Table 1.
Description of the considered datasets.
Fig 1.
The Rhone-department road network.
Maps throughout this research article were created using open-source data from OpenStreetMap and OpenStreetMap Foundation, which is made available under the Open Database Licence. Map tiles are from OpenStreetMap cartography, which is licensed as CC BY-SA (see https://www.openstreetmap.org/copyright). (a) Rhone-ROADS: the Rhone-department network, graphically shown as an undirected and unweighted graph (b) Rhone-OBS: The Rhone-department sub-network shown on top of the Rhone-ROADS. The graph only includes edges with at least one observed elementary taxi trip (in purple).
Fig 2.
Spatio-temporal characterization of taxi observations.
(a) Hourly number of elementary trips for the typical working day (b) Number of edges from the Rhone-ROADS with median speed (c) Evolution of the network median edge-speed over time.
Fig 3.
Three static topological instances of the Rhone-OBS graph: Comparison of BC values.
(a) Undirected, unweighted (circle size proportional to node’s BC) (b) Directed, length-weighted (circle size proportional to node’s BC) (c) Directed, free-flow-travel-time-weighted (circle size proportional to node’s BC).
Fig 4.
The dynamic taxi graph: Median-speed-to-max-speed ratio at different hours of the day.
Edge color (from black/red to yellow) indicates higher speed-ratio, i.e., reduced congestion) (a) 05:00 (b) 08:00 (c) 20:00.
Fig 5.
Nodes’ travel-time-weighted BC over the dynamic graph in the time range [05:00–10:00].
The size of each circle in the subplots is proportional to node’s BC. (a) 06:00 (b) 07:00 (c) 08:00 (d) 09:00 (e) 10:00 (f) 11:00.
Fig 6.
Evolution over time of the top-BC node for some time slots of the dynamic taxi graph.
(a) Node with the highest BC at 06:00: evolution of its BC over time (b) Node with the highest BC at 08:00: evolution of its BC value over time (c) Node with the highest BC at 10:00: evolution of its BC value over time.
Fig 7.
Per-edge temporal correlation between BC and flow (a) and zoom on a specific region (b,c).
(a) Per-edge temporal correlation (only edges equipped with loop detectors have a non-null value) (b) Zoom on an area with two roads: Qaui Dr. Gailleton and Quai Claude Bernard (c) Evolution of the TTBC and flow on the road: Qaui Dr. Gailleton and Quai Claude Bernard.
Table 2.
Performance evaluation in sequential settings: Comparison of W2C-Fast-BC with respect to Brandes [15] and of BADIOS [42, 43] to Brandes.
Fig 8.
Directed graph weighted with discretized road-lengths.
(a) Nodes’ BC values (circle size proportional to node’s BC) (b) Execution times (W2C-Fast-BC and Brandes BC) with 10 cores (c) Percentage error of W2C-Fast-BC with K-fraction = 0.2 (top-1000).
Fig 9.
Directed graph weighted with discretized free-flow-travel-times.
(a) Nodes’ BC values (circle size proportional to node’s BC) (b) Execution times (W2C-Fast-BC and Brandes BC) with 10 cores (c) Percentage error of W2C-Fast-BC with K-fraction = 0.2 (top-1000).
Fig 10.
The interpolated dynamic taxi graph: Median-speed-to-max-speed ratio at different hours of the day.
(a) KNR-interpolated graph at 08:00 (b) Top-1000 nodes’ BC values at 08:00 (c) Execution time of W2C-Fast-BC vs Brandes-BC at 08:00 (d) KNR-interpolated top-1000 BC percentage error at 08:00.
Fig 11.
Dynamic evolution of the mean absolute percentage error with W2C-Fast-BC (K-fraction = 0.2) on the interpolated dynamic graph (on top-1000 nodes’ BC values).
The shaded portion of the graph corresponds to measured values of standard deviation at each time step.
Table 3.
Accuracy of W2C-Fast-BC with respect to Brandes on the noisy graph.
Mean and standard deviation are reported for each accuracy indicator as obtained by aggregating over the 10 instances of the noisy graph for each value of Ψ.
Table 4.
Sensitivity of W2C-Fast-BC to different levels of random noise introduced on G08:00.
Table 5.
Sensitivity of Brandes to different levels of random noise introduced on G08:00.