Fig. 1
Illustration of the subgraphs and related indicators for the London and Paris metro networks. Plots (a) and (b) show the dimension of the subgraph with respect to \(t_c\). The red markers show the dimension of the maximal clique vs. cut-off time \(t_c\). Dark, medium and light blue markers show the dimensions of the k25, k50, and k75 cores, respectively. Plots (c) and (d) show the evolution of the access graph (grey), k25-core (dark blue), k50-core (medium blue), k75-core (light blue) and maximal clique (red), together with the idealised maximal clique (yellow) for London and Paris networks, respectively. The cumulative accessibility indicators used in the analysis are shown in plots (e) and (f) for the respective networks. Dimension of the idealised maximal clique is shown in black markers. We set the upper integral boundary to \(\min (t_{max}, 2\cdot t_{max}^{ideal})\), where \(t_{max}\) is the maximum travel time in the network and \(2\cdot t_{max}^{ideal}\) is twice the maximum idealised travel time. The minimum is taken, as in some networks \(t_{max} < 2\cdot t_{max}^{ideal}\). This happens in the London network, while the \(2\cdot t_{max}^{ideal}\) threshold is shown in (f) for Paris. The cumulative accessibility indicator represents the shaded red area relative to the shaded grey area. The latter represents the idealised cumulative accessibility against which the actual cumulative accessibility is measured. The values of selected indicators for both networks are: \(S^{(MC)}=0.76\), \(S^{(k50)}=0.77\), \(\tau _{(k50)}=0.34\) for London; and \(S^{(MC)}=0.53\), \(S^{(k50)}=0.57\), \(\tau _{(k50)}=0.50\) for Paris.
