## 3 Methodology
### Calculating Centrality Measures to Feature DEA
#### 3.3.2 Airline Analysis Using Centrality Network Metrics  
There are many different metrics that make it possible to uncover some interesting individual behaviors and global properties in a network. We claim that in the domain we consider (airlines), centrality metrics are necessary to shed light on the importance of the position of an airline in the network generated. In other words, centrality metrics attempt to measure how well placed an airline is in relation to the other companies in the airspace--in this case, European airspace. The concept of centrality encapsulates micro measures that allow us to compare nodes and to say something about how a given node relates to the overall network [57; 58]. Our hypothesis lies in finding out whether deep knowledge of the position of the airlines in the context of the overlying network generated from the relationships that emerge in the European airspace may bring about certain benefits when studying the efficiency. Let us now set out the different metrics we use to accomplish this task. **Degree centrality**. The simplest way of measuring the position of a node in a network (henceforth we use the terms 'node' and 'airline' synonymously when referring to an entity in the network) is to consider the so-called degree centrality. This represents the number of links that a node has. Formally:  
\[DC^{v_{i}}=\frac{d(v_{i})}{|V|-1}, \tag{2}\]  
where \(d(v_{i})\) denotes the degree of node \(v_{i}\) in the network. This metric shows how well a firm is connected in terms of direct links. We can also provide an average degree centrality value as follows:  
\[\overline{DC}=\frac{1}{n}\sum_{v_{i}}\frac{d(v_{i})}{|V|-1}. \tag{3}\]  
This value represents an average estimate for the number of relationships between the companies as a macro indicator of its presence in Europe. Although degree centrality may be an important factor when analyzing an airspace network, it misses certain aspects that should be considered. While it captures the geographical distribution of an airline perfectly, it does not reflect the strength of its position within the network--i.e., the importance of the firm relative to the whole airline industry. In any case, it may be that an airline has relatively few relationships but lies in a critical location within the network. In other words, different airlines might yield different performances depending on the airports they operate in, even if the number of links they have remain the same. We use eigenvector centrality to overcome this issue. **Eigenvector centrality. ** Eigenvector centrality (we also use eigencentrality interchangeably throughout the paper), proposed by [PERSON] [59], is able to represent the importance of a node in the network. Let \(EC(G)\) be the eigenvector centrality associated with a network: the crux is that the centrality of a node is proportional to the sum of the centrality of its neighbors. Formally:  
\[EC^{\mathbb{C}_{i}}=\frac{1}{\lambda}\sum_{j=1}^{n}A_{j,j}EC^{\mathbb{C}_{j}}, \tag{4}\]  
in which \(\lambda\) is a scalar called eigenvalue and \(A\) is an adjacency matrix representing the network. The Perron-Frobenius theorem from linear algebra guarantees that when we work with an undirected network, being a connected component--as occurs in our case--iterating over Equation (4) always converges on a fixed-point equivalent to:  
\[\lambda\;EC(G)=A^{T}\;EC(G). \tag{5}\]  
As our airlines' network is undirected, then \(A=A^{T}\), so resulting in:  
\[\lambda\;EC(G)=A\;EC(G). \tag{6}\]  
As a set of separate equations (with \(a_{ij}\) being elements of the adjacency matrix \(A\)), we obtain:  
\[\left( \begin{array}{cccc}a_{11}&a_{12}&\cdots&a_{1n}\\ a_{21}&a_{21}&\ldots&a_{2n}\\ \vdots&\vdots&&\vdots\\ a_{n1}&a_{n2}&\cdots&a_{nn}\end{array} \right)\left( \begin{array}{c}x_{1}\\ x_{2}\\ \vdots\\ x_{n}\end{array} \right)=\lambda\left( \begin{array}{c}x_{1}\\ x_{2}\\ \vdots\\ x_{n}\end{array} \right) \tag{7}\]  
\[