Datasets from five cities (London, UK; Boston, MA; Denver, CO; Minneapolis, MN; and Washington, D.C.) were filtered to cover the same range of months (April-October inclusive) in order to sample corresponding seasonal effects. Of course, the climate of each city is distinct and different, but each resides in the northern hemisphere so summer occurs at approximately the same time of year. Some of the schemes (Denver, Minneapolis and Boston) have closures over winter months, so this and data availability limits our reporting period. All data are taken from 2011, with the exception of Boston, which is drawn from 2012 data (which covers March 2012 – September 2012 inclusive) supplemented with October 2011 data.

Table 1 displays summary statistics about the datasets used. In terms of total journeys and stands, London is by far the most active; London also has a smaller average minimum distance between stands, which will tend to influence the proximity statistics below. Each scheme has 213 days of data and over 150,000 journeys.

As part of the initial data exploration, we explicitly visualised bike journeys across the cities on particular days. The purpose of this was both to visualise the data as an output in itself, and to begin to identify any strong patterns in the large volumes of source data. Each data point included the origin and destination stand IDs, as well as the start and end times of the journey, and the ID of the bike used by the journey. However, because the bikes themselves do not have location/GPS sensors, precise route information is not available.

To create the theoretical cycle routes between each stand, a software package, Routino, was used, with an OpenStreetMap extract of each built-up area, including dedicated cycle infrastructure as well as the road network and path links. Routino uses a configuration profile for each user type. Cyclists will avoid motorways and move on other routes at a constant speed of 20 km/h, except paths and steps where they will move at 5 km/h. Road desirability is factored in, with each road type given a score – trunk roads being given a score of 35%, primary roads 80%, and the smallest roads 95%. Cycleways are given a score of 100% to reflect their attractiveness to bicycle sharing system users. Road routing information, such as one-way streets and turn restrictions, are also applied.

Routino generates a series of latitude/longitude waypoints for the bike’s journeys, allowing multi-stage linear interpolation to create a continuous journey across the city. Individual bikes are drawn as small ellipses at their current position throughout their journeys.

A screenshot from London [Video S1 in Appendix 3 of File S1] is show in Figure 1; similar plots were generated for Boston [Video S2 in Appendix 3 of File S1] and Minneapolis [Video S3 in Appendix 3 of File S1]. Here, a semi-opaque background wipe retains “trails” created by each bicycle, providing a sense of continuity and tracing out the street networks. Self-journeys were represented by traces which orbited the origin/destination stand three times over the journey duration, and then disappeared. These visualisations can be animated for a specific day, or aggregated over multiple days to show a “Monte Carlo-like” picture of the system behaviour e.g. on weekdays [Video S4 in Appendix 3 of File S1] (see also [15] for a complementary visualisation tool).

We generated networks for each system, by counting the number of bikes for each of the source-destination pairs over the reporting period. We further split the data into weekdays and weekends, where we expected different users and patterns due to working (commuter) and leisure users (which we expected to be more dominant over weekends). In each case, this aggregation served to generate a matrix of source-destination flow volumes – we label the origins with the letter i and the destinations with j; it is notable that this typically contained strong “diagonal” elements (the “self-journeys” mentioned previously, which start and end at the same location). This matrix can be thought of a lookup table of total flows of bikes from origin i to destination j, and is a mathematical description of a network in the same way that our visualisations provide a spatial or geometrical description of the networks. We will label this object as having N nodes (columns/rows).

These datasets are rich and complex, and can be disaggregated by time of day, season, weekday/weekend, and other factors as well as space. Figure 2 shows, by way of example, the seasonal variations in the duration of journeys in Washington D.C., in terms of both raw journeys (Figure 2a) and data normalized by area under curve (Figure 2b). The winter months show a distribution of slightly shorter journeys, and fewer journeys as a whole. The initial analyses we carry out in this paper sum data over a number of months in order to create an aggregate picture of the network; analyzing individual networks drawn from particular months and/or times of day may illuminate further patterns.

Researchers at London’s City University [16] considered a series of techniques for visualizing these complex systems of flow data, including pseudomatrices, edge bundling, Kernel Density Estimation-type methods and Bezier curves. We adopted a similar method to their Bezier curve techniques, without the explicit size ordering, and using using opacity (alpha - the opposite of “transparency”) as the main weight variable, scaled linearly from 0 to 175 (out of 255) with the edge weight relative to the maximum. This creates a visual grammar where journeys “start” in a straight line and curve into their destination. Representing these very complex datasets across a variety of cities with a variety of edge weight scaling laws is involved, and a detailed discussion is beyond the scope of this paper; all of the static visualizations of network utilize this Bezier curve formalism to denote direction (Subfigures a and b in Figures 3–12) A circle around a stand denotes a self-journey; in some networks, these were the most popular routes.

The routing generated for the animated visualisations was not used for analysis purposes, as it was judged that it builds in a set of assumptions about route choice, and while reasonable, it infers more information than is available. While a crude measure, Euclidean/Great Circle distance is at least free from these additional assumptions. Self-journeys are problematic in this formulation, as they have zero net displacement but indeterminate journey length. This is not ameliorated by applying routing mechanisms, when there are an astronomical number of redundant paths for a closed loop.

We examined the typical journey distances for the cities in question. Aggregating the journey frequency by distance travelled and then dividing by the total number of journeys yields an initial estimate of journey frequency as a function of proximity (weekday and weekend data are shown in Figures 13a and 13b respectively). However, bike stands are not homogeneously distributed through space, and as such the above distribution function might be skewed by a route which moves between a source with a lot of bikes leaving and a destination with a lot of bikes arriving. We might expect that, given a maximum entropy allocation of journeys (i.e. one that assumes as little prior knowledge as possible), the route between these two locations would be well-used, and that this would lead to a commensurate over-representation of this separation in Figure 13(a-b). If instead, we wish to probe an individual’s propensity to travel a location of certain proximity (assuming a constant distance decay function which to first order is independent of absolute start/end location or time), we need to decompose the probability. We can define a spatial (proximity) model of the flows Y^(s) such that

1)

where Y_ij^(s) is the predicted weight (i.e. number of journeys) between origin i and destination j, and f(d) is some displacement-decay function representing people’s preference to travel to some destination displacement d away (note that f(d) is not a probability – if we take Y_ij^(s) to be in units of “bikes travelling on a route”, f(d) will take 1/bikes as its unit). O_i and D_j are defined by

2)

and

3)

which in spatial interaction modeling can be referred to as the marginal sums, or in network theory, the out- and in-strength of the node respectively. This is analogous to a naive “gravity” model, and a more sophisticated approach to this problem would be to construct a maximum-entropy model and introduce additional balancing factors into (1). However, given an unknown form of the spatial function f(d), Figure 13(a-b) suggests that it may have a more complex form than an exponential or power-law decay. We instead took an approach favoured by[11], constructing and empirical distance-decay function f(d):

4)

Where Y^(s)(d) is the aggregated modeled flows for all journeys associated with displacement between d and d+ Δd (where Δd is the chosen bin size). Rearranging (4) yields the distance decay function f(d), shown in 13(c-d) (for weekdays and weekends respectively), normalized by area under curve in order to compare different cities.

Community detection tools allow the identification of network subregions within the bikeshare flow networks which are linked to one another more strongly than nodes from other subregions. One of the simplest methods in terms of implementation is the cluster-algorithm method [14], whereby the flow matrix is treated as a series of column (or row) vectors; then two nodes are judged as belonging to the same community if their links to other nodes are similar (or more simply, if they both have links to the same nodes), in this case based on a Euclidean similarity measure. Agglomerative hierarchical clustering methods allow choices of splits, so it is important to apply a quality measure to assess how meaningful these clusters are. For a directed weighted network described by Y_ij, we can use the following, usually called the modularity after [17] and [18]:

5)

where m is the sum of all edge weights in the network (the time-aggregated version of equation (2)) and the Kronecker delta acts on the community ID for each node {c_i} (following [19]). The O_iD_j/m term represents the aspatial (directed) null model, the hypothesis that edges will be formed in proportion both to the outstrength {O_i} of the source node and the instrength {D_j} of the destination node. In a graph in which no communities exist in fact, the number of links within a community that we identify would be exactly the same as the global characteristics of the network; then the modularity for each community would be close to zero, and so would the total modularity. For communities with a large number of links within them and few to other groups, there are preferential factors to link within that community that go beyond linking between “popular” nodes; so Y_ij will not resemble the null model, and modularity will tend to be large and positive.

We used this modularity to select an appropriate number of clusters for each city, as it provides a measure of how meaningful these communities identified are, independently of the methods used to detect the communities. As discussed previously, self-journeys were set to 0, to avoid numerous single-stand clusters appearing as a result of such journeys. A Wards hierarchical clustering algorithm was implemented using standard Matlab libraries (linkage and cluster methods) applied to a matrix defined via a call to the MySql database (Table 3).

Expert et al [11] extended this model to account for the likelihood that, for nodes embedded in physical/geographical space, proximity as well as in/outstrength will dictate the probability of linkages being formed, and an interesting research question is what linkages form above and beyond this spatial/volume dependence. They use a spatial null model to find clusters -equivalently, we can form a network scaled by spatial considerations, so that

6)

which we expect to be >0 if two nodes are linked more strongly than pure proximity/strengths would dictate. This is represented graphically in Figures 8–12 (subfigures a-b). Here, blue indicates a positive value and red a negative value, and the intensity and weight the numerical value. This is a visual representation related of the method of residues, which has been applied to urban modeling for over thirty years - [20] provides an overview– and creates a link to more recent spatial network theory.

It is possible to form clusters on the basis of this scaled network, using the same cluster analysis as before. This rescaled edge weight allows for rapid calculation of modularity:

7)

which we can again use to choose the optimum number of clusters/communities. Results are shown in Figures 8–12 (subfigures c-d). Again, self-journeys are set to 0 because distance displacements are not reasonably calculable for these journeys.

Significant variations exist between flow volumes over the course of a day; all cities considered exhibit this characteristic trimodal distribution on weekdays (Figure 14a), with strong peaks at the morning and evening “rush hour” commuting periods, and lunch times, contrasting with the more unimodal peak that dominates weekends (Figure 14b) and we infer to be the hallmark of tourist and/or leisure activities.

We can define departure vectors based on the number of bicycles leaving each stand in each hour-of-day period and their destinations, creating an hour-on-hour network. In order to determine the distribution of edge weights and outstrength (here defined as the total number of bikes leaving a node at within some time period) as a function of time, we can use entropy, in its standard definition. Entropy is a measure derived from statistical physics, and latterly from information theory, which can be understood as the level of equality of distribution in a system or dataset; a high entropy corresponds to a state where all entities have similar values, a low entropy where this distribution of values is unequal. In our system, the entities are edges and the values are weights (or nodes and outstrengths). If all edges had equal weight, the entropy tends to a maximum value of ln(1/n); if all edge weights but one were zero, the entropy is a minimum at 0.

If at time t the edge weight distribution is {Y_ij(t)}, where i is an integer labeling the origin, and j an integer labeling the destination:

8)

where

9)

Notably, these entropy measures (not shown) do not vary by more than +/– 10% throughout the course of the day, so the equality of distribution of bikes on routes as a proportion of total bikes being used at that time does not vary by a large margin over a typical day. Note that the details of which particular routes are popular may change over time; the entropy is a systemic measure which does not relate to individual edges. The variation of the proportional distribution of the bikes to origin stands varies even less (closer to 4% for London). Similar calculations for stand occupation (not derivable from these datasets) yield large variations in entropy as commuters redistribute bikes from one highly ordered state to another at peak times (in London, this tidal effect is very clear from stand occupation data - see for example [21]).

London (Figure 3) shows marked concentrations around the major commuter stations Kings Cross and Waterloo in the weekday data (Figure 3a), as these stations “feed” the financial district in the east (and the centre/”west end”) in the case of Kings Cross. There are marked differences between weekdays (Figure 3a) and weekends (Figure 3b), primarily via a de-emphasis of the financial district at weekends, and a relative increase in traffic in west London and the Hyde Park area (the diamond-shaped flow network to the left of the visualization).

In Boston (Figure 4), the link between North and South Stations in the east of the city is one of the strongest features in the weekday data (Figure 4a); in the west of the city (directly south of the river) there is a fairly strong east-west flow (along Commonwealth Avenue, a station with a high concentration of both Boston University locations and metro stations). At weekends (Figure 4b), the flows to and from North and South station become less dominant, consistent with this relating to commuter behaviours.

Denver (Figure 5) shows very uneven clusters of stands, and the weak flows between the downtown areas to the northwest of the map and the areas to the east and southeast could be linked to the large geographical separations. The flows around the University of Denver campus in the southeast of the city are much more important in the weekday data (Figure 5a) than the weekend (Figure 5b), and at the weekend self-journeys relating to one or two stands are the most popular journeys, suppressing the other routes in the visualisation.

Minneapolis’s weekday flows (Figure 6a) are dominated by movements between two stands named “Social Sciences” and “Kolthoff Hall” - further investigation reveals that this flow occurs across a bridge connecting the two halves of the University of Minnesota Twin Cities (UMN) campus, and so this weekday flow could be largely accounted for by students travelled between lectures and classes. The main apparent difference over the weekend (Figure 6b) is that this flow all but disappears, and so could be related to teaching activities; but a self-journey in the southeast of the city dominates.

Washington DC (Figure 7) shows less dramatic usage changes at the weekend (Figure 7b); during both weekdays and weekends, Washington D.C. itself (which lies in the north part of the network, between the two rivers) is separated from the two more geographically distant cluster in Arlington (south and slightly west on the map) and that the stands in Anacostia (southeast from Washington D.C. and over the river) are much less used. South Arlington/Pentagon City is less well used at the weekend (Figure 7b) than during the week (Figure 7a), consistent with a predominately commuter-led usage.

The systems show similarity in the distribution of journey displacements and durations (13), despite differing climates and spatial extents; Table 2 summarises median travel durations and stand proximities. In the weekday data, Denver shows a much larger median travel distance, despite not being one of the schemes with a larger spatial extent. All of the schemes exhibit a larger median proximity at the weekend apart from London and Washington D.C., even though all schemes show an increase in median journey duration. Notably, this duration change is smallest for Washington D.C. and London, perhaps suggesting that weekday and weekend behaviours are less similar than in other cities. For example, the presence of the Hyde Park cluster in weekday and weekend London data points (Figure 3) reinforces the possibility of leisure and tourism users in both weekday and weekend periods. For each city, the time-behaviour shows consistency with the proximity function, in that there is a nonzero mode; it makes little sense to cycle for a very short distance (much less than 1km), so the function rises to a maximum before decreasing for larger distances/times.

Figure 15 shows the outstrength distribution {O_i} of the five systems; the data is divided by the maximum outstrength of each scheme, in order to compare them directly. Instrength or combined strength (O_i+D_i) could equally well be displayed, as over these aggregation time periods O_i∼D_i (although this is obviously not the case instantaneously). This figure displays a rank/value distribution [22] (which can also be related to the cumulative frequency distribution). Because London’s system has many more stands than the other cities, it exhibits a marked long tail. What is more surprising is the apparent similarity of outstrengths in the top 50 stands in the weekday data (inset, Figure 15a). London, Minneapolis, Boston and Washington D.C. all exhibit similar rank/outstrength distributions in their most used stands. At weekends (inset, Figure 15b), the schemes also appear similar, with the possible exception of London, which has a “flatter” distribution than the other cities. A power law fit does not seem plausible for these datasets – each has a complex structure with linear and nonlinear regions. Nor does it convincingly resemble a log-normal distribution. Ranked edge weight is shown in Figure 16 – these curves do not conform readily to a power law or log-normal behaviour, although the curves imply a quadratic dependence of log-edge weight on log-rank.

Communities are shown in subfigures c-d of Figures 3–7. In the London weekday data (Figure 3c), one can broadly identify groups corresponding to central-east London (where financial services are traditionally based), central-west (more associated with retail), east and west. These seems to cross the river, whereas the weekend data (Figure 3d) exhibits clusters which appear to form more distinct geographical regions, and which are more divided by the river.

In Boston, the weekday data (Figure 4c) shows a distinct east-west split, possibly the eastern region corresponds to commuters to the civic and financial district that lies in the northeast corner of Boston, but a time-analysis would be needed to elucidate that behaviour. Weekend clusters (Figure 4d) are more complex, with a central-south grouping as well as a community in the northeast. For Denver (Figure 5c-d), the clusters shown demonstrate geographical overlap; the weekend data fragments into 4 clusters with no distinct geographical character.

Minneapolis’ weekday data (Figure 6c) show a series of clusters, including one corresponding to the university campus area; a cluster of similar size to the northwest; and a larger group which covers most of the city but lies predominantly to the east. The larger group to the southwest of the centre may be linked the “the wedge”, a region of Minneapolis favoured by young professionals. If so, this cluster could highlight commuters from “the wedge” to downtown, and its associated businesses and entertainments. Weekend data (Figure 6d) splits the city into 3; a large, generic cluster; a small southwestern cluster; and one in the centre/CBD. Washington D.C. (Figure 7c-d) shows a number of complex and geographically overlapping communities. A southeastern cluster (just to the north of the river) sits within the Capitol Hill area; the cluster to the northwest is close to Georgetown and George Washington universities, and may be linked to student movement. These groups appear within both weekend and weekday data. The central cluster covers a range of residential and commercial areas, so it is difficult to speculate on its significance; it may represent commuter behaviours or people travelling to meetings. Again, a time-dependent network analysis might shed some light on this.

A trend that is strongly apparent in the spatially scaled/residual networks (subfigures a-b of Figures 8-12) is that edges in the city centres are less used that would be expected – the red regions occur in the centre of each of the cities. This could be in part due to a higher density of stands, resulting in the same amount of traffic being spread over a larger number of potential routes; the “gravity” model incorporate codependence on stand activities and hence implicitly on activity density, but it is possible that the correct functional form is not the simple linear one used.

In weekday London (Figure 8a), Waterloo Station predominantly creates flows which terminate in the centre-east of London, known as the City, where the financial services industries are based, and flows perpendicular to that motion (i.e. northwest/southeast) are strongly under-represented. This points to the simplicity of the assumptions built into our spatial interaction model; a more accurate model might incorporate office space or job location. However, such a model would also need to account for user demographics or land use at the source and destination stands. Weekday and weekend data (Figure 8b) show that journeys within Hyde Park in the west are over-represented, which would be consistent with tourism and leisure usage. The large flows to and from Waterloo and Kings Cross station that we believe to be commuter-driven flows are notably absent in weekend data. The spatial communities for weekday data (Figure 8c) show a fairly distinct community around Hyde Park and one in the centre of London with smaller communities in the east and south of this group. The eastern community seems to be contained within the financial district, which could point to inter-business transit. Weekend spatial communities (Figure 8d) show a more complex pattern, retaining a group around Hyde Park and identifying new groups in West London.

In Boston (Figure 9a-b), we see flows in the western part of the city that we associate with the region around Boston University (BU); during the week (Figure 9a) we see stronger flows to and from South and North Station than at the weekends (Figure 9b). The spatial community detection picks out the western BU locations in the weekday data (Figure 9c) as well as identifying North and South Station as separate clusters – suggesting their characteristics are unique. This may be due to use of Euclidean distance measures and their unusually high flow volumes – future work could use other community methods, or cosine distance to effectively normalize flows vectors.

Denver’s weekdays (Figure 10a) show a strong overrepresentation from the university area, absent at weekends (Figure 10b). The resulting communities (Figures 10c-d) are correspondingly complex. In weekday Minneapolis (Figure 11a), similarly, the intracampus journey is highlighted; this scaling reinforces the message that this route is much more popular than is explained by a spatial interaction model allocation of cycle journeys, even when weighted for the high stand proximity and stand volumes (in essence, maximum entropy methods spread the flows from a stand evenly across all possible outputs, weighted for destination proximity and destination flow volumes. For these unusually popular journeys, they dominate stand popularity and not vice versa; so an even distribution of the source stand’s outputs is not an appropriate analytical approach). It’s likely that students have to travel (e.g. between lectures) more frequently than other sectors of society, overweighting this stand pair. This route is still present at weekends (Figure 11b), but is less dominant; southwestern portions of the city exhibit high volumes. Communities in the weekday data (Figure 11c) tend to cluster around the centre, with the rest of the city being identified with one community. Weekend data (Figure 11d) yields a more complex picture. Washington D.C. (Figure 12a-b) show some small clusters of reciprocal journeys suggesting some hyperlocal usage patterns not identified in this paper; Pentagon City/South Arlington appears again as a predominantly weekday feature (Figure 12a). Weekday data yields a spatial cluster in the centre-west (Figure 12c) and a series of unique stands. Weekend data (Figure 12d) highlights more geographically distinct communities to the centre-east and centre-west.