## 5 Results  
### Topological Analysis of Madrid Metro Network  
Firstly, network parameters introduced in Section 2.1 were computed. Madrid metro network consists of 243 nodes connected by 280 edges. The diameter of the network is \(D=44\), therefore the longest distance between any par of nodes is 44, and the average path length is \(L_{avg}=14.682\), which means people can reach their destination by on average traveling 14-15 stations. The radius of the network is 22 and the centre is Principe Pio station, that means that the maximum distance from this station to any other on the network is minimal. Moreover, the average clustering coefficient is \(\widetilde{C}_{CLU}=0.008\). The most important global network indicators are summarized in Table 1.  
To inspect and analyze network parameters, we applied the HJ-Biplot to our \(243\times 7\) data matrix, using only centrality measures variables. The first three axes of the HJ-Biplot analysis explain 86.27% of data variability, achieving 54.406% in the first principal plane (Table 2). From the results shown in Table 2, it can be deduced that there is a dominant axis (axis 1) that takes 54.406% of the total inertia of the system. This axis is characterized by the variables degree centrality, eccentricity, closeness and eigenvector centrality (Table 3). In Table 3 the contribution of each factor to the element is shown. The variables that characterize axis 1 (variables that receive a strong contribution of axis 1 and low in axis 2 and axis 3) are: degree centrality, eccentricity, closeness and eigenvector centrality. The variable that characterizes axis 2 is clustering and finally the one that characterizes axis 3 is betweenness. \begin{table}
\begin{tabular}{l l} \hline \hline
**Network Indicator** & **Value** \\ \hline Number of stations, \(N\) & 243 \\ Number of links, \(M\) & 280 \\ Network diameter, \(D\) & 44 \\ Radius, \(r(G)\) & 22 \\ Average shortest path, \(L\) & 14.682 \\ Clustering coefficient, \(\widetilde{C}_{CLU}\) & 0.008 \\ Entropy, \(E\) & 5.431 \\ Modularity, \(Q\) & 0.875 \\ Number of communities & 15 \\ Assortativity & 0.296 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Summary statistics of network indicators. \begin{table}
\begin{tabular}{l c c c} \hline \hline
**Axis** & **Eigenvalue** & **Explained var.** & **Cummulative var.** \\ \hline Axis 1 & 789.976 & 54.406 & 54.406 \\ Axis 2 & 263.913 & 18.176 & 72.582 \\ Axis 3 & 198.745 & 13.688 & 86.27 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Eigenvalues and explained variance. The factorial graph of the plane defined by axis 1 and 2 is shown in Figure 5. The variables with the greatest variability on the first axis are degree centrality, eccentricity, closeness and eigenvector centrality (its associated vectors have the highest length). Degree centrality, closeness and eigenvector centrality characterize the positive side of axis 1 while eccentricity characterizes the negative side. So, the stations located on the left side of the factorial plane are characterized by having high eccentricity values (they project far away in the direction of the vector eccentricity). For instance, stations Arroyo Culebro, Henares, Las Suertes have the highest values in eccentricity. In contrast, the ones on the right side have very low values in eccentricity, this is the case of stations like Puerta del Angel and Noviciado. In addition, those that are located in the first quadrant have high values in closeness because they project far away in the direction of the vector closeness. This is the case of stations like Plaza de Espana, Principe Pio, Noviciado and San Bernardo. Finally, the stations that are located on the right side of the fourth quadrant are those with the highest values in clustering, degree, closeness and eigenvector centrality. Examples of these stations are: Callao, Diego de Leon, Tribunal, Av. de America and Alonso Martinez. This analysis also shows that closeness and eccentricity are strongly negatively correlated (flat angle).