreover, some exhibit a community structure, and one can observe that the most connected cities do not have the largest betweenness [17].
In contrast, the weighted air transportation network is not much studied.It is particularly true for the world air transportation network.In the following, we briefy report related studies of the national, regional, and worldwide weighted air transportation networks.
In [20], the author studied the weighted airport network of India.Te weight indicates the number of weekly fights between two airports.Te distribution of strength reveals the heterogeneity of this network.Moreover, the author showed that the strength of a node correlates well with its degree.Comparing nodes' unweighted and weighted clustering coefcients indicates that the hubs tend to form interconnected groups.However, the airport network in India is disassortative.Te weighted assortativity shows that the hubs with many fights are more connected.
In [21], the authors investigated the air transportation network of the United States from 2002 to 2005, by quarter.Tey explored 16 networks in this period using topological and weighted metrics.Cities are the nodes, and routes between two cities are the edges.Tey considered three types of weight: the nonstop distance between cities, the average passengers, and the average one-way fare.Te degree and the strength distribution of these three attributes exhibit a scalefree behavior.Additionally, they are correlated.Ten, highdegree nodes tend to have high-strength links.Te analysis of the degree-degree correlation shows a rich-club phenomenon.Indeed, there is large trafc among the hubs.Moreover, the interconnected hubs make long distances at an accessible price due to the multiplication of point-topoint fights.Finally, the authors proposed a weighted network model.
In [22], the authors studied the Australian airport network's structure and dynamic fow.Te weights correspond to the number of fights between two airports.Te authors found that the strengths of the nodes evolve in the same direction as the degree.Sydney airport, the most prominent hub, handles the highest fraction of trafc.Te weighted clustering coefcient is lower than the unweighted.Consequently, the edges with low weights form the topological clustering.Te weighted and unweighted degree-degree correlation reveals a disassortative behavior.2 Complexity In [19], the authors analyzed the characteristics of the Asian international passenger aviation market in 2014 and 2018.Te number of passengers weights the links between two airports.To conduct their analysis, they considered 28 Asian top airlines, with 7 low-cost carriers and 21 full-service carriers.Te airports of the low-cost carriers increased, and their degree distribution shows the evolution of the low-cost carriers into a hub and spoke system.Developing countries like China and Vietnam infuence the air transport network of this region.To characterize the airports' infuence, they explored eight centrality measures: the degree, mean association (normalized strength), betweenness, weighted betweenness, page rank, weighted page rank, reverse page rank, and weighted reverse page rank.For most centrality measures, Changi Airport, Incheon Airport, Narita Airport, and Hong Kong Airport are in the top position in 2014 and 2018 [19].
In [23], the authors proposed to extend the defnitions of the clustering coefcient and the assortativity to a weighted network.Tey used the world air transportation network as a typical example.Te link weight is the number of seats available on the fights between two airports.Tey showed that the degree and strength distributions are heavy-tailed.Moreover, there is a linear relationship between a node's degree and strength.Te network exhibits the rich-club phenomenon.Indeed, the interconnected neighbors of the hubs handle the most signifcant proportion of the trafc.Te weighted degree-degree correlation is assortative.
In summary, previous studies on weighted networks have focused on analyzing national, regional, and global air transportation networks.Notably, when studying the weighted world air network, the analysis has primarily concentrated on macroscopic properties [23].Tis paper investigates the weighted world air transportation network in light of its component structure.We perform an extensive comparative analysis of the route (unweighted) and trafc (weighted) networks at the macroscopic, mesoscopic, and microscopic levels.

Data and Tools
Tis section describes the data used to conduct our experiment.Ten, two weighted community detection algorithms are applied to the data.Te purpose is to compare their community structure.Finally, we also compare the communities from the unweighted and weighted world air transportation network.
3.1.Data.Tis network of 2734 nodes and 16665 links originates from FlightAware [24].It collects the fights between May 17, 2018, andMay 22, 2018 [25].Nodes represent airports, and links are direct fights between airports.Te link weight is the number of fights between two airports.Table 1 reports basic properties (unweighted and weighted).One needs at least 12 fights to reach the most distant airports.On average, four fights are required to reach any destination.Te network is not very dense, and globally, there are few triplets by airports with many fights (C w ≥ C).Te unweighted assortativity reveals that the hubs tend to connect to the airport with few connections.In addition, the global weighted assortativity shows numerous fights between these hubs and these types of airports (K w ≥ K).

Tools.
Here, we briefy recall the algorithm to extract the component structure.As it relies on the communities to form the local components, we describe two community detection algorithms that uncover these dense parts of the network.Finally, we present the evaluation measures.

3.2.1.
Extracting the Component Structure.Te process of extracting the component structure described in [8] consists of three steps: (1) Extracting the dense areas of the network by using a community or multiple core-periphery detection algorithms.
(2) Extracting the local components by removing all the links between the dense areas.(3) Extracting the global components by removing all the links within the dense areas.
Note that a node can belong to the local and global components.In this paper, weighted community detection algorithms are used [26].Indeed, they take into account the dynamic fow in the network.

Extracting the Communities.
We use two popular community detection algorithms: Louvain [27] and Combo [28].Tey are weighted nonoverlapping community detection algorithms based on modularity.Our purpose is to evaluate the impact of the community structure variations induced by the community detection algorithms on the component structure.
(1) Louvain.Louvain constructs the communities in two phases, repeated iteratively.At frst, each node is a community.Ten, one evaluates the modularity after grouping neighboring nodes.Te second step considers the group of nodes (community), maximizing the modularity as new nodes.Te algorithm stops when there are no more modifcations, and the modularity can no longer be maximized.Note that the defnition of modularity depends on the nature of the network (unweighted or weighted).
(2) Combo.Combo combines three strategies to uncover the community structure.Indeed, at the initial stage, all nodes form a community.Ten, for several iterations, one can merge communities, split communities, or recombine nodes between communities according to the optimization of the objective functions until any gain is possible.One can use two objective functions (modularity and code length).In this paper, we use the algorithm based on modularity.

Quality Metrics.
We use the classical metrics such as the modularity and the mixing parameter [29][30] to measure the quality of partitions.In addition, we use the NMI to quantify the similarity of the partitions from the Louvain and Combo weighted community structures.

Modularity.
Modularity is the most popular one.It compares the actual community structure with a null model without community structure.Its values range between −1 and 1.Several community detection algorithms aim to optimize modularity.Te best partition is the one that is nearer to 1. Te modularity of weighted networks [27] is defned as follows: A ij is the weight of link between i and j; k i is the strength of node i; c i is the community that i belongs to; the Kro-

Mixing Parameter.
For a weighted network, the mixing parameter μ w of a node i is the proportion of weight pointing outside its community.Te mixing parameter of the network is the average of the nodes' mixing parameters.Te communities are well separated when the average mixing parameter is near 0. Te smaller the mixing parameter, the stronger the community structure.Te mixing parameter of a network is defned as follows: where μ w i � w ext /w i , w ext is the sum of weight from node i to the communities that do not contain i, and w i is the strength of node i.

Normalized Mutual
Information.Te normalized mutual information measures the similarity of two partitions.Te partitions are similar when the NMI is near one and almost independent when close to 0. Te NMI of two partitions P 1 and P 2 is defned as follows: where I(P 1 , P 2 ) is the mutual information between P 1 and P 2 .H (P) is the entropy of P. Here, the sum of entropy normalizes the mutual information.
3.2.7.Jaccard Index.Te Jaccard index is used to compare the similarity of two sets.It is defned as follows: When the two sets are identical, the Jaccard index equals 1.It is equal to zero if the two sets have no element in common.

Community Structure Analysis
Tis section frst compares the weighted community structures uncovered by Louvain and Combo.Ten, we compare the communities of the weighted and unweighted world air transportation network uncovered by Louvain.We use the modularity and the mixing parameter as quality measures of the community structures.We also perform a comparative qualitative evaluation highlighting the similitudes and diferences between weighted and unweighted networks.

Comparing the Weighted Community Structure Uncovered by Louvain and Combo.
Te number of communities extracted from the networks by the algorithms is quite diferent.Louvain uncovers 17 communities.Te largest contains 725 airports, whereas the smallest includes two airports.Combo identifes seven communities.Te largest includes 703 airports, and the smallest consists of 70 airports.Table 2 displays the quality metrics of the two community structures.Teir modularity is identical.Its value of 0.47 indicates that the communities are dense, with a medium proportion of intercommunity links.Te community structure of Louvain contains few connections between the communities with strong weights.In contrast, Combo reveals more intercommunity links with smaller weights.In both cases, the mixing parameter values demonstrate that the communities are well separated.
Although the Louvain algorithm uncovers more than two times more communities than the Combo algorithm, their community structures have numerous similitudes.Indeed, Combo groups some communities of Louvain.Te high value of the NMI (0.87) confrms their similarity.Five communities are very similar at frst glance.Tey cover the same geographical areas.In Figure 1, these areas correspond to the communities with the same color.Tey are in North and Central America-Caribbean, Europe-Russia-Central Asia, East and Southeast Asia-Oceania, Africa-Middle East-Southern Asia, and South America.In addition, Table 3 shows their Jaccard Index.Its value for all similar communities is greater than 0.85.For North and Central  4 shows that the regrouped weighted communities are very similar to the unweighted communities.Tese results reveal several fights between Europe and Russia and East-Southeast Asia and Oceania, even if the number of connections between these regions is limited.Apart from grouping communities of the unweighted network, the communities of the weighted network present some particular singularities.Indeed, some airports belong to communities far from their geographical area.Te airports in North and West Africa are in the Europe-Russia-Central Asia community in the weighted network, while they are in the Africa-Middle East-Southern Asia in the unweighted network.Indeed, these airports have more fights to Europe despite sharing more connections to the other airports in Africa and the Middle East.John F Kennedy Airport in the United States is in the Europe-Russia-Central Asia community.Likewise, Frankfurt Airport in Germany and London Heathrow in the United Kingdom are, respectively, in the Africa-Middle East-India region and the North and Central America-Caribbean area.Tese examples confrm that the communities of the weighted network correspond to areas of infuence, while in the weighted network, they correspond to geographical regions.
Te large communities of the weighted network include several small communities of the unweighted network.Moreover, like large communities, a few small communities of the unweighted network are grouped in the weighted network.
Te modularity and the mixing parameter values reported in Table 5 show a higher community structure strength for the unweighted network.Nevertheless, in both cases, the community structure strength is in a medium range indicating a clear community structure.Te mixing parameter values corroborate these fndings.

Component Structure
We categorize the uncovered components as large or small.Te large components include more than 100 airports and cover large geographical areas.In this section, we describe their features and compare them to the component structure of the unweighted network [8].

Local Components.
Te local components are the dense parts of the networks.Tey correspond to the 17 communities uncovered by Louvain in the weighted network.Tere are fve large and twelve small local components.Te large local components do not refect strict geographical divisions.Tey correspond more to political, cultural, historical, and economic divides.For example, some African airports in Morocco, Tunisia, and West Africa belong to the European component.It is because of the solid economic and historical ties these countries share with Europe.Te small local components are in a single country (the United States, Canada, French Polynesia, Greenland, Israel, Australia, and United Arab Emirates) or a few countries (Caribbean).

Large Local Components.
Te large local components cover (1) North and Central America-Caribbean (725 airports), (2) Europe-Russia-Central Asia (683 airports), (3) East-Southeast Asia-Oceania (630 airports), (4) Africa-Middle East-Southern Asia (313 airports), and (5) South America (201 airports).Altogether, they regroup more than 93% of the world's airports.One can distinguish two typical behaviors when comparing the large local components of the weighted and unweighted network illustrated in Figure 2. In the frst case, the components are very similar.In the second case, separated components in the unweighted network merge into a single component.
Tere are three similar components (North and Central America-Caribbean, Africa-Middle East-Southern Asia, and South America).We quantify their similarity using the Jaccard index.Te higher the similarity is, the closer the Jaccard index is to 1.
Te North and Central America-Caribbean weighted component contains 10% more airports than the unweighted.Teir Jaccard index is high (0.81).Note, however, Te Jaccard index of the unweighted (215 airports) and weighted (201 airports) South America components is 0.78.Tirty-two airports in this unweighted disappeared in the weighted.Tese airports are in Chile and Peru.As aforementioned, the airports of these two countries are now in the weighted North and Central America-Caribbean component.Te airports appearing in the weighted South America component are in Venezuela, Colombia, and Cuba.
Tere are two merged components.Te Europe-Russia-Central Asia component regroups the European and Russia-Central Asia-Transcaucasia components from the unweighted world air transportation network.Similarly, the "East and Southeast Asia-Oceania" component includes the unweighted network's East and Southeast Asia and Oceania components.We join the unweighted European (493 airports) and Russia-Central Asia-Transcaucasia (112 airports) components and the unweighted East and Southeast Asia    Te unweighted and weighted East and Southeast Asia-Oceania components are very similar.Indeed, their Jaccard index equals 0.97.Tese two regions, through the largest airports, have signifcant trafc.In the weighted component, twenty-three airports in French Polynesia disappeared, and three airports from Russia and India emerged.Vladivostok and Yuzhno-Sakhalinsk airports are near China and North Korea.Teir exchanges are with China, Japan, and South Korea.Te Trichy airport in India has heavy trafc with Malaysia, Singapore, and Sri Lanka. 3 presents the small local components uncovered in the weighted (Figure 3(a)) and unweighted (Figure 3(b)) network for comparative purposes Tey are either in a single country (the United States, Canada, French Polynesia, Greenland, Israel, Australia, and United Arab Emirates) or cover a few countries or subregions (Caribbean).Te biggest small component in Alaska contains 30 airports.Te smallest includes two airports.In the following, we concentrate on small components with a size greater than fve airports.

Small Local Components. Figure
Alaska has three small components with comparable sizes (around 27 airports).Tese components have a star shape.Indeed, most of the trafc goes through a leading airport.Fairbanks Airport, in Northeast Alaska, leads the frst component.Te Nome Airport dominates the Northwest.Te third component in the Southwest is centered around the Bethel Airport.Tese components are also in the unweighted world air transportation network.It reveals the isolation of Alaska in terms of trafc and connections.
Canada possesses two small components.Te largest contains 25 airports, mainly on Nunavut and Quebec coasts.It is dominated by Iqaluit and Quujjuaq airports.Te second includes ten airports in the Northwest Territories.Te Inuvik airport serves several fights in this component.Note that the unweighted network merges these two components.
Te signifcant small component in Greenland contains ten airports.Te most frequent fights operate through Godthaab/Nuuk, Kangerlussuaq, Ilulissat, and Sisimiut airports.Tis component is in the unweighted network.Indeed, in Greenland, air transport is a must.
Twenty-two airports in French Polynesia constitute a small local component.Tis country is a set of islands.Tus, air transportation is very developed.Te major airport of this country, Faa'a airport, reaches 18 airports with 547 fights.Te Bora Bora airport is the second most important, with 302 fights.Tis component is not in the unweighted network.So, the fow of fights between the airports of French Polynesia is essential, even though the connections are not dense.
Te last signifcant small component includes French Antilles, Quebec, and Ontario airports.Indeed, there is a strong community from French Antilles based in these two cities of Canada.Terefore, there is high trafc between these airports.Te major airports in Trinidad and Tobago (Piarco Airport and Tobago-Crown Point Airport) capture most of the trafc.Tis component also exists in the unweighted network.
Te 13 small unweighted components that disappear in the weighted network are aggregated into the weighted large components.Among them, four are located in North America, fve are located in Europe, and four in Africa.
Globally, the small weighted local components cover 6.6% of airports.Tey are in North and Central Each color is associated with a component.Te North and Central America-Caribbean component is red (1).Te Africa-Middle East-India component is green (2).Te South American component is brown (3).Te Europe-Russia-Central Asia component is black (4).Te Europe-Russia-Central Asia component is black (4).It is divided into Europe in black (4) and Russia in purple (7) in the unweighted network.Te East and South-East Asia-Oceania component ( 5) is divided into East and South-East Asia in blue (5) and Oceania in orange (6) in the unweighted network.America-Caribbean (3 in Alaska and 2 in Canada and 1 in the Caribbean), Europe (1 in Greenland and 1 in Israel), East and Southeast Asia-Oceania (1 in French Polynesia and 1 in Australia), and Africa-Middle East-Southern Asia (1 in the United Arab Emirates).Among them, fve components include less than fve airports.Te Jaccard index between the weighted and unweighted large global components is not high (0.63).Te large global component of the weighted world air network contains 14 more airports than the unweighted.However, their content is very diferent.Indeed, 100 airports disappear from the weighted largest global component, and 144 new airports integrate it.Airports disappear in the largest weighted global component because Oceania and Russia merged with their neighboring regions reducing the intercomponent links.New airports appear because of the singularities in the North and Central America-Caribbean, Europe-Russia-Central Asia, and Africa-Middle East-India components.Indeed, highly connected airports not localized in their natural geographical components, such as London Heathrow and John F Kennedy, increase the links between the local components attracting new airports in the global component.Moving an airport from one local component to another modifes the large connected component drastically.

Global Components.
Te small global components include 36 North and Central America-Caribbean airports and East and Southeast Asia-Oceania.Teir size ranges from 4 airports to 2 airports.Canada contains most of them (11).Only two components are shared with the unweighted world air transportation network.In the following, we neglect these components.

Topological Properties of the Large Components
Tis section investigates the clustering coefcient and degree-degree correlation.We recall the defnitions of these topological properties in weighted networks.

Clustering Coefcient.
Te clustering coefcient C i of a node i refects the cohesiveness of its neighbors.Te closer it is to 1, the more interconnected its neighbors are.Whether C(k) ≈ k −1 , the network has a hierarchical organization [32].Te weighted clustering coefcient C w of a node i is defned as follows [23]: One can defne C(k) and C w (k) as the average clustering coefcient of nodes with degree k for, respectively, unweighted and weighted networks.Te relation C w > C indicates that the interconnected triplets tend to be formed by links with high weights.Te opposite C w < C shows that lower-weight edges produce interconnected triplets.Te average unweighted clustering coefcient decreases monotonically with the degree.Low-degree nodes have a large clustering coefcient.In contrast, hubs are less cohesive.Tis characteristic is shared with numerous networks such as the India air transport network [20], the actornetwork, and the World Wide Web network.Fitting C(k) by the law k −c , we observe that c ≈ 0.3 for the Africa-Middle East-Southern Asia and the South America components.For the other components, c ≈ 0.2.

Large Local Components.
In contrast, the average weighted clustering coefcient is independent of the degree.Tis result difers from previous analysis with real-world networks [20,23].Moreover, weighted average clustering coefcients are always lower      Complexity 9 6.2.Degree-Degree Correlation.Te degree-degree correlation k nn (k) assess the relation between neighbor nodes.It measures the probability that a node of degree k connects with a node of degree k′.If high-degree nodes tend to connect to high-degree nodes, the network is said assortative.It is disassortative if high-degree nodes tend to connect to low-degree nodes.Te weighted degree-degree correlation is defned as follows [23]: One can compare weighted (k w ) and unweighted (k nn,i ) degree-degree correlation.If k w > k nn,i , edges with high weights tend to connect high-degree nodes.If k w < k nn,i the edge weights connect the degree nodes with opposite magnitudes (low or high).

Large Local
Components. Figure 6 presents the average degree-degree correlation evolution versus the degree for the large local components.One can observe that the weighted degree-degree correlation is greater than the unweighted degree-degree correlation.It indicates a high fraction of trafc transit between hubs in the local components.
However, the curve trend of the unweighted and weighted degree-degree correlation as a function of k is diferent for the large local components.Indeed, above a threshold k, the weighted and unweighted degree-degree correlation of the North-Central America-Caribbean (k ≈ 70), and South America (k ≈ 25), decreases.Te highdegree nodes tend to connect to low-degree nodes.Tus, the disassortative aspect of these unweighted and weighted components appears clearly.Te Europe-Russia-Central Asia component's degree-degree correlation decreases slowly compared to the South America and North-Central America-Caribbean components.Moreover, the degreedegree correlation of the high-degree nodes is higher.It is also less disassortative.In the East and Southeast Asia-Oceania and the Africa-Middle East-Southern Asia components, the unweighted degree-degree correlation shows a characteristic that is slightly disassortative.Te average weighted degree-degree correlation of these components is relatively constant.

Large Global
Component.Figure 6 reports the average degree-degree correlation as a function of degree for the global component.Te unweighted network curve decreases.Terefore, the global component is disassortative.Indeed, the largest hubs (k > 100) connect with low-degree nodes (k < 20).Te weighted network curve exhibits a similar evolution.However, K w (k) > K(k) for most of degree node k.Terefore, even though the hubs connect with smalldegree nodes, they accumulate more fights.

Large Local
Components. Figure 7 represents the strength distributions of the large local components.We perform a goodness-of-ft evaluation with the Kolmogorov-Smirnov test (KS) using the power law, truncated power law, log-normal, and stretched exponential distributions.Results reported in Table 6 reveal that the log-normal distribution better fts the large local components.
Moreover, they are heavy-tailed, like the degree distribution of the unweighted large local components of the world air transportation network.One can expect this result.Indeed, the higher the node degree, the higher its weight.Figure 8 shows this behavior.One can see the relation between the strength and the degree for each large local component.Indeed, the average strength and the degree are linked by the relation (s(k) � k β ).Even though for all the large local components, β ranges between 2.1 and 2.3, this organization is similar to results reported in [20] where β ≈ 1.43.
One can diferentiate three types of large local components if we focus on the exponent of the strength versus degree curve (β).Te Europe-Russia-Central Asia component forms the frst category.It includes several hubs with almost the same trafc.Indeed, the trafc increases slowly as a function of degree.Te second category consists of the North-Central America-Caribbean and the East-Southeast Asia-Oceania components.Tese components also contain several hubs.However, the trafc concentrates in a few hubs.Te Africa-Middle East-Southern Asia and the South America components are in the last category.Tere are fewer hubs and less trafc compared to the other types.Te strength as a function of degree increases faster.Consequently, few hubs accumulate most of the trafc.7 and 8 show the strength distribution and the strength as a function of k of the large global component.Like the large local components, the log-normal law better approximates its strength distribution according to the KS test.One can also see that the strength distribution is heavy-tailed.In addition, the distribution parameters are similar to those of the East-Southeast Asia-Oceania component.As found with the degree-degree correlation, the degree's strength shows that the higher the node degree is, the more trafc it accumulates.Airports with fewer connections have at least more than 100 fights.It is not the case in the large local components.Te β exponent is comparable to those of North and Central America-Caribbean and the East-Southeast Asia-Oceania components.

Topological Properties of the World Air Transportation
Tis section investigates the topological properties of the world air transportation network.We also perform a comparative analysis with the large components.7.2.Degree-Degree Correlation.Figure 9(b) illustrates the evolution of the degree-degree correlation as a function of degree k for the weighted world transportation network.For both weighted and unweighted networks, one can see that the distribution of the points is not monotone.Tus, one cannot conclude about the degree-degree correlation of nodes.Tis result difers from previous results showing an assortative behavior for the weighted world air transportation network [23].Te Africa-Middle East-Southern Asia component is the only component with similar behavior.Indeed, all the others exhibit disassortative behavior.

Strength Distribution.
Like the components, the lognormal distribution best fts the strength distribution represented in Figure 10.In addition, the strength as a function of degree exponent (s(k) � k β with β ≈ 2.177) is in the range of the local components.

Centrality Analysis
Centrality analysis investigates the most infuential nodes in a network.Tere are multiple defnitions of centrality that exploit either local or global characteristics of the networks [33,34].Here, we perform a comparative analysis of the strength (number of fights in an airport) and degree (number of routes in an airport) centralities of the various components.Tis analysis is in line with recent works considering the community structure to defne new centrality measures [35].
Figure 6: Average degree-degree correlation versus degree of the large components.Te blue dots represent the average degree-degree correlation for the unweighted network.Te red dots represent the average degree-degree correlation for the weighted network.Te weighted degree-degree correlation is more signifcant than the unweighted degree-degree correlation.A high fraction of trafc is concentrated between major hubs in the local components.Tis concentration of trafc at these hubs is essential for optimizing the overall efciency and connectivity of the air transportation network.
Complexity the number of fights and destinations.Dallas Fort Worth Airport, the second most connected airport, ranks fourth according to the number of fights.All the airports mentioned above have more than 100 connections and 100000 fights within this component.Tis is not the case for the ffth important airport, the Ronald Reagan Washington Airport.It is a national airport in the capital city of the United States.It is a hub for American Airlines and receives several million passengers.Altogether, these airports handle almost 15% of the regional fights.London Heathrow ranks ninth.It operates around 74000 fights.Denver and Houston airports which are in the top fve hubs in terms of connections rank, respectively, eleventh and sixteenth [8] when considering the number of fights.
In the Europe-Russia-Central Asia component, four out of fve top airports are in Europe and one in the United States.Dublin Airport in Ireland is the frst with more than 4500 fights.It is the central hub of the important low-cost carrier, Aer Lingus Airlines.Very connected, the Barcelona Airport is the second with numerous fights.Indeed, it is a tourist city receiving several million passengers

Complexity 13
Europe.Even though it has a higher number of links in this component, Amsterdam Schiphol Airport ranks ffth.It is the principal airport of the Netherlands.Te Munich Airport and the London Stansted Airport, which are in the fve highest hubs in terms of routes [8], rank, respectively, sixth and sixty-ffth in terms of fights.Te Frankfurt Airport, in the same group as the two mentioned above, is now in Africa-Middle East-Southern Asia.Te most vital airport in Russia, the Pulkovo Airport, is the fourteenth.Te Dubai Airport, which belongs to this component, ranks twentyffth, although it is the biggest airport in the Middle East.
In the East and Southeast Asia-Oceania component, three out of fve most busy airports are in China.Te others are in Singapore and Hong Kong.Te top airport is Beijing Capital Airport, China's capital city.It is the only one to have more than 40000 fights.Singapore Changi Airport ranks second, even if it is not one of the fve most connected airports in this component.It is the primary hub of Singapore Airlines.Te third airport is the Shanghai Pudong Airport in Shanghai, a populated city in China.Guangzhou Baiyun Airport is another central hub in China, ranking fourth.Indeed, this airport has the second-largest number of connections and carries millions of passengers.Te Hong Kong Airport, with less than 30000 fights, ranks ffth.Chengdu Shuangliu Airport and Taiwan Taoyuan Airport, in the top fve destinations, rank, respectively, eleventh and ninth [8].Te Sydney K Smith Airport, the most signifcant Oceania airport, ranks eighth.
Te Indian airports dominate the Africa-Middle East-Southern Asia component.Indeed, they are in the top three busiest airports.Te two others are in Saudi Arabia and Germany.With its numerous links, the Indira Gandhi Airport handles the highest trafc with more than 20000 fights.Te second airport with almost 17000 fights is the Chhatrapati Shivaji Airport, located in Bombay, a metropolis of India.One of the primary airports in India, the Kempegowda Airport serving Bangalore, ranks third.It is a hub of Air Asia.Te most connected airport, King Abdulaziz, ranks fourth.Te Frankfurt am Main Airport is ffth with 30 internal connections and more than 11000 fights in this component.Te most connected airport in the region, the Dubai Airport, is in the Europe-Russia-Central Asia component, and the ffth in terms of destinations, the Addis Ababa Bole Airport, ranks nineteenth in terms of fights.
In the South America component, among the top fve, four airports are in Brazil and one in Colombia.Tey have less than 8000 fights.Tancredo Neves Airport is the frst.Situated in Belo Horizonte metropolitan area, it is a hub of Azul Brazilian Airlines.It does not have a high number of connections, but its trafc is very dense.Te second, Guarulhos G A F Montoro Airport, is the most crucial airport in Brazil.It serves the largest economic and tourist city, São Paulo.El Dorado Airport, located in the capital of Colombia, ranks third.Rio G-T Jobim Airport in Rio de Janeiro, one of the most populated    Te Frankfurt am Main Airport dominates the interregional fow of fights in the Europe-Russia-Central Asia region.Being the most important airport in Germany, it has around 100000 fights with 211 connections.Te Charles de Gaulle Airport ranks frst with almost 70000 fights.Located in the capital of France, it is the busiest airport in this country.Very connected, the London Heathrow Airport is a third of the list.It serves several fights to diferent regions of the world.Te busiest airport in the Netherlands, the Amsterdam Schiphol Airport ranks fourth.Te Munich Airport is the ffth airport with the largest inter-regional trafc.It is the second busiest airport in Germany.All these airports mentioned above are in the top 10 of the global component.One can say that the European airport leads the inter-regional fights.
Te top fve busiest inter-regional airports in the East and Southeast Asia-Oceania region do not include any airport in Oceania.Indeed, the Narita Airport in Japan's capital city is the busiest inter-regional airport with about 40000 fights.Te Beijing Airport ranks second.It is the most connected, with trafc comparable to the Narita Airport.Te Suvarnabhumi Airport, the most critical airport in Tailand, is the third, with almost 30000 fights.Te Incheon Airport, the most signifcant hub in South Korea, is the fourth.Its trafc is around 30000.Te ffth airport with the largest international trafc is the Hong Kong airport, with almost 26000 fights.Only Narita and Beijing airports fgure in the top 10 airports in the large global component.
While the Indian airports dominate the regional trafc, the busiest inter-regional airports of the Africa-Middle East-Southern Asia region are scattered in the Middle East.Te Dubai Airport is by far the most dynamic in this region.Its position and big area promote it.Located in the capital of Qatar, the Hamad airport ranks third.Te second-largest airport in the United Arab Emirates ranks fourth.Cairo Airport in Egypt is ffth in this ranking.It has a comparable number of fights to the third and fourth airports (between 12000 and 14000).Te frst inter-regional airport in this region is not in the top 10 of the large global component.It ranks nineteenth.
Te inter-regional airports handling the highest trafc in the South America region are in diferent countries.Te Guarulhos Airport in Brazil is the busiest.El Dorado Airport in Colombia follows.Tese two airports are also regional hubs.Te Ministro Pistarini Airport, located in the capital of Argentina, ranks third.Te regional hub and the second most important in Brazil, the Rio G-T Jobim Airport, is the fourth inter-regional airport in this area.No airport in this component is in the top 20 of the large global component.Indeed, the Guarulhos Airport ranks twenty-fourth.

Comparison with the World Transportation Network.
Table 9 lists the 25 busiest airports in the world regarding the number of fights.It also includes their local and global strength rank and their degree rank.It shows that the North and Central America-Caribbean region controls a big part of the world's trafc.Indeed, 19 are in this area.Five are in the Europe-Russia-Central Asia region, and one is in the East and Southeast Asia-Oceania region.If we compare the airports' strengths with their degree, one detects that European airports usually deserve more destinations worldwide while North American airports have more fights.
Although 19 airports in the North and Central America-Caribbean area are in the top 25 worldwide airports for their trafc, only six are in the top 25 inter-regional airports.In contrast, they are all in the top 25 regional airports.Tese results demonstrate that the North and Central America-Caribbean region focuses more on regional trafc.Te airports of New York (John F Kennedy and Newark airports) rank, respectively, 3 and 6 for inter-regional trafc.Terefore, New York City is the US gateway.Los Angeles is the busiest airport in the world, but the trafc is well distributed between local and global destinations.Chicago O'Hare and Hartsfeld J Atlanta are in the top 5 airports in the world, but it is mainly due to their regional position.Note that the rank 16 Complexity An air transportation network's strength centrality is ranked by its worldwide strength rank.It is computed using the total strength (internal and external).Local rank is the strength centrality rank in the large local components.Internal strength is used to calculate it.Global rank is the strength centrality rank in the global component.External strength is used to calculate it.No indication is given when an airport does not belong to the global component.Top inter-regional airports are in bold.Top regional and inter-regional airports are in italics.Others are top regional airports.
18 Complexity of the regional airports in the world air transportation network can be misleading because they are not the most active in inter-regional trafc.London Heathrow Airport ranks third worldwide.It is mainly due to its position in the inter-regional trafc.In contrast, Charles de Gaulle Airport, Frankfurt am Main Airport, Amsterdam Schiphol Airport, and Munich Airport rank 6, 7, 12, and 21 in the top 25 airports of the world.Tey are all in the top 25 regional and interregional airports.However, they generally exert a stronger infuence at the regional level.Note that London Heathrow belongs to the American-Caribbean component.Its trafc is very dense compared with the airport in this component.One can make a similar remark about the Frankfurt am Main Airport, which belongs to the Africa-Middle East-Southern Asia component.
Te Beijing Airport is the only one from the East and Southeast Asia region in the top 25 crucial airports.It is the eighteenth worldwide according to the number of fights.In addition, it is the frst regional airport and the most connected to the world.
Dubai Airport is the forty-ffth in the world, the frst from Africa-Middle East-Southern Asia, even though it deserves many routes.Unexpectedly, it is the twenty-ffth infuential regional airport of the Europe-Russia-Central Asia component and the nineteenth inter-regional airport.Te most infuential airport in South America, the Guarulhos-Governador André F M Airport in São Paulo, Brazil, is the seventy-ffth airport in the world in terms of trafc.In comparison, it is the frst in the South America component and the twenty-fourth important inter-regional airport.
To summarize, North America leads the trafc in the world air transportation network.Te component structure shows that most of this trafc is regional.Indeed, the large global component, which captures the inter-regional fights, exhibits numerous airports from diferent world areas essential to the inter-regional trafc.In addition, the large local components display the infuential regional airports hidden in the world air network.

RBO Analysis.
Te ranked-biased overlap (RBO) [36] quantifes the similarity of two ranking lists.A parameter enables the prioritization of higher ranks over lower ones and the extension of the evaluation's depth.Its value ranges between 0 and 1. Te higher its value, the more identical the lists.We compare strength with degree centrality of the airports of the large components [8].We tune the RBO to give equal importance to all ranks.Figure 11 shows the evolution of the RBO of the topranked airports for the large components and the world air transportation network.Te curves cover the range from top 5 to top 45 sampled with a step of 5.
One can distinguish two types of curves.In the frst category, the RBO increases monotonically as the number of airports increases.It includes the Europe-Russia-Central Asia, Africa-Middle East-Southern Asia, East-Southeast Asia-Oceania, and South America components.Flights in these components do not concentrate on the main hubs.Indeed, the top 5 degree and strength airports are diferent.In the second category, the RBO decreases to a minimum, and then it increases monotonically.North and Central America-Caribbean, East-Southeast Asia-Oceania, and the global component belong to this category.Teir top 5 airports' ranking according to degree or strength is highly similar.Tere are more diferences between the top 10 and 15 airports, so the RBO decreases.Beyond this value, the strength and degree rankings become more homogeneous, so the RBO increases.Looking at the top 45 airports, one can rank the components according to the similarity of strength and degree rankings.East-Southeast Asia-Oceania is the component where trafc (strength) and hub size (degree) are more similar.Te Europe-Russia-Central Asia component exhibits the most diferent ranking.So, in the former, trafc is generated by hubs, while in the latter, airports with few connections can handle a high share of trafc.
Te RBO between top strength and degree airports increases monotonically in the world air transportation network.Diferences are more pronounced compared to the components.Indeed, the top 5 airports based on degree are concentrated in Europe, while the top 5 based on strength are in the United States.Tis diference illustrates the diferent focus in the two regions.Indeed, in the USA, airlines focus on regional trafc, while in Europe, serving many international destinations is a must for companies.

S-Core Analysis
Te s-core [37,38] analysis is the generalization of the kcore [39] analysis to weighted graphs.Te k-core [40] of a graph is the subgraph obtained by recursively removing all the vertices of degree smaller than k until the degree of all remaining vertices is larger than or equal to k.By extension, the s-core of a graph is a subnetwork in which a node has at least a strength s.One can extract the maximum s-core by removing nodes iteratively from the network.Indeed, the s min(si) -core, where each node has at least a strength 1, is the whole network.One forms the next level, by removing all the nodes with the minimum strength s min(si+1) .Te remaining nodes form s min(si+1)core, and so on until one reaches the core number max s n -core for which it is impossible to obtain the s min(sn+1)core.
Tis section reports the max s-core analysis of the large weighted components and a comparative investigation with their corresponding max k-core.Additionally, it presents a similar analysis of the world transportation network.9.1.Regional Analysis.Te max s-core contains almost half as many airports as the max k-core.However, they share 22 airports (84.6% of the max s-core airports).Te trafc and destinations of these airports are therefore closely related.London Heathrow in the United Kingdom, Portland and Windsor Locks in the United States, and Montreal/Pierre Elliott Trudeau have more trafc than destinations with the other airports of the max s-core.
Te max s-core of the Europe-Russia-Central Asia component has 27 airports, listed in Table 11.Covering several countries in Europe, they are generally in the capital cities.Pulkovo Airport is the only one in the Russian subregion.Ben Gurion Airport has the lowest trafc with 10747 fights.John F Kennedy in the USA has the highest number of fights.Indeed, it is the main connection between Europe and the USA.Charles de Gaulle in France, Munich in Germany, Amsterdam in the Netherlands, and Zurich Airport in Switzerland are not in the max s-core.Indeed, the trafc of these airports is more directed towards interregional destinations.Five airports in this max s-core are not in the 27 top strength airports.Tese airports are Venice, Marco Polo in Italy, Eleftherios Venizelos in Greece, Budapest Ferenc Liszt in Hungary, Ben Gurion in Israel, and Helsinki Vantaa Airport in Finland.
Te max s-core contains three times fewer airports than the max k-core.All airports in the max s-core are in the max k-core.John F Kennedy Airport in the USA has a lot of trafc and connections to airports in Europe.Otherwise, no country dominates the max s-core, while the max k-core airports are mainly in Germany, the United Kingdom, France, Italy, and Spain.
Table 12 contains the 18 airports of the max s-core of the East and Southeast Asia-Oceania component.Tese airports are in diferent countries, mainly in capitals and megacities.Only four are in Oceania.Kansai Airport in Japan, with its 7687 fights, has the lowest trafc.Note that the airport of Beijing, with the most signifcant trafc of this component, is the third airport with the weakest trafc in this max s-core.Indeed, it has many fights with low-strength airports.Tere are fve airports (Brisbane Airport in Australia, Auckland Airport in New Zealand, Tan Son Nhat Airport in Vietnam, Ngurah Rai Airport in Indonesia, and Kansai Airport in Japan) in the max s-core that are absent from the 18 top strength airports.Te max s-core contains half as many airports as the k-core.Nevertheless, China dominates the max k-core, and it is not the case for the max s-core, which includes airports from Oceania.
Table 13 lists the nine airports in max s-core of the Africa-Middle East-Southern Asia component.Tey are in three countries (5 in India, 2 in Saudi Arabia, and 1 in Germany).Indeed, there is a lot of trafc between Saudi Arabia and India because these many Indians work in Saudi Arabia.With 4609 fights, Rajiv Gandhi International Airport is the airport with the lowest trafc in the max s-core.Tese airports form a complete graph.Tey concentrate a large part of the trafc of the component.Indeed, except for Rajiv Gandhi Airport, they are all in the top 9 strength airports.Te max k-core is twice larger, and it includes the airports of the max s-core.Tus, Frankfurt Airport is very connected in terms of trafc and destination with the hubs of this component.Nevertheless, India dominates the max score and the max k-core.
Table 14 gives the ten airports in the max s-core of the South America component.Tey are all in Brazil and mainly on the east coast.Rio Galeão Tom Jobim Airport has the fewest number of fights.Only two airports, Santa Genoveva Airport in Goiania and the Deputado L E Magalhães Airport, are not present in the top 10 strength airports.All the airports in the max s-core are in the max k-core, which contains almost twice as many airports.Furthermore, Brazil dominates the max s-core as it does with the max k-core.
To summarize, we observe two typical behaviors for the max s-core.In the frst one, the airports forming the max score are mainly in a single country.Indeed, the United States, India, and Brazil dominate the max s-core of their component.In the second case, the airports in the max score are more evenly distributed in the component.It is the case for the East and Southeast Asia-Oceania and Europe-Russia-Central Asia components.
Te max s-core always contains far fewer airports than the max k-core, typically one-half to one-third.In addition, most airports in the max s-core are in the max k-core.Tus, airports with high trafc tend to have also many connections.9.2.Inter-Regional Analysis.Figure 13(a) shows the max score of the large global component, and Table 15 lists its 22 airports.Tese airports are in North America, Europe, and East and Southeast Asia.Tey are located in capital cities and are the main airports of their country.Suvarnabhumi Airport in Tailand has the lowest trafc with 13196 fights among the max s-core airports.Tree US airports in the max s-core are not in the top 22 strength airports (General Edward Lawrence Logan Airport, Dallas Fort Worth Airport, and Seattle Tacoma Airport).Tus, the airports that concentrate the majority of inter-regional trafc share many fights.Te max s-core and the max k-core have comparable sizes.However, the max k-core includes more countries.Tree airports (General E L Logan Airport, Dallas Fort Worth Airport, and Seattle Tacoma Airport) are in the max s-core and not in the max k-core.Tese airports have many Complexity fights and a moderate number of destinations compared with the other airports in the max s-core.16 in the max score of the North and Central America-Caribbean region.Minneapolis-St Paul Airport, with its 24088 fights, has the lowest trafc.Seven airports of the max s-core of the world air network are not in the top 33 strength airports.Te max s-core global air network has half as many airports as the max k-core.Moreover, the airports in the max s-core are mostly absent in the max k-core.Indeed, the max k-core is more concentrated in Europe.Tese results confrm the orientation for high trafc in the USA and high number of destinations in Europe.Overall, with their high trafc, North American airports lead the weighted world air transportation network, while European airports dominate the unweighted world air transportation network with their high number of destinations.Te component structure eliminates this disparity and reveals other important airports worldwide.

Discussion and Conclusion
Tis paper investigates the relationship between the weighted and unweighted worldwide air transport network and its impact on its component structure.Table 17 summarizes the main fndings.Overall, the weighted network contains fewer components.In both cases, the large local components cover distinct geographical areas.However, their geographical repartition difers slightly.Indeed, the weighted network has fve large local components, while the  Airports not in the top 22 inter-regional airports with the highest strength are in bold.Te core strength of a node is its number of links in the maximum s-core.Global rank is the rank in the largest global component.Airports are ranked from largest to smallest strength.
24 Complexity unweighted network contains 7. Tree components are quite similar (North and Central America-Caribbean, Africa-Middle East-Southern Asia, and South America).Te two other components (Europe-Russia-Central Asia and East-Southeast Asia-Oceania) group neighbor components of the unweighted network.Russia-Central Asia is attached to Europe, while Oceania joins East-Southeast Asia.Tis is due to the substantial trafc between these regions linked to their economic integration.One can observe the same type of phenomenon more locally.Indeed, some major airports integrate regions with which they share a high proportion of their fights.For example, John F Kennedy Airport in the United States belongs to the Europe-Russia-Central Asia component, London Heathrow is in the North and Central America-Caribbean component, and Frankfurt Airport in Europe is in the Africa-Middle East-South Asia component.
Although the weighted network has more global components than the unweighted, there is also a single large global component in both cases.Although they are of comparable size (around 20% of the world's airports), their content is quite diferent.Indeed, when components merge, many airports that were linking them disappear from the global component.Additionally, new airports emerge when an airport does not belong to its natural geographical area.For example, most of the airports linked to John F Kennedy located outside the Europe-Russia-Central Asia component integrate the global component.
Analysis of the average weighted clustering coefcient reveals that it is independent of the degree for the large local components.Furthermore, their values are always lower than their unweighted equivalent.Low-trafc airports form triplets in the components and the global air network.In contrast, the average unweighted clustering coefcient decreases monotonically with the degree.Indeed, low-degree nodes tend to have a higher clustering coefcient than hubs.It refects the lack of adequacy of the hub and spoke confguration for rerouting passengers when necessary.Te Africa-Middle East-Southern Asia and the South America components have fewer interconnected hubs than the other local components.Consequently, their confguration is less hub and spoke.Te global component exhibits similar behavior.However, the hub and spoke efect is more pronounced.Indeed, as it contains more hubs, it also has fewer triplets accentuating the rerouting inefciency of the component.
According to the evolution of the average weighted degree-degree correlation as a function of degree, one can classify the components into two categories.Te frst category includes the Africa-Middle East-Southern Asia Component.In that case, the degree of the nodes does not exert a signifcant infuence.Te second category contains the four other local components and the global component.Te weighted degree-degree correlation tends to decrease in these components as the degree increases.Te high-degree nodes link more and more with small-strength nodes.In other words, as the degree increases, these components' hub and spoke confguration is more pronounced.Africa-Middle East-Southern Asia component departs from this behavior because air transportation is less mature in this region.Te world air transportation network exhibits similar behavior to the Africa-Middle East-Southern Asia component.Te same observations are also valid for the unweighted network suggesting that weights are not essential for this property.Additionally, for all the local components and the global air network, it appears that K w (i) ≥ K(i).Consequently, airports with high degrees manage a higher number of fights.
Te strength of the components follows a log-normal distribution with heavy-tailed characteristics.Te strength as a function of degree shows that numerous hubs in the Europe-Russia-Central Asia component manage a comparable number of fights.Some hubs have a lot of trafc in the North-Central America-Caribbean and East-Southeast Asia-Oceania components.Compared to the other components, Africa-Middle East-Southern Asia and South America have fewer hubs and less trafc.However, these hubs handle a high fraction of fights.Te strength distribution of the global component and the world air transportation network is also log-normal.Moreover, their hubs also manage most of the trafc.
Te centrality analysis shows that the top fve highstrength airports are usually in the leading countries of their region (the USA in North and Central America-Caribbean, China in East and Southeast Asia-Oceania, India in Africa-Middle East-Southern Asia, and Brazil in South America).It is more homogeneous in the Europe-Russia-Central Asia component.Furthermore, airports included in components far from their geographical areas are in the top fve high-strength airports.Te top high-degree airports serve higher number of fights in the North and Central America-Caribbean and the East and Southeast Asia components.It is not the case for the other local components.
Te top high-strength inter-regional airports are in several countries except for the USA, which dominates the inter-regional trafc in the North and Central America-Caribbean area.Te top fve strength inter-regional airports difer from the leading regional airports.It shows that airports are more or less specialized in regional or interregional destinations.Furthermore, with a high fraction of the top inter-regional airports, Europe leads the interregional trafc.In addition, the top high-degree airports and high-strength airports are similar.Centrality analysis of the world air transportation network shows that North American airports dominate the trafc while Europe-Russia-Central Asia dominates the destinations.
Two categories appear in the max s-core analysis of the local components.Te frst includes the Europe-Russia-Central Asia and the East and Southeast Asia components, with the maximum s-core covering several countries.In the second category, a few countries (one to three) concentrate on the high trafc of the component.Te max s-core of the local components contains far fewer airports (half or one-third) than the max k-core.Tus, few airports share numerous fights among the airports in the local components with several routes.Te maximum s-core of the global component covers several countries from the North and Central America-Caribbean, Europe-Russia-Central Asia, and the East and Southeast Asia-Oceania regions.Tese three regions serve the most inter-regional fights.In contrast to the local components, the maximum s-core and k-core of the global component have comparable sizes.Consequently, the inter-regional hubs of these components concentrate several numbers of fights.
Te max s-core of the global air network is mainly located in the USA, while the max k-core airports are in Europe.Ten, the most important airports in the s-core and the k-core are blurred when considering the global air network instead of the component structure.
Tis comparative analysis illustrates the essential contribution of the component structure representation for uncovering the regional and inter-regional similarities and diferences of the world air transportation network.Indeed, typical network properties have been designed for networks with a homogeneous density.Because of their local density variations, some characteristics can be blurred, hence the importance of decoupling local from global analysis.Tis representation opens multiple research directions.In future work, we plan to exploit the component structure to gain a better understanding of the robustness of the air transportation network against targeted attacks.Indeed, one can design tailored attacks on the components and inspect their global, regional, or inter-regional impact.

Figure 1 :
Figure 1: (a) Te communities the Louvain community detection algorithm identifes.It includes eighteen communities.(b) Te communities uncovered by the Combo community detection algorithm.It contains seven communities.For both, each color represents a community.Similar communities have similar colors.One can observe that the Combo algorithm regroups the communities of the Louvain algorithm located in Canada and Alaska.

Figure 2 :
Figure 2: (a) Te airports in the large components of the weighted network.(b) Te airports in the large components of the unweighted network.Each color is associated with a component.Te North and Central America-Caribbean component is red(1).Te Africa-Middle East-India component is green(2).Te South American component is brown(3).Te Europe-Russia-Central Asia component is black(4).Te Europe-Russia-Central Asia component is black(4).It is divided into Europe in black (4) and Russia in purple(7) in the unweighted network.Te East and South-East Asia-Oceania component (5) is divided into East and South-East Asia in blue(5) and Oceania in orange(6) in the unweighted network.
Figure 4  shows the global components extracted from the weighted world air transportation network.Tere are one large and 11 small global components.Te large global component contains 557 airports (20.44% of the world's airports).It covers the world.Te small global components, which include 36 airports, are principally in the North and Central America-Caribbean region.
Figure 5 represents the average (weighted and unweighted) clustering coefcient versus degree for the large local components.

Figure 3 :
Figure 3: Small local components in the weighted (a) and unweighted (b) world air transportation network.Te red circles are the components disappearing in the weighted network.Te blue circles are similar small components in both networks.Te green circles are the components that do not appear in the unweighted network.Te orange circles are the components splitting in the weighted network.Geographical areas outside the fgure do not contain small components.

Figure 5 :
Figure 5: Average clustering coefcient versus degree for the large components.Te blue dots represent the average clustering coefcient for the unweighted network.Te red dots represent the average clustering coefcient for the weighted network.Te blue curve is the estimated k −β function.Weighted average clustering coefcients are lower than their unweighted equivalent.Te interconnected triplets tend to be formed by low-weight edges in the large local components.

Figure 4 :
Figure 4: (a) Te airports in the large global component.Tey are distributed all over the world.(b) Te 11 small global components are circled.Teir size ranges between 2 and 3 airports.Most of them are located in North America.

7. 1 .
Clustering Coefcient.Figure 9(a) presents the average clustering coefcient as a function of degree k for the weighted and unweighted world air transportation network.Similar to the large components, the unweighted clustering 10 Complexity coefcient decreases monotonically as the degree increases.Te function k −c (with c ≈ 0.27) gives a good approximation of their relation.Te exponent value estimated is identical to the Europe-Russia-Central Asia component.Te weighted average clustering is very low and almost independent of the degree.As it is always below the unweighted average clustering coefcient, one can conclude that edges with low degrees tend to constitute the triplets.

Figure 7 :
Figure 7: Strength distributions of the large local components and the largest global component.Dots denote empirical distribution, and lines denote estimates.Te distributions under test include power law (PL), truncated power law (TPL), log-normal (LN), and stretched exponential (S-EXP).Te values in bold are the best-ft parameters according to the Kolmogorov-Smirnov test.Te large local components have heavy-tailed characteristics.

Figure 8 :
Figure 8: Strength as a function of degree of the large local components and the largest global component.Dots denote empirical distribution, and lines denote estimates.Te more links an airport has, the more trafc it receives.

Figure 9 :
Figure 9: (a) Te average clustering coefcient as a function of k.Te exponent value estimated is identical to the Europe-Russia-Central Asia component.(b) Te average degree correlation of degree k.For both fgures, the red points are the values of the weighted world transportation network, while the blue points are the unweighted world transportation network values.Te distribution of the points is not monotone.Tis behavior is similar to the one of the Africa-Middle East-Southern Asia component.

Figure 10 :
Figure 10: Te world air transportation network's strength distribution (a).Dots denote empirical distribution, and lines denote estimates.Te distributions under test include power law (PL), truncated power law (TPL), log-normal (LN), and stretched exponential (S-EXP).Te parameters that best ft the data, as determined by the Kolmogorov-Smirnov test, are shown in bold.Te strength as a function of the degree (b) of the large global component.Te log-normal distribution is the best ft for the strength distribution.

14 Complexity Table 7 :5
Te internal strength of an airport is determined by the number of fights within its dominant local component.Te local rank of an airport is determined by its internal strength, ranked in descending order within its local component.Te cumulative fraction of connected airports indicates the proportion of airports that the top x local hubs can access within their respective local components.

Table 8 : 9 :
Strength centrality in the large global component.the top fve inter-regional hubs in each region.An airport's external strength is the number of connections it has with airports in the large global component.Te global rank in a region is determined by decreasing external strength in that region.Airports' rank in the global component is based on decreasing external strength.ComplexityTable Te 25 largest nodes in the air transportation network are ranked descendingly according to their strength centrality.

Figure 11 :Figure 12 :
Figure 11: Te RBO of the large components (a) and the world air transportation network (b).Te top 50 hubs of the unweighted (degree) and weighted (strength) are compared by step of 5. Flights in the Europe-Russia Central Asia, Africa-Middle East-Southern Asia, East-Southeast Asia-Oceania, and South America components do not concentrate on the main hubs.In the other large components, the hubs accumulate the trafc.

Figure 13 (
b) represents the max s-core of the global air transportation network.With 27 airports, North America dominates.Tere are also fve airports in Europe and one in Japan.Tere are 11 airports listed in Table

Figure 13 :
Figure 13: (a) Te 33 airports in the maximum s-core of the weighted world air transportation network.(b) Te 22 airports in the maximum s-core of the weighted large global component.S is the maximum s-core value, and K is the maximum k-core value.Blue points represent the airports belonging to the maximum s-core and the maximum k-core, the red points are the airports exclusively in the maximum s-core, and the yellow points are the airports belonging only to the maximum k-core.

Table 1 :
Basic topological properties of the world air transportation network. is the network size.|E| is the number of edges.d is the diameter.L is the average shortest path length.µ is the density.ζ and ζ w are, respectively, the unweighted and weighted average clustering coefcients.λ and λ w represent, respectively, the unweighted and weighted assortativity, also called the degree-degree correlation coefcient.η is the hub dominance. N

Table 2 :
Modularity, mixing parameter, and NMI of the community structures discovered by the community detection algorithms of Louvain and Combo.Middle East-Southern Asia component is also high (0.87).Te 35 airports of the unweighted component that disappear in the weighted are in the Middle East and West Africa.In contrast, eleven new airports emerged (6 are in Kenya, 2 in West Africa, and 3 in Europe) in the weighted component.Te Frankfurt am Main Airport, the largest airport in Germany, is unexpected in this component.Indeed, it has high trafc with Saudi Arabia and India.Te Rzeszów-Jasionka airport in Poland and Araxos airport in Greece are also in this component.Both airports have dense trafc with Frankfurt am Main.
in Canada, Alaska, Peru, and Chile.London Heathrow, the most important airport in the United Kingdom, also appears in the weighted component.Indeed, it has several destinations in the United States, and numerous fights from North America land at this airport.Te most unexpected airports in this component are the Marshall Airport in the Marshall Islands and the Osmani Airport in Bangladesh.Te frst collaborates with United Airlines situated in the United States.Te second has fights to and from London Heathrow.Te Jaccard index of the unweighted (336 airports) and weighted (313 airports) Africa-

Table 3 :
Te Jaccard index of the fve similar communities uncovered by Louvain and Combo.

Table 4 :
Te Jaccard index of the communities uncovered by the weighted and unweighted Louvain algorithm.
Tree are similar (North and Central America-Caribbean, Africa-Middle East-Southern Asia, and South America), and two are regrouped by the weighted algorithm.Te East and Southeast Asia-Oceania regroups the East and Southeast Asia component and the Oceania component.Europe-Russia-Central Asia regroups the Europe component and the Russia-Central Asia component.

Table 5 :
Te Jaccard index of the European-Russia component is 0.87.Nine airports present in the unweighted disappear, while 87 new airports appear in the weighted component.Tey are mainly in Norway, West and North Africa, Iran, the United Arab Emirates, and the French Antilles.Beyond their geographical localization, the countries of this weighted component have political, historical, and economic relations.Indeed, from the airports in Moscow and St. Petersburg, the Russian airports join the other world regions throughout Europe.North Africa, West Africa, and the French Antilles are associated with Europe politically and historically.It translates into high trafc between these regions.
Quality metrics of the community structures uncovered by the unweighted and weighted Louvain community detection algorithms: modularity, mixing parameter, and NMI.
Table 7 reports the top fve airports in descending order of the number of fights with airports in their local component (internal strength centrality).Te top fve airports in the North and Central America-Caribbean component are in the United States.Tey are in densely populated states such as Georgia, Illinois, Texas, California, and Washington D.C. Hartsfeld J Atlanta Airport captures the higher number of fights.It is also the most connected.It is one of the densest in terms of passengers.Located in the second most populous city in the United States, the Los Angeles Airport is the second busiest in terms of fights, although it has a low ranking in terms of the number of routes.Chicago O'Hare Airport ranks third in

Table 6 :
KS test for the strength distribution.
Distributions under test are power law, truncated power law, log-normal, and stretched exponential.Te smallest value (bold) corresponds to the best ft. 12 Complexity per year.Te largest hub, Charles de Gaulle Airport in the capital city of France, ranks third with considerable trafc.Unexpectedly, John F Kennedy Airport in the United States ranks fourth with 40 links within this component.Tis airport has more fights to Europe than several local airports.Indeed, it is the main gate between the United States and Complexitycities in Brazil, is fourth.It is less connected among the top fve.Te ffth airport, the Presidente J Kubitschek Airport, is located in the capital of Brazil.8.1.2.Inter-Regional Analysis.Table 8 lists the top fve airports participating in the inter-regional trafc in each large local component.Tese airports also belong to the large global component.Four of the fve most essential airports for interregional trafc in the North and Central American-Caribbean region are in the USA.Te other one is in Canada.Te frst two airports in this ranking are the John F Kennedy Airport and the Newark Liberty Airport.With more connections (120), John F Kennedy airport's (nearly 100000) fight fow is much denser than Newark Liberty airport's fight fow (39 fights and almost 70000 fights).Tese airports are located in New York state.Indeed, this state, located in the Northeastern United States, is one of the most important ones, with the largest number of residents.Moreover, it is near other regions and includes numerous international institutions and multinational corporations.Te Chicago O'Hare Airport and the Los Angeles Airport rank third and fourth.Tey are in other populated cities, Chicago and Los Angeles, in the United States.Tese two airports have comparable degrees and trafc.In Canada, the busiest airport, Lester B. Pearson Airport, ranks ffth.It is a gateway to this country and the USA.Te top three in this region are in the top 10 of the large global component.
20 ComplexityTe max s-core of the North and Central America-Caribbean component contains 26 airports.Fort Lauderdale Airport, with 19801 fights, has the lowest trafc.Table10lists these airports.Only three are outside the United States: (1) Montreal/P E Trudeau Airport in Montreal, the busiest airport in Canada; (2) Cancun Airport, the second busiest in Mexico; and (3) London Heathrow Airport in the United Kingdom.Teir trafc is so high with the US that they can be considered part of the country.Most of the airports are on the coast of the United States.None deserves New York.Indeed, its airports' main trafc is with other world regions.Four airports (Raleigh-Durham, Pittsburgh, Windsor Locks, and Cancun) in the max s-core are not among the top 26 airports in terms of trafc.Tese airports primarily serve other airports in the max s-core.

Table 15 :
22 max s-core airports of the global component.