## 5 Results
### Analysis of Vulnerability  
A basic study on the robustness of Madrid metro network was presented in [29]. In addition, in this work we analyzed the communities that exist within Madrid metro network using the concept of modularity. Transit operation zones in transport networks can be identified through communities [5]. In transport networks with high modularity, stations in the same community have dense connections but sparse connections between stations in different communities. This implies that flow of passengers is more efficient within each community than between communities. As many as 15 communities were identified in Madrid metro network (see Figure 11). The modularity is high (\(Q=0.875\)); it results in Madrid metro network being robust against breakdowns. In Table 5, the distribution of stations per community and line is shown. The excessive size of some lines in Madrid metro network leads to their fragmentation in various communities. For example, \begin{table}
\begin{tabular}{l l l} \hline \hline  & **Mean** & **Standard Deviation** \\ \hline Global & 2,882,126.11 & 3,080,341.64 \\ Cluster 1 & 3,299,972.89 & 2,715,313.06 \\ Cluster 2 & 1,518,573.90 & 790,653.57 \\ Cluster 3 & 8,981,036.46 & 6,606,708.17 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: ANOVA analysis of the flow passengers. Figure 10: [PERSON]’s correlation coefficients of the centrality measures and boarding count. centrality. Specifically, the HJ-Biplot method has been used because of its advantages when inspecting multivariate data matrix. Using the coordinates obtained from the HJ-Biplot, a classification of the stations according to their centrality measures has been carried out. As a result of HJ-Biplot analysis, positive correlation has been found between degree and clustering, degree and closeness, degree and betweenness and degree and eigenvector centrality. Instead, closeness and eccentricity are strongly negatively correlated. This implies that to classify the stations it would not be necessary to use all centrality measures. From the Cluster analysis we have obtained a classification of the stations according to their centrality measures in accordance with their remoteness to the city center. Three clusters have been found so that, the stations farthest from the city center have the lowest centrality measures and belong to a cluster, those located around the city center have medium values and belong to another and those from the city center have the highest values and belong to the other one. Finally, an ANOVA analysis was performed and the relationship between the centrality measures of a station and the flow of passengers of the station has been demonstrated. From the Pearson's correlation analysis it was found that the most central stations are those with the highest passenger flow. Regarding the robustness analysis, Madrid metro network has a highly agglomerated community structure. On the other hand, the excessive length of some lines of Madrid metro network leads to their division between some communities. Although this is an usual characteristic of these transportation networks, it can result in some robustness problems that affect most peripheral stations. In this work, the importance of Complex Network Analysis and Statistics has been highlighted when offering sustainable solutions. There is no doubt about the vital role that transport networks play in the development of smart cities and how the above techniques can help to achieve this. However, they are not only useful in this area, but they can also be applied to other areas of science in the context of Applied Sciences. In future works, the robustness of Madrid metro network will be analyzed in more detail. For this, the passenger flow which would be affected by a possible failure in the network will be taken into account. In addition, other factors such as socio-demographic and land-use factors, will be introduced in the study, so that the relationships between them and centrality measures can be analyzed.