Demand-driven design of bicycle infrastructure networks for improved urban bikeability

Cyclist route choice model

The benefit of bike paths fundamentally depends on their usage and in turn on the routes of cyclists. We map cyclists’ route choices to a shortest-path problem on a preference graph G = (V, E) with N = ∣V∣ number of nodes (intersections) and M = ∣E∣ number of edges (street segments). We derive the preference graph G from the physical street network G^street. Both graphs share the same set of nodes V. Each edge e_ij ∈ E in the cyclist preference graph represents a street segment that connects intersections i, j ∈ V and is assigned a perceived distance

Here, ({l}_{ij}^{{{{rm{street}}}}}) denotes the physical length of the corresponding street segment ({e}_{ij}^{{{{rm{street}}}}}) in the street network and ({p}_{ij}in {{p}_{ij}^{B},{p}_{ij}^{0}}) is a penalty multiplicador summarizing cyclists preferences against riding along the street segment e_ij. The set of street segments equipped with bike paths E_B ⊆ E contains street segments e_ij ∈ E_B without distance penalty ({p}_{ij}^{B}=1). Street segments that are not in this set e_ij ∉ E_B have penalty factors ({p}_{ij}^{0} > 1). The value of these penalty factors ({p}_{ij}^{0}) may depend on different characteristics of the individual street segments, representing the perceived safety or convenience for cycling.

Adopting this perspective of a cyclist preference graph, we take cyclists to choose their route based on the shortest path ({{{Pi }}}_{ito j}^{* }={{{rm{argmin}}}}left[{L}_{ito j}({{{Pi }}}_{ito j})right]) between their origin i and destination j, minimizing the perceived trip distance

over their potential paths Π_i→j. Effectively, cyclists choose the most direct path to keep the physical distance of their trip as short as possible but accept detours to avoid busy streets and use bike paths or low-traffic residential streets as alternative routes (Fig. 1).

a, If all major streets (thick edges) are outfitted with dedicated bike paths (thick blue lines), cyclists choose the most direct route (1, solid black arrow) from their origin (pin) to a destination (flag) over alternative paths (dashed arrows). b, If only some major streets are equipped with a bike path, cyclists avoid busy roads without a bike path (thick gray lines) and may prefer a short detour (2, solid black arrow). c, If none of the streets have dedicated bike infrastructure, cyclists movimiento the distance and safety of their route choices and may prefer long detours (3, solid black arrow) via low-traffic residential streets (thin gray lines) to more direct routes with high car traffic.

This simplified route choice model enables efficient calculation of route choice decisions, particularly compared with more complex stochastic models^32,33. To illustrate the concept, we focus here on the effect of the street type and take the penalties ({p}_{ij}^{0}) of a street segment to depend on the volume of car traffic on the respective segment, where higher penalties correspond with larger car traffic volumes (that is, a lower perceived safety or convenience; see Methods for details). In principle, the approach can be extended to include additional factors (see Supplementary Note 1 for a brief discussion) such as slopes, (left) turns or crossings by appropriately modifying the cyclist preference graph (for example, adding more edges with a penalty for left turns) as well as more complex route choice models or heterogeneous preferences among cyclists^27,28.

Network generation

We describe the bike path network of a city as a subgraph G_B = (V, E_B) ⊆ G^street of the city’s street network, in which each street segment e_ij ∈ E may (e_ij ∈ E_B) or may not (e_ij ∉ E_B) be equpped with a bike path. Even in this simple binary model, the number of possible bike path networks G_B scales exponentially with the number M of edges in the street network, as each street segment may or may not be equipped with a bike path (there are thus 2^M possible subgraphs). Testing all of these networks is impossible for real-world cities in reasonable time (see Supplementary Note 2 for a more detailed description of the underlying optimization problem). Recent approaches utilize forward network percolation models to construct bike path networks^24,26 or apply percolation models to a fixed cyclist flow²⁵ to find efficient networks.

Here we employ a complementary approach that follows the idea of pruning links from a network—as previously employed in network community-detection algorithms³⁴ and to study the structure of aviation networks¹⁶. Specifically, we create a sequence ({{{G}_{B}(M^{prime} )}}_{M^{prime} }) of bike path networks where (M^{prime} in {0,1,ldots ,M}) street segments are outfitted with a bike path (see Fig. 2): we start from an optimal bike path network in which every street segment is equipped with a bike path (E_B(M) = E) such that there is no penalty for any street segment (({p}_{ij}={p}_{ij}^{B}=1) for all edges). We then compute the route choice decisions of the cyclists in their preference graph G (as described above) on the basis of their demand distribution n_i→j, which denotes the number of cyclists traveling from nodes i to j. To construct the family of bike path networks, one by one we remove the least important bike path ({e}_{ij}^{* }(M^{prime} )) from the network, ({E}_{B}(M^{prime} -1)={E}_{B}(M^{prime} )setminus {{e}_{ij}^{* }(M^{prime} )}), adjusting the penalty of that street segment from ({p}_{ij}={p}_{ij}^{B}=1) to ({p}_{ij}={p}_{ij}^{0} > 1) in the cyclist preference graph G. We quantify the importance of a bike path ({e}_{ij}in {E}_{B}(M^{prime} )) in the current state of the bike path network (with (M^{prime}) remaining bike paths) as the product ({p}_{ij}^{0},{n}_{ij}(M^{prime} )) of the penalty ({p}_{ij}^{0}) (if the street had no bike path) and the number of cyclists that using that street segment ({n}_{ij}(M^{prime} )). The product represents the graph-theoretical weighted betweenness centrality of the edge in the cyclist preference graph. This approach minimizes the negative impact of each removed bike path on the perceived distance of the cyclists in the current bike path network. After each change to the cyclist preference graph G, we update the route choice decisions of the cyclists, ensuring that the algorithm continually adapts to the cycling demand given the currently available set of bike paths ({E}_{B}(M^{prime} )). The process terminates with an empty bike path network ({G}_{B}(0)=(V,{{emptyset}})) merienda all of the bike paths have been removed.

a, Block diagram of the algorithm. b, Illustration of bike path networks ({G}_{B}(M^{prime} )) with different numbers (M^{prime}) of street segments equipped with bike paths. Edges represent major busy streets (thick lines) or minor residential streets (thin lines), and whether the street is equipped with a bike path (blue) or not (gray). The black dotted lines indicate cyclists’ route choices. (i) We start from a full bike path network G_B(M) = G, in which all M street segments of the network G are equipped with a bike path ((M^{prime} =M)). (ii) We first remove all of the bike paths that are not used by any cyclists, n_ij = 0, leaving us with the smallest subgraph G_B(M₀) that still optimally serves the given demand. (iii, iv) We then remove the least important edges one by one, defined by the smallest product ({p}_{ij}^{0},{n}_{ij}) of the penalty multiplicador and the number of cyclists using the bike path, updating the cyclists’ route choice and recording one network ({G}_{B}(M^{prime} )) for each number (M^{prime}) of bike paths. (v) The algorithm terminates with an empty bike path network G_B(0) merienda all of the bike paths have been removed.

See Supplementary Note 3 for a discussion on the computational runtime of the network generation.

In contrast to iteratively adding bike paths to an initially empty graph and building on the suboptimal cycling routes in networks with few bike paths, this procedure creates bike path networks adjusted to ideal cycling conditions; for example, it keeps bike paths that may not be important in the perfect network if the cyclists start to use them more heavily as the other bike paths are removed (see Supplementary Note 4 for details).

Inputs to our algorithm are: (1) the street network G^street, (2) the penalty factors ({p}_{ij}^{0}) for each street segment not equipped with a bike path, (3) the demand distribution n_i→j and (4) the cyclists’ route choice model. These parameters may either be as-is empirical values, or planned/desired ideal values (for example, describing the desired or predicted demand for cycling in a city). The latter application might be particularly relevant for planning bike path network extensions if urban quarters develop or are repurposed.

Application

We test the proposed algorithm using data from two German cities: Dresden and Hamburg (see Fig. 3). We take the street networks of both cities from OpenStreetMap (OSM)³⁵—using the street classification as a proxy for their expected traffic load—as input data; we also take data from regional bike-sharing services to model the cycling demand. We fix the penalty factors against those of physically protected bike path infrastructure based on the street type classification decoded in OSM (see the Methods for a detailed description of the data).

a, Street network (gray lines), bike-sharing-station locations and station activity (colored circles) in Dresden between November 2017 and March 2020⁴⁴. b, Distribution of station usage, measuring the combined number of in- and outgoing trips per station in Dresden. Bike-sharing usage is strongly heterogeneous and is dominated by two heavily used stations (pink) along the north–south axis between the central train station (center) and university campus (south). Station density reflects this usage pattern and is highest in the central city (north/center) and near the university campus (south). c, Street network, bike-sharing-station locations and station activity in Hamburg between January 2014 and May 2017⁴⁵. d, Distribution of station usage, measuring the combined number of in- and outgoing trips per station in Hamburg. The station activity distribution is homogeneous across a broad spectrum of total number of trips. This homogeneous usage is also reflected in a more homogeneous distribution of bike-sharing stations, which is slightly denser only in the inner city (south).

Source data

The two cities are representatives of two archetypes of regional demand constellations: spatially homogeneous all-to-all demand and confined few-to-few demand. Bike-sharing usage patterns in Hamburg indicate a regional demand structure that refers to the first archetype (see Fig. 3c,d), whereas corresponding data for Dresden hint at the latter archetype, which is reflected by the dominance of trips between the university and main train station (see Fig. 3a,b).

Algorithmic generation of bike path networks

We generate families of bike path networks ({{{G}_{B}(M^{prime} )}}_{M^{prime} }) for both cities. We chose a network in which all primary and secondary (P + S) street segments (as per their classification in OSM) are equipped with bike paths and compare this network to our generated network with the same total length ({{Lambda }}(M^{prime} )={sum }_{ein {E}_{B}(M^{prime} )}{l}_{e}^{{{{rm{street}}}}}) of bike paths such that ({{Lambda }}(M^{prime} )={{{Lambda }}}_{{{{rm{P+S}}}}}). Taking the installation and maintenance cost of the bike path network proportional to its length, we thus compare networks with the same budget. Due to some antiparallel one-way streets, our algorithm may place slightly more bike paths along other streets than effectively exist in the P + S network. As we assume bidirectional paths, our algorithm may equip only one of the antiparallel streets with a bike path, instead of both as in the P + S network.

Figure 4 illustrates both types of networks for Dresden and Hamburg. The network generated by our algorithm largely coincides with the primary and secondary roads due to the high penalty if bike paths are removed; however, we observe strong differences in the density of the bike path coverage. Especially in Dresden, the resulting bike path network is much denser along the central north–south axis of high station density and bike-sharing usage, indicating that our algorithm correctly adapts the network to the input demand conditions (Fig. 4c). The differences for Hamburg are smaller due to the comparatively homogeneous demand across the city, although our algorithm introduces bike path shortcuts through residential areas in cases of high demand or to connect stations to the bike path network.

a–f, Networks for Dresden (a–c) and Hamburg (d–f) with bike-sharing-station locations (purple, compare with Fig. 3). a,d, Bike path networks generated by the algorithm (blue) with the same total length as all of the primary and secondary streets, ({{Lambda }}(M^{prime} )={{{Lambda }}}_{{{{rm{P+S}}}}}). b,e,Networks for the scenario in which only the primary and secondary streets (as per their OSM classification) are equipped with bike paths (black). c,f, Comparisons between both networks. The networks generated by the proposed algorithm largely coincide with the primary–secondary networks (orange edges in c and f) but more accurately reflect the input demand structure by also keeping highly used tertiary or residential streets equipped with bike paths and exhibiting a higher density of bike paths in high-demand areas.

Source data

To quantitatively compare the bike path families for both cities, we normalize the length of bike paths (lambda ={{Lambda }}(M^{prime} )/{{Lambda }}({M}_{0})) with respect to the length Λ(M₀) after removing all of the unused bike paths (see Fig. 2b). We define the total perceived distance of all trips in the cyclist preference graph as

where L_i→j(Π, λ) = ∑_e∈Πl_e(λ) (compare with equation (2)). Here, l_e(λ) denotes the effective length of the street segment e in the cyclist preference graph G, given a set of bike paths ({E}_{B}(M^{prime} )) with normalized length λ (that is, including penalties only for those streets from which we have removed the bike path); ({{{Pi }}}_{ito j}^{* }(lambda )) denotes the shortest path in this cyclist preference graph and thus the route chosen by cyclists going from i to j. To compare the total perceived distance across both cities, we measure the overall performance b(λ) of the resulting network as the bikeability

where we again normalize the absolute values to the best- (λ = 1) and worst-case (λ = 0) scenarios; b(0) = 0 describes the network with no bike paths, whereas b(1) = 1 is the optimal network with bike paths along all of the shortest paths. We remark that our definition of bikeability differs from past measures³⁶ in that it quantifies the efficiency of the bike path network with respect to a specific demand distribution. The total difference between the physical trip length of cyclists and the direct shortest paths is only on the order of 10%, consistent with the empirical observations of cyclists’ route choice behavior^27,28.

Figure 5 illustrates the bikeability across the generated sequence of bike path networks. Interestingly, a small fraction of bike paths with a small relative length of λ > 0.1 is sufficient to achieve more than 50% of the maximal bikeability in both cities. The larger area under the bikeability curve for Dresden compared with Hamburg is consistent with the differences in the demand structure between the two cities: we achieve a faster improvement in Dresden due to the more concentrated demand distribution, whereas we have to cover most of the city of Hamburg due to the more homogeneous demand. See Supplementary Fig. 3 for a brief overview of the bikeability of a further twelve cities.

a,c, The resulting bikeability b(λ) (blue, equation (4)) and relative cost of the network (gray) for Dresden (a) and Hamburg (c) as a function of the normalized length of the bike path network λ. b,d, Corresponding fraction of total distance traveled on streets with and without bike paths, in networks with bike paths with the same relative length λ_P+S, and the primary–secondary (P + S) comparison network. Adapting the network to the demand structure achieves over 89% of the distance traveled on bike paths (blue), also reducing the distance traveled on residential streets (light gray). For our algorithm, a negligible fraction of the total distance is cycled on tertiary (dark gray) and secondary (black) roads without a bike path (not visible in the bar chart).

Source data

A comparison with the bikeability of the primary–secondary bike path network with the same relative length λ_P+S of bike paths highlights the better adaptation to the demand structure in our algorithm. The bikeability is already high when all large roads are equipped with bike paths (about 0.87 for Dresden and 0.82 for Hamburg). Yet our algorithm manages to further increase this value to about 0.97 for Dresden and 0.95 for Hamburg (capturing more than 70% of the remaining potential of an optimal network b(1) = 1). Moreover, by adjusting the network to the route choice behavior, cyclists keep to streets equipped with bike paths for more than 89% of their total trip distance, compared with only about 60% in the primary–secondary network (see Fig. 5b,d). A negligible but non-zero fraction of the distance is cycled on tertiary and secondary streets without a bike path. See Supplementary Notes 5 and 4 for a comparison with static and dynamic forwards percolation approaches, respectively.

Impact of demand structure

We attribute the difference between the two cities in the above analysis to the structure of the bike-sharing demand distributions. To quantify the impact of the demand structure on our bike path network families and their bikeability curves, we compare the above results to synthetic bike path networks with homogenized demand. We create these homogeneous demand settings by first distributing demand equally between all stations and then distributing the stations as equidistantly as possible in the street network (see Fig. 6a and Methods).

a, Illustration of the demand homogenization for Hamburg. Starting from the empirical demand (i), we create uniform demand between all stations (ii) and then distribute the stations approximately equidistantly across the city (iii). b,c, Comparison of the bikeability b(λ) (equation (4)) from the empirical (compare with Fig. 5) and homogenized demand data for Dresden (b) and Hamburg (c). The comparatively smaller area between the curves (shaded gray, equation (5)) for Hamburg suggests a more homogenous empirical demand, in line with our observations in Fig. 4.

Source data

Comparing the bikeability curves b(λ) and b_hom(λ) in the empirical and the homogeneous demand settings, respectively, we find a comparatively large difference for Dresden and a much smaller difference for Hamburg (Fig. 6b,c). We quantify these differences by the area

between the two bikeability curves, which describes the impact of the patterns—as well as the structure of the station and demand distribution—on the bikeability (β_hom ≈ 0.015 for Dresden, β_hom ≈ 0.007 for Hamburg; compare with Fig. 6). This confirms our above analysis that the heterogeneous, centralized demand and station distribution in Dresden enhances the bikeability as fewer streets have to be equipped with bike paths to cover a large fraction of the total demand, whereas there is a much smaller effect in Hamburg. A similar approach may be used to quantitatively compare the impact that different street networks, or different types of cycling or desired demand distributions may have on the resulting bike path networks.