Abstract
The growth of transportation networks and their increasing interconnections, although positive, has the downside effect of an increasing complexity which make them difficult to use, to assess, and limits their efficiency. On average in the UK, 23% of travel time is lost in connections for trips with more than one mode, and the lack of synchronization decreases very slowly with population size. This lack of synchronization between modes induces differences between the theoretical quickest trip and the ‘timerespecting’ path, which takes into account waiting times at interconnection nodes. We analyse here the statistics of these paths on the multilayer, temporal network of the entire, multimodal british public transportation system. We propose a statistical decomposition – the ‘anatomy’ – of trips in urban areas, in terms of riding, waiting and walking times, and which shows how the temporal structure of trips varies with distance and allows us to compare different cities. Weaknesses in systems can be either insufficient transportation speed or service frequency, but the key parameter controlling their global efficiency is the total number of stop events per hour for all modes. This analysis suggests the need for better optimization strategies, adapted to short, long unimodal or multimodal trips.
Introduction
Although the coupling between different transportation networks is fundamental^{1}, most of the studies on Public Transport Networks have been performed considering only one single transportation mode: private cars^{2,3,4}, taxis^{5,6,7,8}, Subway^{9,10,11,12,13}, Train^{15,16,17,18,19,20}, Bus and Trams^{18,21,22,23}, and at a worldwide scale, airline networks (see^{24} and references therein). However, most transportation systems are coupled to each other and as it was recently shown in^{25}, interconnections can have dramatic consequences on the behavior of the whole system. This finding triggered a wealth of studies^{26,27,28,29,30,31} on multilayer networks — also coined multiplex networks — providing a new paradigm for studying these coupled systems. Public Transport Networks belong to this class and provides a paradigmatically example of spatial^{32}, temporal^{33}, and multilayer Network^{30} where each layer corresponds to a single transportation mode.
A few studies only considered many modes merged in an unique network^{34}, but this aggregation might hide important structural features due to the intrinsical multilayer nature of the network^{35}. In particular, in the case of urban transport, not considering the connection times can lead to unprecise estimates for the network's navigability^{36}. We note also that interchanges are not symmetrical: railtobus and bustorail waiting time are different and are independent from the actual traffic volume^{37} (at least as long as capacity limits are not taken into account^{14}). In addition, the existence of alternative trajectories on differerent transportation modes enhance the system resilience^{36}.
Inter and intramodal connections can be intensively optimized just through modifications and offsetting of the existing timetables, allowing to reduce waiting times at transfer points of a city like Washington D.C. of about 26%^{38}. A better knowledge of the structure and layout of the Public Transport System would impact a wide range of areas. Indeed, mode choice is one of the fundamental steps in transportation forecasting^{39} and has represented in the past a perfect experimental field for the study of individual choice behavior^{40}. Developments in the availability of urban public transport has the potential to improve significantly the air quality in metropolitan areas^{41} and directly influences the social geography of a city^{42}. However, multimodality not only means the existence of more and better alternative options, but having to deal with all these alternatives at the same time. From the users point of view, the difficulty of dealing with the enormous amount of information needed for describing and taking advantage of the public transport of a city is such that it is no more managed by personal experience and habits, but by services offered by major information technology companies. From the transport agencies point of view, the managing task becomes significantly harder because: i) different modes are run by separate agencies and both data handling and optimization tasks have to cross high organizational barriers; ii) it is not trivial to identify aspects of the system that are relevant for service optimization.
Therefore, in order to help decision makers, new quantitative approaches are needed to highlight the limits of the system and to assess the impact of new infrastructural development. Our capacity to understand transfer behavior and to evaluate transfer improvements are indeed limited by the lack of proper analytical tools, as we have to take into account many important aspects simultaneously^{43}. If we want the public transit system to become a viable alternative to automobile, it is crucial to design costcompetitive and reliable public transportation systems that guarantees both short travel times and a travel experience comparable to those of car trips^{44}.
Another important difficulty in the study of transportation systems is the data availability. In particular, it is usually very difficult to obtain traffic related data, and we take advantage here of the availability of another type of data which will enable us to assess the structural efficiency of the system. This opendata information consists in the set of timetables for all transportation modes in the United Kingdom, except for Northern Ireland (see Methods and the Supplementary Information for more details). We will focus on the urban scale and identify key quantities characterizing the efficiency of the system, providing directions to improve urban Public Transport Systems. More precisely, our goal is to determine:
how far is an urban, multimodal public transportation system from optimum,
how the temporal aspects impact the structure of quickest paths,
how important are the multilayer aspects,
the key differences between transportation systems in different cities.
Our study is based on the statistical analysis of the quickest paths on the multimodal transportation networks. We assume that origins and destinations are uniformly and independently distributed on the location served by the transport system and we do not take into account access time at the departure location. This uniform demand does not take into account how flows are actually allocated over the network (a piece of information that is usually not directly available) but allows us to focus directly on the structural features of the network, and not on its actual use and on the qualities perceived by the average user. In this sense, the weaknesses and optimization that we discuss here, concern an ideal optimum where all possible routes with all possible origins and destinations would be improved. The methodology developped here can however be very easily adapted to the case where origindestination matrices are known.
Results
We first define different types of paths in these systems. In particular, in order to understand how far are urban transportation systems from an ideal optimum, we compare the quickest timerespecting paths with the minimal path. The minimal path is the quickest one, computed by using the largest speed observed on each link and by neglecting waiting times, and represents an unreachable condition, equivalent to having all the existing transportation systems perfectly synchronized for the specific trip under consideration. In contrast, the timerespecting path is the quickest path but where we use the real timetable and where walking and waiting times are taken into account. The timerespecting path is by definition longer than the minimal path and as we can see on an example shown in figure 1, they can be extremely different from each other. In addition, real trips are bound to the transportation system and do not follow a straight line of euclidean distance d(a, b) from the origin a to the destination b. The topographical and infrastructural constraints induce differences of the transportation network topology between cities. A consequence of this is that the length (a, b) of the quickest (timerespecting and minimal) trips on the network might be very different from d(a, b), and this difference can be measured by the detour that can be interpreted as a costbenefit ratio^{45}. In order to compare the availability of routes in different networks, we use the quantity R = max_{d}_{>1} r(d) for a fixed d subset (see Supplementary Information). The values of R in different cities are strongly anticorrelated (−0.95) with the static network normalized cyclomatic number^{46} M_{N} = (E − N − 1)/N, reflecting the fact that the more loops are present in the network and the less the detour. In the following, we would like to exclude this topological influence and in order to compare various cities, we will use as a spatial metric the effective length on the network.
Only large cities can afford significant railbased elements (trains, metro, tram) in their public transport systems and therefore can have a high propensity to interchange^{47}. Other transportation modes such as ferries and coaches, play a secondary role at an urban level (and air transportation is naturally out of the game). Coaches emerge for minimal paths in certain cities, but their low frequencies are completely excluding them from timerespecting paths. Other forms of road transportation are usually more accessible and, for this reason, bus is the dominant layer for short distances. If cities have enough suitable street space dedicated to Bus and Bus Rapid Transit systems, they are even able to outperform metro and rail systems^{44}.
Each transportation mode is characterized by its cruise speed, departure frequencies and accessibility. A consequence of these peculiarities is that, depending on the length of the trip, the public transport system offers different optimal timerespecting solutions. At the national scale (Figure 2, Left), different strategies emerge at different spatial scales. We will not consider very short trips for which the origin and destination are closer than 1 km, because distances so short could be easily covered by walking and usually do not rely on transportation systems. Above this scale, the vast majority of short trips are made within the bus layer, and the rail system becomes dominant for interurban trips of length larger than approximately 40 kms. Air transportation emerges naturally for longer distances above 200 kms, and its importance increases significantly for distances of order 400 kms (e.g. GlasgowBirmingham) and d ≈ 500 kms (e.g. GlasgowLondon Luton), and becomes finally dominant for trips longer than 700 kms, connecting for example the southern part of England with the northern part of Scotland.
At the urban level, transportation modes that capture a significant fraction of the timerespecting paths are Bus, Railways and, when available, the Metro and Tramway layer (Figure 3). The bus stops represent the vast majority of the possible origins and destinations and are almost always used in our paths. This bus layer contains in general the largest part of both minimal and timerespecting paths. The use of the fast transportation modes emerges progressively with increasing (Figure 2, right), with a higher rate in larger cities such as London, Manchester and Birmingham, where the MetroTramway systems are present. As this transportation mode has a high frequency and a fast speed, and is not affected by congestion, it is naturally used as a quickest alternative to buses across city centres. Nevertheless, due to its limited accessibility, the largest fraction of short trips are done in the bus layer, also in cities where the system has an extended offer of metro lines (Figure 3). The metro layer is in competition with the rail layer, which has higher speed but lower departure frequencies. In cities with high multimodality (i.e. high average number of modes used per trip), the rail network attracts the largest part of the mobility at distances much lower than at the national level. Indeed, for London (Figure 3, left) and Manchester (Figure 3, right), the length done by train overcomes the one by buses at and .
Comparing minimal with timerespecting paths
We first analyze the multimodal aspect of trips, quantified by the numbers Λ_{m}_{(t)} which represent the number of different modes for the minimal (m) or timerespecting (t) paths. For some cities, the timerespecting paths display a larger Λ_{t} than for minimal paths, while for others, it is the opposite (Fig. 4a–b). The relative loss in multimodality due to synchronization can be measured by The larger Δ and the larger the difference between minimal and timerespecting paths. We see in Fig. 4c that Δ is positive when the average speed of the alternative (ie. nonbus) layers V_{nb} is sensibly larger (>2.5 times) than the average speed V_{b} of the bus layer (see Supplementary Information for more details). The quicker the rail and metro layers are, the more multimodal the minimal path would tend to be. Indeed, for minimal path, the use of fast nonbus layers is only limited by their accessibility, i.e. by the extratime needed for reaching the interlayer connection point. For timerespecting paths, multimodality also implies the importance of synchronization, and it appears that in cities where metro or rail are sensibly faster, their frequency is also lower (Fig. 4d). In other words, in cities where the fast layers are extremely advantageous in term of speed, the system suffers from synchronization problems. This empirical finding suggests the existence of a structural limit to transportation systems' possibilities that policy makers should take into account in the search of a efficient optimization strategy.
As a consequence, if the rail and metro layers are relatively quick, they are used for minimal paths while additional waiting times due to mode change can be too costly for the timerespecting paths. On the other hand, in cities where the bus layer is fast but with a frequency as low as for faster layers (eg. London, Liverpool, Cardiff), minimal paths tend to use buses only, while the timerespecting paths face the synchronization limits of the bus layer itself (see for example figure 1). More generally, the factors responsible for the time difference between timerespecting and minimal paths are: (i) waiting times (both intra and interlayer); (ii) the fact that the optimal riding times used to compute minimal paths may differ from the riding times at a particular hour; (iii) a long walking time for connecting different modes in a wide stop area. In order to quantify the differences between minimal and timerespecting paths, we introduce the synchronization inefficiency δ, computed as the ratio of timerespecting travel time τ_{t} and minimal travel time τ_{m} For all cities, δ reaches its maximum δ_{max} for short trips, where waiting times are long compared to the travel time, and then decreases with the distance according to the following function, valid for all cities (see Fig. 5, left) where ν ≈ 0.5. The collapse observed for δ for all cities suggests that there is an underlying process describing the accumulation of waiting and walking times along timerespecting paths. The specifics of the different cities appear in the system efficiency in both the worst δ_{max} and best δ_{min} limits. We note here that this Eq. 3 is consistent with a simple argument based on the central limit theorem leading to ν = 1/2.
Timerespecting paths are however not completely different from the minimal ones and we can measure the similarity of two paths by using their spatial overlap q, defined as the fraction of length of edges they have in common. The overlap is q = 1 for extremely short trips (if a single edge is used, waiting times are not playing any role), and then decreases with for all cities as (Fig. 5, right) where is the scale parameter for each city. The function converges to a limiting value q_{min} in the range [0.15, 0.33]. This minimal overlap is due to the limited number of good options available, especially close to the origin and the destination. The constraints due to the local connectivity and optimal cruise speeds make the minimal path the best option also when time causality starts playing a role. The exponential decay of the overlap with suggests that there is a typical ‘branching’ length for each city, which sets the probability of having alternate routes. In other words, the probability for the timerespecting path to deviate at each from the minimal path is proportional to .
Anatomy of a trip
We have seen so far that the length governs the behavior of most quantities characterizing a trip. In order to identify the role of the temporal and multilayer aspects of the network in the structure of the timerespecting paths, we detail how the total travel time can be decomposed into different components: the riding time (with any mode), and waiting and walking times at interchanges. In addition, in order to take into account the multimodal aspect of the network, we discriminate riding times per layer and we separate intralayer from interlayer waiting time. This wide spectrum of temporal quantities forms what we call the ‘anatomy’ of a trip, and is represented in Figure 6 a–c for different cities. This figure allows for a quick understanding of how the temporal structure of trips varies with distance.
We first note that the travel speed grows with (see Supplementary Information) which implies that travel time for timerespecting paths grows sublinearly with the distance covered. Another important contribution in trips is due to walking between modes in the multilayer network, which represents a fixed cost of multimodality (in addition to interlayer waiting times). We naturally expect walking times to grow with as the number Λ_{t} of layers used in timerespecting paths (see Supplementary Information for more details). Finally, waiting times are one of the two main contributions to the synchronization inefficiency δ, the other being the difference between optimal and actual edges' riding times. These times will play a relatively minor role for long distances, as their relative importance compared to riding times decreases.
The analysis of these anatomy plots shows the following. At short distance, in all cities but London, most of the travel time is spent in intralayer waiting time. Most trips start at the bus layer, and the first connections within this layer are those that make the system extremely inefficient. We therefore define for each city a distance such that for trips shorter than the waiting time represents more than 50% of the travel time. For , the lack of synchronization is dominant and the temporal network is far from being optimal. This distance interval corresponds to values of where the overlap q is larger than 50% (Fig. 6d). For short trips, we have few alternative paths and cannot avoid waiting times due to synchronization problems. Timerespecting paths are thus very similar to minimal paths, and (large) waiting times are directly added to optimal riding times.
For long distances, we already saw that the multimodal nature of the systems becomes important. The use of fast transportation modes becomes advantageous only when the difference in speed compensates for the time necessary to reach the rail or metro network. In order to measure this effect, we define the distance such that for the largest part of the trip is done with a transportation mode different from the bus. Trips with are then essentially made within the bus layer, and most of the travel time is due to actual transfer (walking or riding). We observe for large cities like London, Manchester, or Birmingham, a finite value of indicating that at a certain point the bus layer loses its dominant role. This is in contrast with smaller cities where is larger than the city radius, implying that fast layers always play a marginal role in these cases.
If we take into account time respecting paths with at least one interlayer connections only, we find that on average for all cities considered in this work, the time spent in connections (walking and interlayer waiting times) represents a significant fraction (23 ± 6%, see Supplementary Information) of the total travel time. The different regimes identified in figure 6d suggest that different strategies might have the better impact for each city for optimizing the transport time of trips of specific distances. Short trips are indeed dominated by intralayer waiting times, while long trips by riding times. In the cases where the multimodality becomes dominant, interlayer waiting and walking times, together with the fast layers' cruise speed, become instead the most relevant quantities for the optimization task.
The role of the total number of stop events
An interesting question concerns the characterization of a multilayer, temporal network such as the transportation systems that we consider here. Obviously, the number of modes and their frequency play important roles in their efficiency. A simple, natural quantity is then given by the average number of stop events per unit time where C_{α} is the number of stop events in the layer α and Δt the duration of the time interval considered in the analysis of the temporal network (see Methods). In this study, we considered a starting time of 8:00 am (monday) and a duration Δt = 16 h which covers a whole day of mobility. The quantity Ω represents a global measure of the transportation service offered in a city, of the infrastructural cost of the transport network, and is indeed proportional to the cities population (Fig. 7a). In order to improve the transportation system and to serve more people, one may add new lines, new connections, increase the frequency of a line, or even introduce a new transportation mode in the network, and the quantity Ω integrates all these modifications.
As we will see, it is actually remarkable and unexpected that a single network indicator such as Ω is enough to explain the behavior of many key quantities characterizing the public transport network of different cities. For example, the interplay between temporal and multilayer aspect of the public transport network is highlighted in Fig. 7b, showing that the fraction λ_{t} of timerespecting paths using more than one mode^{1} is larger for cities with a larger number of stop events. If we assume that the average number of possible alternative to bus layer path (which is always an available option) is , the expected fraction of unimodal trips is . The average interdependency of the timerespecting paths is then 1 − ω Using this form to fit the data shown in Fig. 7b, we obtain hour/stops.
Similarly, Ω is related to the cruise speed (where τ_{cruise} is the time spent in a moving vehicle) and therefore also to the time respecting paths travel speed (Fig. 7c). Indeed, we can assume that the fraction ω of unimodal paths is traveled at the average bus layer speed , while the fraction 1 − ω of multimodal paths is traveled at a speed , where represents the ratio of lengths on the nonbus and bus layers ( is the average speed for the fast layers — see Supplementary Information). The average travel speed grows then with Ω as Using the value for obtained above, we minimize the variance between the estimated and empirical values of and we find the optimal value: k ≈ 0.8 (see Fig. 7c). This result shows that for all cities considered here (and under uniform demand), approximately 45% of timerespecting paths are on nonbus transportation modes.
In addition, the quantity Ω also characterizes the efficiency of a public transportation network in terms of synchronization. Indeed, we observe that the average synchronization inefficiency measure decreases with Ω as a power law (see Fig. 7d) where μ ≈ 0.3 ± 0.1. The expected decrease is naturally due to the fact that larger values of Ω implies larger frequency and thus a better synchronization between modes. The small value of μ is however bad news in terms of efficiency: in order to divide by a factor 2 we need to multiply Ω by a factor of almost 10. We can however hope that when exact origindestination matrices are known, a better optimization of the system can be obtained through targeted improvements. It is not unusual to observe power law behavior in urban systems^{49,50}, and although the fits are not perfect (essentially due to the small number of available decades), this result Eq. 8 could be useful for constructing coarsegrained models of transportation in cities. Besides this, we note here that the city of Edinburgh is an outlier in all figures 7(a–d) and, for this reason, has been excluded from the bestfit of figures 7b and 7c. Indeed, even if Ω is relatively high for this city, Edinburgh's public transport system seems to use a significantly different strategy in managing the mobility demand, characterized by an extremely high busfrequency. The network is therefore extremely efficient in terms of synchronization but not performant in terms of cruise speed (see fig. 4d), as can be seen with timerespecting paths that are mostly composed of slow unimodal bus trajectories (see figures 7b and 7c).
Discussion
We identified the total number of stop events per hour Ω as the key quantity which characterizes the efficiency of a transportation system and its efficiency in terms of speed, multimodality and synchronization. Naturally, Ω is not the only parameter at play: multimodality depends also on the different cruise speed and departure frequencies in the different layer. In the UK transportation system, these two quantities are anticorrelated, as if a city system might try to optimize rail and metro systems, with respect of the bus system, either making them faster or more frequent. This relationship has important practical applications, as it constitute a limit that policy makers need to take into account in their system optimization, and can serve as a support for evaluating alternative Public Transport Systems' designs.
The temporal aspect of the Public Transport Networks appears to be influential for trips covering all distances. Short timerespecting paths tend to be mostly similar to the minimal ones, and waiting times are directly added to the riding times of the associated minimal paths. Waiting times then represent the largest fraction of the total travel times, and at this scale an increase of bus departure frequency, or methods like timetables offsetting^{37,38} of the bus service may represent a good optimization strategy. Longer timerespecting paths tend instead to diverge from minimal ones, and very large waiting times can be avoided thanks to the availability of alternative routes, and when it is possible, longer trips are progressively taking advantage of the multimodality of the system. For cities with a large level of multimodality, as it is the case for London, Birmingham Manchester, it becomes hard to disentangle the temporal and multilayer aspects of the system. Waiting time (together with walking time) does not represent a simple cost to minimize, but a price to pay to access to fast transportation.
The value of waiting and walking times are perceived as higher than the time spent travelling^{52}, in particular because walking demands a greater physical effort^{51}. Waiting time has an higher perceived cost because of the frustration due to the sheer inconvenience of waiting^{52}. All these costs have to be integrated with those related to the time needed for accessing the network^{44}, the stress of the transfer experience^{43}, breaking personal habits^{53}, scheduling costs and those caused by the unreliability of arrival times^{52}. In order to optimize the travel experience and to minimize the perceived mobility cost, it is then necessary to consider the full anatomy of trips and to distinguish between transportation modes and between the nature of time spent (riding, waiting, walking). In this respect, we believe that the tools and the methodology developed here will allow for an integrated view of these systems and will be helpful for testing and finding specific optimization strategies.
Methods
Data
The land transport timetables used in these papers are provided by the National Public Transport Data Repository^{54} under Open Government licence. A snapshot of every public transport journey is recorded for all services running in Great Britain (England, Scotland, Wales) during a full week in October 2010. The raw files contain the information available in the travellines web sites and callcentres during the selected week. For road transport, transportation agencies take into account the average traffic conditions at different hours and days for the design of timetables, so that they implicitely contain congestion effects.
The modes covered and identified are bus, coach, train (national rail), ferry and metro (including Underground, tram, light rail and nonnational rail trains). All routes are referenced to stops coded using the NaPTAN scheme (National Public Transport Access Nodes) data^{55}. In the NaPTAN scheme, every UK rail or metro station, coach terminus, airport, ferry terminal, bus stop or taxi rank is associated to at least one Stop Point. Not all Stop Points are actually used, so only those that were present in the timetables are considered active and have been taken into account. Stop point are then organized in Stop Areas representing facilities (Airports, Bus/Metro/Coach/Railway Stations) or possible interchange points. The definition of these Stop areas has been taken as a basis for defining a multilayer network from the timetable data. A further process of data cleaning and aggregation has been performed to have a consistent definition of intermodal exchange points (see Supplementary Information). To complete the spectrum of transportation modes, we use detailed schedules of all nonstop UK domestic flights, provided by Innovata LLC^{56} for the week of 18–24 October 2010. Each of these flights has been associated to the Stop Points of the arrival and departure airport (and eventually to a specific terminal). The multilayer temporal network dataset derived from these data is publicly available at http://www.quanturb.com/data.html.
Multilayer temporal network
The intermodal exchange points are identified by (i) original NaPTAN Stop Areas, (ii) new Stop Areas obtained by a spatial aggregation of Stop Points (see Supplementary Information). To be an exchange point, journeys of different transportation modes should stop in that Area and to correctly define a Multilayer network^{30}, we associate all Stop Points to a layer α, representing a specific transportation mode. If a Stop Point belongs to a Stop Area, the point is not represented in the network and all vehicle stops in that point are associated to the area. Both Areas and Points are identified by an id i. As buses and coaches may stop in the same location, a copy of the same Stop Point can be defined in two different layers, and thus associated to two different vertices v_{iα} and v_{iβ} in the multilayer network. Similarly, if an Area i has associated points belonging to a set of layers α, β, γ, …, a vertex representing that Area is defined in each of those layers (v_{iα}, v_{iβ}, v_{iγ}, …). Interlayer edges connects all couples of vertices associated to the same Point or Area in different layers in both directions (i_{α}, i_{β}) and (i_{β}, i_{α}). If the connection from a layer α to a layer β is performed by walking, a walking distance is assigned to each of these edges (i_{α}, i_{β}) which is calculated as the average distance between all couples of active Stops Points in i belonging to the two different layers α and β. The travel time has been then computed using a standard walking speed of 5 km/h^{44}. In addition to the walking times, additional 30 minutes are added to the interlinks from the airflights layer to all the others, in order to take into account the characteristic waiting times in airports. Similarly, two hours of checkin and security control times are added to the interlinks towards the airline layer (which corresponds to the time suggested by airlines to be at the airport before departure time).
In each layer α, we thus have a set of N_{α} vertices, representing stops locations. The timetables define a set of events occurring in these vertices. Each vehicle departure can be associated to a directed connection between two vertices v_{iα} and v_{jα} that occurs at a certain time. These events can be represented as C_{α} quadruplets (i, j, t, δt), where i, j ∈ V_{α}, t denotes the departure time and δt the riding time for that specific trip^{57}. Besides the temporal network, we can also study the static topology of the public transport network by defining a set of E edges, where the edge (i_{α}, j_{α}) exists if there is at any time at least a connection between v_{iα} and v_{jα}. For each of these edges, we compute the minimal riding time observed at any time δt_{min}. We define the minimal path as the shortest path on this static network, where the cost associated to each link is the minimal riding times. We use these minimal paths as a benchmark which represents the optimal mobility though the multi layer network. All the measures performed in this paper are limited to the largest strongly connected component^{46} of the static network associated to the corresponding area.
Timerespecting paths
Paths performed through the network must respect the timeordered sequences of contacts. For this reason, a journey has to follow causal temporal paths defined as a sequence of connections with nondecreasing times^{33}. We define the travel duration τ_{ab}(t) as the shortest time needed to reach b starting from a connection from a departing at a time t′ ≥ t. The duration is not static but depends upon t. In this paper, we focus on the morning rush hour, and thus we chose t_{0} = Monday, 8:00am. The temporal distance is measured starting from the actual beginning of the trip, without taking into account the first waiting time t′ − t. Furthermore, to limit the contribution of a small number of location from where connections are extremely rare, we introduce a waiting time cutoff Δ_{c} = 2 h limiting the maximum delay allowed for a single connection^{57}. Even while working on a static connected component, this cutoff limits the number of allowed paths. At a national scale, approximately 16% of the trips in the largest strongly connected component of the static network have been excluded because unreachable with this choice of t_{0} and Δ_{c}.
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Acknowledgements
The authors are supported by the European Commission FETProactive project PLEXMATH (Grant No. 317614).
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Institut de Physique Théorique, CEA, CNRSURA 2306, F91191, GifsurYvette, France
 Riccardo Gallotti
 & Marc Barthelemy
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R.G. and M.B. designed, performed research and wrote the paper.
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