## 3 Methodology
### A Network-Based Approach to Refine DEA  
As mentioned, DEA proves to be a rather good method to assess the efficiency of organizations. In particular, DEA outputs an efficiency frontier that may be used to determine a reference for efficiency performance. However, this efficiency frontier usually contains more than one unique efficient organization. In this paper, we adhere to the approach by [PERSON] and [PERSON] [60] to better discriminate between the efficiency of the efficient organizations involved in the DEA process. This approach transforms DEA output results (organizations' efficiency) into a weighted directed network and then uses eigenvector centrality to rank the importance of each organization, so better distinguishing between efficient entities. Since we use a completely different domain of study, we have adapted [PERSON] and [PERSON]'s approach as follows in the next description.  
**Stage 1**. Run DEA with the different proposed models to obtain the efficiency of the studied organizations.  
**Stage 2.** For each model, benchmark the individual efficiencies of each airline with regards to the efficiency frontier (i.e., efficient airlines). This process outputs a lambda value of \(\lambda_{i,j}\) if organization \(i\) is an exemplar of organization \(j\), with \(\lambda_{i,j}\in[0,1]\). In other words, lambda values represent estimates to achieve target efficiency. A value of \(\lambda_{i,j}=0\) means airline i is not a model of reference for airline \(j\). If \(\lambda_{i,j}>0\), then there exists an endorsement relationship between airlines i and \(j\). Note that \(\lambda_{i,j}\) does not have to be confused with algebraic eigenvalues \(\lambda\) in Section 3.3.2. Since notation for both terms is accepted worldwide, we prefer not to change either of them.  
**Stage 3.** We aggregate the lambda values resulting from each model for which we ran DEA, so obtaining a lambda matrix as follows:  
\[\left(\begin{array}{cccc}\lambda_{11}&\lambda_{12}&\cdots&\lambda_{1m}\\ \lambda_{21}&\lambda_{22}&\cdots&\lambda_{2m}\\ \vdots&\vdots&\vdots&\vdots\\ \lambda_{n1}&\lambda_{n2}&\cdots&\lambda_{nm}\\ \text{Model 1}\end{array}\right)+\left(\begin{array}{cccc}\lambda_{11}& \lambda_{12}&\cdots&\lambda_{1m}\\ \lambda_{21}&\lambda_{22}&\cdots&\lambda_{2m}\\ \vdots&\vdots&\vdots&\vdots\\ \lambda_{n1}&\lambda_{n2}&\cdots&\lambda_{nm}\\ \text{Model 2}\end{array}\right)+\cdots+\left(\begin{array}{cccc}\lambda_{11}& \lambda_{12}&\cdots&\lambda_{1m}\\ \lambda_{21}&\lambda_{22}&\cdots&\lambda_{2m}\\ \vdots&\vdots&\vdots&\vdots\\ \lambda_{n1}&\lambda_{n2}&\cdots&\lambda_{nm}\\ \text{Model 1}\end{array}\right)=\left(\begin{array}{cccc}\sum\lambda_{11}& \sum\lambda_{12}&\cdots&\lambda_{1m}\\ \sum\lambda_{21}&\sum\lambda_{22}&\cdots&\lambda_{2m}\\ \vdots&\vdots&\vdots&\vdots\\ \lambda_{n1}&\lambda_{n2}&\cdots&\lambda_{nm}\\ \text{Model 1}\end{array}\right)\]  
**Stage 4.** We use the aggregated lambda values from the previous step to generate a network in the following way:  
Airlines are represented by network nodes.  
A link between two nodes \(i\) and \(j\), denoted as \((i,j)\), is created if there exists a lambda value \(\sum\lambda_{i,j}>0\), which is, in turn, the weight of the link between the two nodes.  
**Stage 5.** Calculate the eigenvector centrality for every node in the network. This step will return a ranking, from which a better discrimination of the efficiency of the airlines involved will be easily extracted.