Illustration of the user-based multi-edge P-space representation. (a) Two geo-localized crossing PT lines ℓ1 and ℓ2 are shown on the map of central Paris. (b) Schematic of the P-space multi-edge representation for a section of the network: all pairs of nodes corresponding to stops on the same line are connected by edges with the same label.
Scatterplot of time versus physical distance associated with the shortest time paths for each origin–destination pair. The points are coloured according to the number of points in the area considered. Scatterplots are shown for the cities of (a) Paris, (b) Toulouse, (c) Nantes and (d) Strasbourg. Colours indicate the logarithm of the number of origin–destination pairs in a given range time–distance bin.
Travel time factors with respect to distance travelled. The factors have been computed using the lowest 1, 2 and 5% of the time distribution for each distance travelled by public transportation for the following cities (including their surrounding areas) (a) Paris, (b) Toulouse, (c) Nantes and (d) Strasbourg.
Pattern detection using the multi-edge P-space representation. (a) Geographical representation of graph GSP, where links correspond to the 1% best shortest paths of the whole public transportation network. (b) The normalized BiCross validation error computed for the adjacency matrix XSP(10 km,11 km) (solid black line) of the same graph, for the associated random matrix XSPrandom(10 km,11 km) (dotted line). The selected number of structures ks is assigned by a red rhombus. (c,d) Two of the structures revealed in the PT system of Paris. Green dots are ingoing, while red dots are outgoing affiliated.
Similarity between commuter flows and PT privileged connections in French municipal areas. For each of the urban agglomerations considered (Toulouse, Nantes, Strasbourg, Paris area and Paris Petite Couronne), the bar chart’s height indicates the weighted Jaccard index sTM between the commuter flow network GTMcom and the PT structural pattern network GC (for further explanation see text). The total number of commuters within each city using either car or PT is indicated below each bar.