Social and economic flows across multimodal transportation networks in the Greater Tokyo Area

The hex network is formed by connecting each hexagon to its six (or less if it is on a border) neighbors on the grid. This creates a transportation network representing local travel, usually walking but may also represent cycling, local driving, or other slow means of transportation. As such, the 3 min time weights for these edges are based on a 5 kph (walking) speed across the 250 m interhex distance. For all walking links we use a capacity of 2500 people per hour, which we convert to 10,000 to represent the total value on the basis that the capacity’s effect is spread out across the main commuting hours. The hex with the greatest job surplus (jobs minus population) is 15,665 and the greatest population surplus is 2024, and each hex has six neighbors, so these capacities are more than adequate to allow sufficient flow to reach all needed areas.

The hex network serves two main purposes: (1) some hex grid spaces are inaccessible directly via any other transportation network, so this ensures a connected graph (among adjacent hexes at least), and (2) in many cases using purely transportation links based on the closest station/stop/intersection leads to unnatural travel time differences. For example, for many people in the suburbs, the closest station is one that only has local trains. So it is often faster to walk (or ride a bicycle) to a slightly further station with express trains than to use the closest station.

Because the hexes permeate the whole geographic region, there are hexes in locations (such as mountain tops and rivers) where there is no population, there are no transportation routes, and even walking would be impossible. In the future we plan to remove hex network links representing impassible route (e.g., crossing rivers and train tracks directly, without using a bridge/gate). Considering the 250 m interhex distance, and the nature of our home-work transportation analysis, this particular abstraction is unlikely to be responsible for any inaccuracies.

Our train data combines all rail services (surface trains, subways, and streetcars; public and private) and schedule types (e.g. local, rapid, commuter, express). One natural representation of a rail network is to connect each line that stops at a given station to a single node representing that station (a space of stops Barthélemy (2011)). Because multiple lines, and multiple types of routes on the same line, connect to adjacent stations, capturing the station as a single node requires the use of either a layered graph (with each line on a different layer) or a multigraph (that allows multiple edges between pairs of nodes) (Goczyłla and Cielatkowski 1995). We originally used a multigraph construction; however, for our analysis the transfer times between trains/lines is crucial to the total travel times. Using a layered graph requires inter-layer links to connect the lines, and these are essentially train transfer edges. After our initial construction in this way, we realized the importance of not just inter-train transfer times, but also initial waiting and access times. In order to seamlessly integrate all these interstitial times into our network and algorithms we decided to include them directly as part of the train network.

To do this we first create a node representing each physical station in the system. Then for each type of train of each line that has a stop at that station, we create a node representing its platform. We then connect each physical station node to each line’s platform node at that station via a train access link. We also directly connect all the platform nodes at a station to each other with train transfer links. Finally, we connect the platform nodes along each line using route links. We represent each schedule type (local, express, etc.) as a separate line. In this way, express trains that skip stations are captured by links directly connecting the stations actually used by that route. Express trains and local trains on the same tracks have distinct platform nodes in order to capture the transfer time of switching trains.

This set of abstractions produces a flat (non-layered, non-multi) rail system network. Even though there are multiple kinds of edges, the difference is in the interpretation/meaning of the model rather than in the function of the model. Certainly other formulations are possible; however, utilizing an intuitive physically explicit network allows us to keep all cost weights and all usage levels as edge values. This fosters simpler, faster, and clearer algorithms to process flows and measure network features.

For this project the transportation networks are modeled as directed. The route links are weighted by the traversal times for a link of that type on that segment using data from (Ekitan: Train and Bus Schedule Data 2019). In the case of daily average train route times, most edges are reciprocal with identical properties. However, we make use of the directedness to introduce an asymmetry in the traversal of the train access links. Exiting a station (the links from the platform nodes to its physical station node) takes 1 minute (the average walking time to exit the station), while the reverse direction (from the physical entrance to boarding a train) takes 5 minutes to account for both walking and waiting times². Transfer links have a uniform (for now) time-weight of 5 min. A simplified example of the directed train network construction is shown in Fig. 3.

The access and transfer times approximately reflect walking and waiting times between trains and platforms without overly complicating our station specification (Hibino et al. 2005). Based on this structural foundation we can later incorporate heterogeneity in transfer and access times according to specific lines, locations, distances, time of day, congestion conditions, etc. if/when the data becomes available. Realistically, adding such details would likely only alter the total trip times by a couple of minutes for average daily traversal times, and so they are unlikely to make much of a difference for our identification of aggregate traffic patterns. However, if one wishes to make highly accurate predictions of specific traffic volumes, especially in a time-dependent analysis, then these details matter. The complete trail network for the Greater Tokyo Area includes 1546 stations, 131 distinct lines, and 10,536 directed links capturing all routes of all types. A map of Tokyo overlaid with the full train network can be seen in Fig. 4.

Tokyo trains are often overcrowded, reaching nearly 200% of their comfortable riding capacity during the morning rush (Ministry of Land 2018). The maximum number of passengers varies by the number of cars, the specific make of train, and the number of trains per hour, but only a couple of lines ever exceed 80,000 people per hour (in two-way traffic, although one direction can greatly outweigh the other at certain times of day, at least in the suburbs). For simplicity we use this figure as the basis for the 240,000 capacity of each train edge in the system (multiplying by four to include core commuting hours). Keep in mind that this is not intended as an estimate of the capacity of each line for predicting crowdedness, but an approximate figure to use as the maximum capacity for network flow algorithms.

The bus network is constructed similarly to the train network so that physical bus stops (stations) are separate nodes from the stops along the lines (platforms). Again, constructing the bus network in this way allows us to include waiting times and transfer times in our model as edge attributes, thus fostering the use of standard network flow algorithms. Traveling from the physical bus stop node to the bus line node (waiting to enter the bus) takes 5 min, while the opposite direction (exiting the bus) takes just 1 min. Transfers from one bus to another are assumed to take 5 min in average waiting and transit time. Just like the train network, the bus route (platform) nodes for express and local bus routes are distinct so that transferring from one to the other type of bus on the same line incurs the 5-min transfer cost. Traversal times are set from the bus schedules using the average traversal time for each link. This time does not include heterogeneity in traversal times as the result of time-dependent road congestion, loading and unloading times, differences in speeds from skipped stops, or other factors. Also, similarly for the trains, some buses only run at particular hours of the day, and at irregular frequencies, but these details are not included here for daily traffic. As a result, some supposed morning commuters may be simulated riding a bus that only runs once a day after midnight (incorporating schedule data can alleviate this issue).

There are 5084 unique bus lines using 32,901 physical bus stops (stations) and 181,574 directed links included in our data for the Tokyo area. The local bus stops are typically dense along the route, enabling one to make out the contours of the road from the linear connections between sequential bus stops (see Fig. 5). Express buses (ones that run the same paths as local buses but skip several stops or directly connect two distance points, such as airport shuttles) are modeled as edges directly connecting the bus stops where the bus actually stops. These are easily identifiable in Fig. 5 connecting to Narita and Haneda Airport from various stations around the Tokyo Area as well as several commuter buses that link Tokyo and other major stations together.

The number of buses running per hour exhibits significant variation both across lines/locations and for the same line at different times per day. Some buses run less than once per hour while others have 13 buses per hour at their peak times. However, our current data does not include this information on a per-route basis. Because the capacity figure is used to cap the traffic along a link, and increasing bus traffic along a link to accommodate need is easy, we use the 10 buses per hour value for setting capacity limits. Each large bus has a capacity of 74–78 people (sitting plus standing)(Toei Transportation Customer Center 2018), although some buses are smaller we use a value of 75 throughout. We again multiple the per-hour capacity by four to capture the maximum load for the core commuting time in each direction. We hope to get more refined line-specific data for future analyses, but as a result of the above approximations we currently assign a capacity of 75×10×4=3000 people for each edge in the bus network. Again. rather than an estimate, this should be thought of as a theoretical maximum capacity.

Our road network is constructed from road segments tagged as tertiary or above (or not specifically labeled and left as ‘road’) in OpenStreetMaps data (OpenStreetMap Contributors 2019). OpenStreetMap labeling is sparse in Japan compared to other developed countries, and we had to make assumptions based on typical values to fill in missing road speed limits (Japan Traffic Safety Association: Rules of the Road 2017), typical drive speeds, numbers of lanes, and road capacity. For approximate drive speeds we adopted a convention of 70 kph for major highways, 30 kph for other major roads, and 25 kph for minor roads. These drive speeds have been reduced from the speed limits to account for traffic congestion, railway crossings, turning, traffic signals, etc. Edge traversal times are calculated from the Haversine distance between its end points and the approximated driving speeds.

The number of lanes is rarely specified in the data, so to approximate capacity we again needed to make assumptions based on typical values while recognizing that the actual road system exhibits heterogeneity within each category. Because our network model is everywhere directed, two-way road segments are modeled as bi-directional with the one-way values in each direction. To calculate the capacity we used standard vehicles-per-hour rates for these road types and speeds (Polus et al. 1991; Highway Capacity Manual 2000): 2000 vehicles per hour per lane for freeways, 1000 for major arteries, and 500 for the remaining roads. As before, we multiply these per-hour rates by four in order to generate the commuting capacity values actually used as edge capacity values for the network simplex algorithm. A table showing the values of each feature used for each road type is available in the Additional file 1.

In Japan it is common for roads to frequently change their thickness and speeds, thus complicating classification efforts. Perhaps for this reason when we included only the tertiary and above links, the road network was too sparse and disconnected. The addition of the unclassified (rather than smaller) roads largely corrected this problem. Although there are still several disconnected segments when leaving out the minor roads; this accurately reflects the infrastructure of Japanese roads. The fragmentation is exacerbated because the road network data used for the current analysis leaves out many of the main expressways. Despite this deficiency, the intermediate road network (between local access roads and major expressways) is an integral part of the transportation network that connect areas otherwise unreachable (except by walking). Figure 6 shows the detail and extent of the included road network. Areas between roads are accessible via the hex network, although most of the large areas without roads are unpopulated mountain and forest regions. In addition to road links, each intersection node is connected to any other intersection node within 173.3m via a walking link. Although these non-road links between road nodes use the walking speed of 5kph, and this is perhaps an underestimated speed even for narrow Japanese residential roads, in this case the links and speed are meant to represent all local movements, including: travel to and from parking, congestion, waiting for pedestrians, and various other factors – and typically only for short distances. We tend to think of the road network constructed in this way as the “taxi network” in consideration of the included links, speeds, and revealed role in Tokyo transportation.

The four networks described above represent the geography and three transportation systems; we use additional connecting links in order to integrate these disparate modes together. One can think of the individual transportation modes as layers, and these connecting links as inter-layer edges (Idri et al. 2017). Or you can think of the whole system as a physically explicit flat network. These two conceptualizations are methodologically identical for the purposes of edge traversal time-costs, especially because the sets of nodes for each mode are unique. In our representation a bus stop and taxi stand in front of a station are represented as distinct nodes from the station itself. We use the actual geographic coordinates for those locations to determine the distances, and from those distances the time costs of intermode transfer. Although such an explicit network increases the number of nodes and links needed compared to a more abstract representation that collapses these distinct nearby nodes into a single node (e.g. using a multigraph), it benefits from representational simplicity, faster algorithm speeds, and the ability to intuitively and directly conduct hypothetical conditional response analyses (e.g., link knockout to represent accidents, construction, etc.)

We connect transportation nodes to hex nodes and to each other (of the same or different mode) whenever they are within 173.3 m (the hex circumcircle radius is 144.34 m) of each other. This method ensures that every transportation node is connected to at least one hex (and often two or three). For the train and bus modes, only the physical station nodes are thus connected; i.e., one must go through a station node to reach its platform nodes. Connecting the modes in this way provides an intuitive geographic foundation for the whole system, and (when analyzed together) the nodes of other transportation networks are directly connected to each other to represent direct intermode transfers (e.g., from a train station to a bus stop or taxi).

Table 2 reports the numbers of nodes and edges for each mode as well as the fully integrated network. Aside from our physically explicit flat graph, there are alternative methods for integrating disparate modes of transportation and capturing the transfers among them. As eloquently stated by Idri et al. (2017), “The representation of this transfer action is treated as one of the key issues in multimodal route planning problem.” One common solution is to generate a “transfer graph” that virtualizes the connections between modes to facilitate interoperability (Ayed et al. 2011). However, because we are not including schedules or other time-dependencies in the intermode (or intramode) transfers, and because our transportation networks are geographically grounded, we solve the intermode transition problem by (simply) adding walking links between nodes of different modes (and sometimes nodes of the same mode) to explicitly model the act of transitioning between them and “glue” the networks together. This simplification of a transfer graph yields an intuitive representation of a multimodal network as a single, simple directed network.

Even though our network is of the common digraph variety, our geographically grounded construction alters the calculation and/or interpretation of network structural measures used in standard transportation network models. For example, the degree of a station node is no longer the number of adjacent stations, but is now the number of line types with stops at that station plus the number of hexes, bus stops, and road intersections within 173.3m. Although the node degree (considering all links or just a subset) may still be related to other graph features, it cannot be interpreted in the same way as previous analyses (Barthélemy 2011; Levinson 2012; Tsekeris and Geroliminis 2013).

As another example, the path length (in terms of edges used) is a less clear indicator of anything useful because there is so much heterogeneity in the physical lengths of edges by mode. We return to this point later, but the crux is that only network measures that can incorporate an edge weight/cost are useful and appropriate for multimodal transportation network analysis. Measures such as the betweenness centrality and anything requiring the average path length are computationally impractical; that is, they are impossible to compute in practice. The reason is not strictly the number of nodes, but also the number of equicost paths along the hex network part of the graph among so many pairs of nodes. One alternative approach to the hex network is to use a Voronoi grid based on the nodes (Ren et al. 2014), but the hex network is superior for providing high-resolution integrated analyses because it (for example) integrates the walking/bus/driving time or the use of other modes to/from stations. Furthermore, having a homogeneous grid size/shape is important when tracing the network flow results back to the location attributes. Rather than lament the analyses and comparisons we cannot perform because of our network construction, we now move ahead to explore the new analytical opportunities that this representations creates.

Of the line segments selected by MLIT, 76 are within our research area. First, we isolate the segments included in the empirical data from our network. We then calculate the Pearson correlation coefficient between the segments’ ridership data and the usage values from simulations limited to the same segments (Ren et al. 2014). In the cases where multiple types of lines run between a pair of stations, we use the maximum usage value among the network edges. We calculate the Spearman correlation and the Kendall τ to evaluate how well our simulations match the ranking of the segments. Table 9 reports the results of these comparison tests.

The generated O-D trips to-work and from-work achieve similar correlation levels for the morning commute data. The reason for this is the particular segments chosen by MLIT; the majority of which are located in or near the central core rather than along commuter lines. Once near enough to the job-dense central region the trains are indeed crowded in both directions due to people (for example) living west of the city but working on the east side of the core and vice versa. In fact, of the 76 segments reported by MLIT, there are 9 lines with two segments and 1 line with three segments. In the empirical data there are examples of lines running in opposite directions with comparable ridership.

Considering the simplifying assumptions made in generating the network, and the strictness of the test of matching ridership on just 76 of the 10,536 segments, the level of agreement between our simulations and empirical data is already reasonably high. Especially in light of the fact that in our simulations bus and taxi travel are also competing options. By adding data on heterogeneous transfer and access times, schedule limitations (e.g., evening only routes), and a cost/time trade-off in route planning, we expect to further improve the accuracy. Although those additions must be left for future work (after we are able to obtain the data), there is something we can do now to broaden our evaluation: test entire lines instead of specific segments.

In the following analysis we extrapolate the empirical segment data to the lines, and compare the aggregated simulated traffic per line using both the mean and maximum. The data from MLIT only covers a tiny percentage of the line segments (0.72%), but it does include at least one segment from approximately half of the lines in the Tokyo area (65 of 131 lines, if not distinguishing by type). Twelve empirical segments appear on shared lines, so to extrapolate the value for each line we first take the mean of the segments’ ridership values for each line. We next isolate those 65 lines present in the empirical data from our simulated data. We then aggregate the simulated usage data for those lines as the mean value of all the segments of that line (including all types of trains of that line). Finally, we perform the same statistical tests as above to determine the level of similarity of the results. Table 10 shows the results of comparing the mean ridership values along each line to the mean empirical values for that line.

As expected, comparing the aggregate across the whole line to the extrapolated line usage improves the resulting accuracy in most cases (only the Pearson correlation of the simplex algorithm is weaker). The Spearman correlation of the generated trips to work exceeds 0.45 while the trips from work are just slightly less. These results using the mean usage across each line make an even stronger case for stochastically generated O-D trips.

That said, the empirical data is reported for the most crowded hour of the included line segments. Because capacity for a given line is largely (but not entirely) constant across time, these are also the maximum ridership values for that line. Within any given hour, the ridership of a train is highly correlated across segments of a route. So the ridership of the whole line is fairly well represented by the segment data we have, especially in terms of ranks. So instead of using the mean to extrapolate and aggregate values along the whole line, we now examine the maximum values. For simulated data each line is assigned the value of the link with the highest usage. Table 11 shows the match accuracy is further improved using the max values compared to the mean values.

The Spearman correlations are slightly below 0.60 for the stochastically generated trips, although still less than 0.40 for the simplex method. Considering the limitations of our transportation model construction, and the methods used to generate O-D trips, and our goal for recreating overall daily patterns rather than time-specific traffic micro-simulations, this result represents a strong relationship between our predicted ridership levels and the empirical ranking of train line ridership.

For the maximum ridership levels to match this much better than the segment-level ridership implies that a different segment of the line receives higher usage in the simulated data. This difference likely results from a combination of the simplifications used to create our current model and the limited scope of available empirical data. Although the current level of accuracy isn’t exact enough to use for policy or routing decisions, the results are highly encouraging for the accuracy including our planned expansions.

One obvious limitation of this analysis is that it only compares traffic volumes on the train network. In the future, we plan/hope to gain access to aggregated GPS-style data of people’s movements that would cover all modes of transportation (even walking) and allow us to better calibrate and verify the model’s flows. Another limitation is the sparseness of the empirical data; the data only contains ridership for 76 of the 10,536 train edges. A larger dataset may not reveal better accuracy, but it would be a more robust test of the accuracy. Yet another limitation is that the data only reports ridership during its peak hour rather than a total for a day, although we expect these are highly correlated at least for ranks. Because we analyze commuting from home to work, and most of this traffic happens during the morning rush, this limitation is probably not problematic for our ranking analysis. However this limitation will become problematic when we include travel for other purposes such as shopping, commuting to school, etc. across a whole day.