Dynamical efficiency for multimodal time-varying transportation networks

Spatial systems that experience congestion can be modeled as weighted networks whose weights dynamically change over time with the redistribution of flows. This is particularly true for urban transportation networks. The aim of this work is to find appropriate network measures that are able to detect critical zones for traffic congestion and bottlenecks in a transportation system. We propose for both single and multi-layered networks a path-based measure, called dynamical efficiency, which computes the travel time differences under congested and free-flow conditions. The dynamical efficiency quantifies the reachability of a location embedded in the whole urban traffic condition, in lieu of a myopic description based on the average speed of single road segments. In this way, we are able to detect the formation of congestion seeds and visualize their evolution in time as well-defined clusters. Moreover, the extension to multilayer networks allows us to introduce a novel measure of centrality, which estimates the expected usage of inter-modal junctions between two different transportation means. Finally, we define the so-called dilemma factor in terms of number of alternatives that an interconnected transportation system offers to the travelers in exchange for a small increase in travel time. We find macroscopic relations between the percentage of extra-time, number of alternatives and level of congestion, useful to quantify the richness of trip choices that a city offers. As an illustrative example, we show how our methods work to study the real network of a megacity with probe traffic data.

path in a multilayered network. Here, we explain the algorithm for both type-I and type-II that decreases dramatically the 23 computational effort required. In Fig. S3, we report the computational time for the all shortest path algorithm in multilayer 24 network compared to the all H-shortest path algorithm defined here. While the classical path searching requires a computational 25 time that grows according to a power law with the number of layers, with our algorithm has a linear growth. 26 We consider H-paths as paths between two nodes that belong only to one or two layers that, for the sake of notation, we 27 indicate here as G [1] and G [2] . We notice that a layer can be the result of the aggregation of two or more single layers where 28 the distance between two nodes of the aggregated network is the fastest possible using the best combination of duplex in the 29 ensemble of networks. As an example, the time to go from a node O to a node D can be, in the aggregated network, the sum of 30 the walking time to a metro station, the travel time spent in the metro and last mile by bike-sharing service, if this was the 31 best combination using this three modes of transportation at that path between two location is not feasible in one layer, we set 32 their relative distance equal to infinity. Once we have the structure of layer G [1] and G [2] and their corresponding link travel 33 times W(t), it is possible to consider the matrix N × N , D [1] (t) and D [2] (t), of the relative node-to-node distances at time t. 34 In order to compute the H-shortest path of type I between a node i and a node j, it is sufficient to sum element by element the We obtain a vector s [1,2] ij (t) = (i [1] (t)) T + j [2] (t), where 36 each coordinate k represents the travel time from i to j passing from layer G [1] to layer G [2] through station k at time t. The 37 minimum value of vector s [1,2] ij (t) is the travel time of the H-shortest path and k * = argmin(s [1,2] ij (t)) the most convenient 38 station where, eventually, to change (if k * = i, j). Moreover, if we consider also the alternative path with a travel time close to 39 the minimum we can obtain the dilemma factor as explained in the main text.

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With this algorithm the computational effort decreases dramatically. In fact, it is sufficient to compute the all shortest path 41 algorithm in every single layer and then, for each couple of Origin-Destination (i, j) to sum 2 vectors of dimension N and find 42 the minimum value. The algorithm is illustrated schematically in Fig. S2.

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For H-shortest path of type−II, that is paths that start from a layer G [1] , changes in k to a second layer G    Dilemma factor. In the main paper, we showed the equation and the general results for the α and β dilemma factors, and also 57 the relation between the two of them. Here, for completeness, we want to show how, at local level, the α dilemma factor (and 58 consequentially the β for the relation revealed in the main paper) is spatially distributed during the day among the intermodal 59 junctions. In our case, the stations are the interchanges between road and metro system and α = 5%. In Fig. S7, the size 60 of the red spots is proportional to the number of alternative stations with an extra travel time less than the 5% than the congestion does not influence the traffic performance of a urban network, the value of ∆Ei = 0, for each i ∈ N . In order to 68 show the difference between efficiency and the link speed values and the betweenness centrality, often used in transportation 69 networks to evaluate the traffic performance, we plot in Fig. 4 of the paper the coloured map of Shenzhen and the correlation 70 plot between these link measures. We notice that there are substantial differences in centrality when we consider the network 71 with link speed value, in particular during the peak hour (6pm). In the scatter plots on the right of Fig. 4 of the paper we 72 observe a weak (R 2 = 0.32) correlation between Dynamical efficiency and link speed values and no correlation (R 2 ≤ 0.03) 73 with the betweenness centrality with ( f ) ) or without ( g) ) considering link speed data.

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Bike layer 75 In this section, we illustrate an extension of the work presented in the main paper. The main purpose of this part is to give In order to test the robustness of dynamical efficiency, we used a link percolation process on the complete Shenzhen's network. \p (t), (continuous line in Fig S12a-b); B) the standard deviation among all the 50 instances 96 (dashed line in Fig S12a-b); D) the max and min value maxt E [c] \p (t) (Fig. S13); C) the dynamical efficiency pattern 97 at time t = 6am, 9am, 12pm, 3pm, 6pm, 9pm. (Fig S14-16).

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Materials 99 Origin-Destination information. From the same dataset of taxis GPS used to estimate the link speeds, we extracted the origin 100 and the destination location of each trip passing through the studied zone of Shenzhen. We considered not only the trips the 101 have origin and destination in that zone but also the trip that start and/or end in the external part, but pass through it.

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The number of trip per hour changes as reported in Fig. S8a. The length of the majority of the trips is around 2km and the 103 distribution is plotted in Fig. S8b. We assign in this case, the origin to the zone of the first point in the map and destination 104 the last zone containing a GPS signal. In this way, we were able to collect around 220k taxis trips for the 7th of September and 105 divided into the microzones as showed in Fig. S9a. The zone-to-zone traffic of the aggregated data is visualized in the circular 106 graph S9b where the colour and the thickness are proportional to the demand between two zones.    Mean Network Speed (mph) Fig. S5. Here is reported the map of the road efficiency for Shenzhen every 2 hours form 2am to 12 am. We can appreciate the emerging pattern of congestion that develops and propagates from the city center during the two peak hours. It reaches its lowest value around 8pm. The difference between the efficiency and the speed map (Fig. S6) it becomes, here, clear: the efficiency values are naturally smoothed and delimit the congested zones. The physical meaning of the dynamical efficiency expresses the concept of reachability of that specific location. Fig. S6. Speed map of Shenzhen during the whole day. The same data have been used to calculate the efficiency in Fig. S5. We notice that in the speed representation of the map it is more difficult to separate the congested zones and the values (corresponding colors) are not spatially smoothed as in Fig. S5.  Fig. S7. Local α-dilemma factor during the day. Here it is represented the α dilemma factor, with α = 5%, for each interchange station. The red spot is proportional to the number of alternatives of that location if we allow the travel time to be until 5% more than the H-shortest path (that changes in k). We notice how the congestion, namely in the morning and evening peak hour, increases also the number of 'close' alternatives. (a) The division of the map of downtown of Shenzhen in 50 homogeneous rectangular zones. Every origin and destination of all taxis trips in our dataset has been assigned to its relative zone. In this way we were able to reconstruct the circular graph in Fig. S9b of the OD demand.

(b)
Here the curves reported in Figure S13a have been divided by the corresponding maximum average value for each case P = 40, . . . , 200 (continuous lines in Fig S13a). The high similarity of the average value of the dynamical efficiency normalized by the corresponding max value supports the proof of the robustness of this measure respect to percolation and network variations. We also notice how the standard deviation among the tested 50 scenarios per case (P) tend to increase with P. Fig. S15. Dynamical efficiency patter after link removal with p = 40 at t = 6am,9am,12pm,3pm, 6pm, 9pm. The pattern here is very similar to the complete Shenzhen's graph used in the results section of the main paper. This is just one of the 50 cases used to deduce the statistics about robustness of dynamical efficiency with 40 links randomly removed (highlighted in dark red). We notice that the minimal value in the scale is very small but not zero. This is due to the fact that at least one extreme of each eliminated link is reachable by some path and, in this particular case, there is no isolated node.

Fig. S16
. Dynamical efficiency patter after link removal with p = 120 at t = 6am,9am,12pm,3pm, 6pm, 9pm. In this case we begin to appreciate some differences in the spatial distribution of dynamical efficiency with respect to the complete graph. The congested zones are less defined and 120 links removed from the networks create some inefficiency roughly homogeneously distributed in all the city network.
Fig. S17. Dynamical efficiency patter after link removal with p = 200 at t = 6am,9am,12pm,3pm, 6pm, 9pm. In this particular scenario where 200 links (≈ 10% of the total network) has been removed from the network the spatial pattern of dynamical efficiency took a very different and defined configuration with respect to the complete graph. In particular, we notice a zone in the top-left part (instead of the top-central zone) of the map that has been heavily influenced by a few links removed around it. Also in the bottom part of the map, instead of the left-bottom part as in the complete graph, we have a right-bottom part considered as inefficient according the dynamical efficiency.