Figure 1

Illustration of the two-layer model with the logical graph $G^{\lambda }$ and the physical graph $G^{\varphi }$. (a) A simple example showing the actual mapping $M(E^{\lambda })$ of $G^{\lambda }$ on $G^{\varphi }$. The logical edge $e_1^{\lambda }$ is mapped on $G^{\varphi }$ as the path $M(e_1^{\lambda })=(v_1^{\varphi },v_2^{\varphi },v_3^{\varphi })$. Assuming that $G^{\lambda }$ is unweighted, the loads of the three indicated nodes are $l(v_1^{\varphi })=3$, $l(v_2^{\varphi })=2$, and $l(v_3^{\varphi })=4$. (b) Two layers in the EU transportation network. The nodes are the train stations. The edges in the physical graph are the existing rail tracks connecting neighboring stations. Every edge in the logical graph connects the first and the last station of a particular train route; its weight reflects the number of trains following this route in any direction. (c)–(f) Basic characteristics of the EU network (WA and CH yield similar results): (c) node degree distribution in the physical graph; (d) node degree distribution in the logical graph; (e) edge weight distribution in the logical graph; (f) the distribution of the lengths of traffic flows, counted in a number of hops $h$ in the physical graph.Reuse & Permissions

Figure 2

EU data set (WA and CH yield similar results). The first three figures present the physical layout of the (a) node degree $k^{\varphi }$, (b) betweenness $b^{\varphi }$, and (c) real load $l$. The size of a node is proportional to the measured value. In (d), we present the scatter plots of the (top) node degree $k^{\varphi }$ and (bottom) betweenness $b^{\varphi }$ versus the real load $l$. In the top left corner of every plot, we give the value of the corresponding Pearson’s correlation coefficient.Reuse & Permissions