Keywords

1 Introduction

In present time, one of the essential transport systems is the air transportation system. And, it is one of the inseparable elements of our present societies for their high level of mobility. So the study of air transportation network plays a vital role for proper maintenance of the system. Many studies have been done to analyze various countries air transportation network such as US [10], China [11], India [2], Brazil [5] etc. The studies have been done to analyze the infrastructure, connectivity, flow of traffic etc. of the network. As the dependency on these networks is increased, so the study of the robustness of the network becomes essential. Robustness is the ability of a network to continue to perform properly when it is subject to failures or attacks [7]. Robustness is one of the most anticipated properties of any transportation network. The level of vulnerability to which a network can expose through random and most central nodes or/and link failures, can be evaluated by studying the robustness of the network during different node or edge failures [1]. The robustness analysis of an airline network shows how vulnerable and fragile the network is when the network facing unintentional errors (random attack) and intentional attacks [12].

In this paper a comparison of three major airline networks of India viz. Indigo, Jet airways, Air India, with ANI based on network parameters is analysed. Firstly, the paper provides an analysis of robustness of the airlines. The daily networks of four different European airlines, during the busiest week, were analyzed by Han et al. [8]. And they have found all airline networks have scale-free and small-world properties [8]. Also Reggiani et al. in 2009 [13] and in 2010 [14] investigated the Lufthansa network (LH) as both the worldwide and European networks; and they also found the networks were scale-free in both the cases. Again, multiple airlines in different alliances and parts of the whole world were studied by Lordan (2014) [12]. In 2015, Wijdeveld studied the robustness of 17 European airline networks for both error or random failure and targeted attacks [16].

The remaining parts of this paper is structured as follows: next section includes some parameters used in complex network analysis followed by an investigation on Airport Network of India (ANI) and three airline networks and then calculation of the values of various metrics of these networks is done. In the next section, the robustness analysis of the three airline networks have been reported, followed by their comparison with ANI. Conclusions are presented in the last section.

2 Some Parameters Used in Complex Network Analysis

Now, some definitions of some network parameters, which are frequently used in complex network analysis, are discussed (Table 1).

Table 1. Some network parameters
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3 An Investigation on ANI and Three Airlines Networks

In ANI and airline networks, nodes are the domestic airports and there exists a link between any two nodes (airports) if there is at least one direct flight connecting them per week. The number of such flight per week between the airports (nodes) is taken as the weight of the link in the network. Some dummy links are also added to the networks to make the networks symmetric. For the analysis of airline networks of India, three major airlines, viz. Indigo, Jet Airways and Air India are considered. All the air movement data for the networks are consider for the year 2016, obtained from Airports Authority of India (www.aai.aero). The values of the network parameters of ANI [3] and the three airline networks under consideration are given in Table 2 (Figs. 1 and 2).

Table 2. Network parameters of ANI and three airlines
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Fig. 1.
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Structure of ANI

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Fig. 2.
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Structure of the three airlines

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4 Robustness Analysis of the Airline Networks

Robustness analysis of the networks can evaluate the effect of targeted attack (e.g. attack by terrorists on an airport) and random attack or failures (e.g. inclement weather) of the networks. In the analysis for targeted attack, we remove five highest degree nodes (airports) from each airline networks and observe the change in the average path length, clustering coefficient, network criticality, Reachability and to study the effect of random failures, we remove any five randomly selected airports from each airline networks and calculate the network parameters. Table 3 shows the change of network parameters (in percentage) of the three airline networks after removal of key node based on degree. The negative sign indicates decrement of the parameter whereas positive sign indicates the increment of the values. We also calculate these measures after removal of the nodes sequentially, based on relatively high degree or high betweenness i.e., in order to evaluate the robustness of the airlines networks, each time a node with high degree or betweenness is isolated (removed), the centrality measures are calculated again for all the remaining connected nodes (airports). Then from the remaining nodes, the airport with highest centrality is selected for removal in the next step and so on. The process will continue until removal of five airports. Figures 3 and 4 show the change of APL and NCC after consecutive removal of five nodes (airports) based on degree and betweenness and consecutive removal of five random nodes (airports) respectively.

Table 3. Change of network parameters after removal of a key node
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Fig. 3.
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Change of APL and NCC after consecutive removal of five nodes (airports) based on betweenness and degree.

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Fig. 4.
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Change of the APL and NCC after consecutive removal of random nodes (airports)

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5 Comparison of Airline Network with ANI

In this section, a comparative study is carried out between the three airline networks and ANI. From structural perspective, we calculate the Correlation coefficient between the network parameters of the three airline networks and ANI. From the Table 4, we observe that all the three airline networks are positively correlated but Indigo network is most strongly correlated with ANI. And from robustness perspective, a comparison is made based on network parameters of the three airline networks and ANI by evaluating the value of R2 (Coefficient of determination) for two cases. First one is evaluated for the values of the parameters after removal of one node and second one is for the values of the parameters after removal of five nodes. In case of removal of one node, Air India seems to have marginally better consensus with ANI than the other airline networks (see Fig. 5). However in case of multiple removals of nodes, we observe that Indigo network has sufficiently higher value of R2 than Jet Airways and Air India (see Fig. 6).

Table 4. Correlation between network parameters of ANI and three airline networks (Structural perspective)
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Fig. 5.
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Comparison based on network parameter after removal of one node from the networks

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Fig. 6.
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Comparison based on network parameters after removal of five nodes from the networks

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6 Conclusion

In this study, an analysis of the three major airline networks is carried out, which gives a fair idea about the whole ANI. It is expected that after removal of high degree nodes, average path lengths should increase and clustering coefficients should be decrease. But only Indigo network shows an increment in average path length after removal of nodes (airports) based on degree. It may be attributed to the fact that in case of Air India and Jet Airways most of the paths are not always routed through high degree nodes (like Delhi, Mumbai), rather these airlines provide direct flights between the not so high degree nodes (airports). Also from the comparison of network parameters of the three airline networks with ANI, we observe that from structural perspective Indigo network is strongly correlated with ANI. From robustness perspective, although for removal of only one node, Air India has shown a slightly better consensus with ANI, in case of multiple removals of nodes Indigo has again shown a way better correlation with ANI than rest of the airlines under consideration. Though in this study we consider the three most popular Airlines namely Indigo, Air India and Jet Airways, Jet Airways stops all its operations in India very recently. And it will be interesting to perform a similar study without Jet Airways and compare the results with this work. For further study, we can also consider some more network parameters for better correlation analysis.