Growing Urban Bicycle Networks

Michael Szell∗a,b, Sayat Mimarc, Tyler Perlmanc, Gourab Ghoshalc, and Roberta Sinatraa,b

aNEtwoRks, Data, and Society (NERDS), IT University of Copenhagen, 2300 Copenhagen, Denmark bComplexity Science Hub Vienna, 1080 Vienna, Austria cDepartment of Physics and Astronomy, University of Rochester, Rochester, NY 14627, USA

Abstract Cycling is a promising solution to unsustainable car-centric urban transport systems. However, prevailing bicycle network development follows a slow and piecewise process, without taking into account the structural complexity of transportation networks. Here we explore systematically the topological limitations of urban bicycle network development. For 62 cities we study different variations of growing a synthetic bicycle network between an arbitrary set of points routed on the urban street network. We find initially decreasing returns on investment until a critical threshold, posing fundamental consequences to sustainable urban planning: Cities must invest into bicycle networks with the right growth strategy, and persistently, to surpass a critical mass. We also find pronounced overlaps of synthetically grown networks in cities with well-developed existing bicycle networks, showing that our model reflects reality. Growing networks from scratch makes our approach a generally applicable starting point for sustainable urban bicycle network planning with minimal data requirements.

Introduction tion, and climate, reveals that in the EU alone cycling brings a yearly benefit worth e 24 billion while Cities worldwide are scrambling for sustainable solu- automobility costs society e 500 billion [4]. These in- tions to their inefficient, car-centric transport systems sights provide further impetus for coordinated efforts [1,2]. One promising, time-tested candidate is cycling. to extend cycling infrastructure as one solution to the It is an efficient mode of sustainable urban transport urban transport crisis and to effectively fight climate that can account for the majority of intra-urban trips change [5, 6]. which are primarily short and medium-distance [3]. In practice, however, bicycle infrastructure devel- Cost-benefit analysis that accounts for health, pollu- opment struggles with a particularly pervasive polit- *Corresponding author. Email: [email protected] ical inertia due to “car culture” [7, 8]: For example, Copenhagen took 100 years of political struggles to develop a functioning grid of protected on-street bi- a cycle networks [9] that continues to be split into 300 disconnected components today [10]. Accordingly, the most developed, influential bicycle network planning guidelines, such as the Dutch CROW manual [11], ac- knowledge that building up bicycle networks happens typically through decades-long, piecewise refinements. Unfortunately, given the advanced stage of the plane- b c tary climate crisis [12], humanity has run out of time – there is now no other choice left than making urban transport sustainable as fast as possible [1, 7, 13]. While there has been historical inertia, the ongoing arXiv:2107.02185v2 [physics.soc-ph] 13 Jul 2021 COVID-19 pandemic has prompted several cities to engage in successful accelerated network development, proving that such efforts are indeed possible [14, 15]. A systematic exploration of city-wide, comprehensive development strategies is therefore urgently needed. Figure 1: The state of existing bicycle networks. a We extract street networks from 62 cities covering differ- Although the prevailing, piecewise application of ent regions and cultures; many are considered modern and qualitative policy guidelines in existing bicycle network well developed. b The distribution of city-wide lengths of planning [11, 16] might have a good track record in bicycle tracks indicates negligible existing cycling infras- e.g. Dutch cities and Copenhagen [9], this process lacks tructure that is also c split into hundreds of disconnected rigorous scrutiny: Are the resulting networks truly op- components. See more details in Supplementary Table 1. timal? Can such policies be replicated in other cities?

1 And are there fundamental topological limitations for Minimum spanning tree Triangulation Fully connected developing a bicycle network? Indeed, an evidence- based, scientiﬁc theory of bicycle network development is missing.

There is a growing academic literature on analyzing Investor's Cohesive Traveler's bicycle networks of specific cities, for instance Mon- optimum planar network optimum treal [17], Seattle [18], or recent data-driven approaches Resilient for Bogota [19] or London [20]. While such stud- Economic ies are invaluable in terms of local enhancements and Figure 2: Optimal connected network solutions. data consolidation for a particular place, here we in- Adapted from Ref. [29]. (Left) The investor’s optimal strat- stead focus on a global analysis, in particular on the egy for a connected network is to invest as little as possible, fundamental topological limitations of bicycle network minimizing total link length [10]. Its solution is a minimum development that are relevant for all urban environ- spanning tree, maximally economic but minimally resilient ments, independent of the availability of traffic flow with low directness, inadequate for travelers. (Right) The data [21]. This approach follows the idea of a Science traveler’s optimum connects all node pairs creating all di- of Cities [22] where we study the topological proper- rect routes. This solution is minimally economic, maxi- ties of bicycle networks that are independent of place mally resilient and direct, inadequate for investors. It also has crossing links and is therefore not a planar network. using computational, quantitative methods of Urban (Center) A both economic and resilient, as well as cohe- Data Science [23]. sive planar network solution in-between is the triangulation. In particular the minimum weight triangulation, approxi- The vast majority of cities on the planet has neg- mated by the greedy triangulation, minimizes investment. ligible infrastructure for safe cycling [6]. Indeed, urban transport infrastructure development worldwide has been heavily skewed towards automobiles since Results the 20th century, today featuring well connected networks of streets for motorized vehicles [10]. Rather The starting point for our analysis is the street net- than uprooting the existing infrastructure and replac- works of 62 cities, see Fig. 1(a) and Supplementary ing it with an entirely new one—an economically infea- Table 1. Here, links represent streets and nodes are sible strategy—we investigate how to retrofit existing street intersections. Being embedded in a metric space, streets into bicycle networks. Sacrificing specificity for these constitute planar graphs [30]. We downloaded generalizability, our formulation contains as a starting and processed these networks from OpenStreetMap us- point two ingredients: the existing street network of a ing OSMnx [31] (see Methods). city, and an arbitrary set of points of interests. With Although many of the covered cities are from well these minimal ingredients we explore different growth developed regions, we observe that they have negli- strategies that sequentially convert streets that were gible bicycle infrastructure, Fig. 1(b). Additionally designed for only cars to streets that are safe for cy- these are split into hundreds of disconnected compo- cling [11, 24]. Inspired by the CROW manual [11], the nents, Fig. 1(c), which has previously prompted anal- objective of all explored strategies is to create cohe- ysis of strategies to merge them [10]. Although such sive networks, i.e. well connected networks that cover strategies make sense in cities with already well es- a large fraction of the city area (see Methods). tablished bicycle infrastructure, they are less useful in most other cities. Further, they produce minimum Across the realistic strategies we report a growth spanning tree-like solutions that are economically at- phase that initially leads to a diminishing trend of tractive but lack resilience and cohesion (Fig. 2), and quality indicators, until a critical fraction of streets are they potentially reinforce socioeconomic inequalities by converted, akin to a percolation transition observed in connecting only already developed areas while ignoring critical phenomena and also present in the growth of under-developed ones [21], prompting us here to grow other forms of transportation infrastructure as well as new networks from scratch instead. patterns of traffic [19, 21, 25–28]. In other words, initial investments into building cycling-friendly infras- Growing bicycle networks from scratch tructure leads to diminishing returns on quality and efficiency until the emergence of a well-connected giant Our process of growing synthetic bicycle networks con- component. Once this threshold is reached, the quality sists of four steps, Fig. 3, starting with the street improves dramatically, to an extent which depends on network and the points of interest. For details see the specific growth strategy. We provide empirical evi- Methods; for an intuitive, interactive exploration see dence that the majority of cities effectively lie below the http://growbike.net. threshold which might be hindering further growth, im- plying fundamental consequences to sustainable urban Step 1) Points of interest (POIs). An arbitrary planning policy: To be successful, cities must invest set of POIs is snapped to the intersections of the street into bicycle networks with the right growth strategy, network. We investigate two versions of POIs: i) ar- and persistently, to surpass a critical mass. ranged on a grid, and ii) rail stations.

2 Figure 3: Growing bicycle networks. Explorable interactively at: http://growbike.net. Illustrated here for Paris. Step 1) Points of interest: A set of points of interests (POI, orange dots) is snapped to the intersections of the street network. Shown are grid POIs, alternatively we investigated rail stations. Step 2) Greedy triangulation: The POIs are ordered by route distance and connected stepwise without link crossings. Reached POIs are colored blue. Step 3) Order by growth strategy: One of three growth strategies (betweenness, closeness, random) is used to order the triangulation links from the strategy’s 0-quantile (empty graph) to its 1-quantile (full triangulation), resulting in 40 growth stages. Shown are the ﬁve quantiles q = 0.025, 0.125, 0.25, 0.5, 1. Step 4) Route on street network: The links in the growth stages are routed on the street network. These synthetic bicycle networks are then analyzed for all 62 cities.

3 Step 2) Greedy triangulation. All pairs of nodes approach is the most local approach possible and are ordered by route distance and connected stepwise leads to a linear expansion of a dense as possible as the crow flies. A link is added only if it does not cross network from the topological city center. an existing link. This greedy triangulation is an easily computable proxy for the NP-hard minimum weight 3. Random – adds links randomly and is used as triangulation [32]. It creates an approximatively short- a baseline. This strategy is not just a theoretical est and locally dense planar network [33], and a con- null model but well resembling how cities build nected, cohesive, and resilient network solution min- their bicycle networks in practice, as we discuss imizing investment, therefore satifsying both traveler later. and investor demands, Fig. 2. Step 4) Route on street network. The abstract Step 3) Order by growth strategy. Each of three links in the 40 stages are made concrete: They are growth strategies is used to order the greedy triangula- routed on the street network. These synthetic bicycle tion links from the strategy’s 0-quantile (empty graph) networks are then analyzed for all 62 cities. to its 1-quantile (full triangulation), resulting in a se- quence of growth stages. To study this growth process Different growth strategies optimize dif- in a high enough resolution we split the growth quantiles into 40 parts q = 0.025, 0.05,..., 0.975, 1. The ferent quality metrics three strategies are: We measure several network metrics to assess the qual- 1. Betweenness – orders by the number of shortest ity of the synthetically grown networks and to compare paths that go through a link. It can be interpreted them with existing bicycle networks. These metrics as the simplest proxy for traffic flow (assuming are: length L, length of the largest connected com- uniform traffic demand between all pairs of nodes). ponent (LCC) LLCC, coverage C, POI coverage CPOI, Thus, growing by betweenness is an approach that directness of the LCC D, number of connected com- aims to prioritize flow. ponents Γ, global efficiency Eglob, local efficiency Eloc. We define coverage as the area of all grown structures 2. Closeness – starts with the “most central” node, endowed with a 500 m-buffer, see light blue areas in i.e. the node that is closest to all other nodes. Fig 6(b) for an illustration. Directness measures the From this seed, the network is built up by con- ratio of euclidian distance versus shortest distance on necting the most central adjacent nodes. This the network only for the LCC, while global efficiency

a b c d

e f g h

Figure 4: Different growth strategies optimize different network quality metrics. The three thick curves show the changes of network metrics with growth following three strategies (betweenness, closeness, or random) averaged over all 62 cities for grid POIs. By construction all curves arrive at the same endpoint, but they develop distinctly before that. For rail POIs and individual cities see Supplementary Figures 4-9. Red curves show the car network’s simultaneous decrease of quality metrics if a five times decrease of speed limits is assumed for cars along all affected streets. Grey dotted lines show metrics for the minimum spanning tree (MST) that connects all POIs with minimal investment. Growth of a length, b coverage, c directness, d global efficiency, e length of LCC, f POIs covered, g connected components, h local efficiency. The yellow arrow highlights the substantial dip in directness until the critical threshold which is more pronounced for random growth than for betweenness growth.

4 a = 0.025 = 0.1 = 0.2 provides a similar but network-wide measure that ac- qB qB qB counts for disconnected components [34]. See Methods for more details.

We ﬁrst investigate how these quality metrics change throughout the growth process averaged over all cities, Fig. 4. The three thick curves show the change of the metrics with growth following the three strategies (betweenness, closeness, or random) for grid POIs. b Similar results hold for rail stations, see Supplemen- tary Figures 4-6. By construction all curves reach ex- actly the same point at the 1-quantile (full triangulation), but their development diﬀers substantially before that. The minimum spanning tree (MST) solution is depicted as a baseline (grey dotted lines). This is the most economic connected solution that reaches all POIs, see Fig. 2; therefore any connected solution that c reaches all POIs must be at least as long as the MST.

From Fig. 4(a) we observe that length grows linearly for closeness and slightly faster for the other strategies because closeness prioritizes close links which typically have similar length, while betweenness and random growth selects distant links earlier. Random growth adds single links scattered randomly across the q city and therefore has the fastest growth of coverage, B Fig. 4(b), followed by the betweenness-based strategy, while closeness leads to a linear growth. Directness, Figure 5: Network consolidation: Bicycle network Fig. 4(c), displays a large dip for random growth, from growth has a dip of decreasing directness. a Three D 0.73 down to D 0.48 at the 0.345-quantile, early stages of betweenness growth in Boston. b Directness initially decreases due to the initial largest connected com- and≈ a smaller dip from≈D 0.79 to D 0.63 for ≈ ≈ ponent (LCC) being short (see the example qB = 0.025 for betweenness growth at the 0.1-quantile. Directness Boston). As the LCC grows in a tree-like fashion, direct- starts lower for closeness growth, around D 0.58 ≈ ness decreases to a minimum (qB = 0.1), then consolidates but quickly overtakes the other strategies at the 0.05- into a single connected component with cycles (qB = 0.2). quantile. Global efficiency, Fig. 4(d), starts at a high The process is similar for most other cities as shown here level, around E 0.7, and grows slightly un- for Montreal, Mumbai, Paris, Tokyo, and also holds for ran- glob ≈ til Eglob 0.82 for both betweenness and closeness. dom growth, see Supplementary Figures 4,6,7,9. c We find Random≈ growth starts instead much lower, around mixed results for global efficiency: Mumbai and Montreal display a single jump, Tokyo is flat, while Boston and Paris Eglob 0.33. The length of LCC, Fig. 4(e), is almost identical≈ as the growth of length for betweenness shown an initial dip before increasing. and closeness because the LCC makes up most of the network here. However, the LCC in random growth has a sigmoid growth pattern as it takes longer for the Network consolidation and non- components to connect, Fig. 4(g). Coverage of POIs, monotonic gains in quality Fig. 4(f), is similar to coverage but more pronounced for random and betweenness growth. On average all The dips observed in directness, see yellow arrow in POIs are covered before the 0.6-quantile. Finally, lo- Fig. 4(c) for random growth, are akin to a phase transition in a percolation process from a disconnected set cal efficiency, Fig. 4(h), is steady around Eloc 0.22 for closeness, but grows fast for both betweenness≈ and of components to a sudden emergence of a giant connected component, as known for e.g. random Erd˝os- random growth from around Eloc 0.05. ≈ Rényi networks [25]. Similar transitions have been observed in generalized network growth [35,36], including To summarize, the different growth strategies opti- in various random spatial networks [30] and sidewalk mize different quality metrics and come with different networks [28], and similar flavors of bicycle network tradeoffs: 1) Use betweenness growth for fast coverage, growth [19]. Figure 5(a) and (b) illustrates this con- intermediate connectedness and directness, and low lo- solidation process for individual cities: Links are added cal efficiency. 2) Use closeness growth for optimal con- one by one, growing the largest connected component nectedness and local efficiency but slow coverage. 3) until a critical threshold at the curve’s minimum (at Use random growth for fastest coverage but low di- qB = 0.1 for Boston), at which the largest connected rectness, connectedness, and efficiency. component consolidates the majority of the network

5 and starts forming cycles that in turn increase direct- a ness. Because connectedness increases around the critical threshold, the evolution of connected components is inverse to the evolution of directness, Fig. 4(g). While the global efficiency averaged over all cities shows an initial increase followed by saturation, see Fig. 4(d), we find mixed trends at the level of individual cities: Mumbai and Montreal track the average trend, Tokyo has a flat global efficiency, while Boston and Paris show a dip before the critical threshold is reached with rapid gains thereafter (Fig. 5(c)). b LCC This network consolidation has important implica- tions for policy and planning. The point at which the transition happens represents substantial investments into building the network. Stopping investments and growth before this point leads to a net loss in investment as measured by infrastructure quality. Indeed, pushing past this threshold leads to substantive gains.

Real Milan Synthetic Milan The effect of bicycle network growth on the street network Figure 6: Synthetic bicycle networks perform several times better than existing ones. a We plot the While it is beneficial for a city to grow its bicycle net- distributions (over cities) of the ratios Msyn/Mreal between work, it is important to ask how this growth affects the network metrics of synthetic and existing topologies fixed network of streets used by cars. The magnitude of this at same length (Lsyn = Lreal), for betweenness growth and effect depends not only on the network topology, but grid POIs (for all other growths see Supplementary Fig- also on the concrete bicycle infrastructure being imple- ure 2). Synthetic networks have on average 5 times larger mented: shared spaces, unprotected cycle lanes, pro- LCCs, 3 times the global efficiency, and higher local effi- tected cycle tracks, bicycle streets, their width, and so ciency. Existing networks only tend to have better coverage on. To consider these factors, leading bicycle planning because they are more scattered. b Illustration of high coverage (light blue area) due to extreme scattering and low manuals consider a plethora of local variables [11, 37], length of LCC (dark blue sub-network) for Milan’s existing for example road category, speed limit, volume of the bicycle network, versus its synthetic version at same length motorized traffic, or car parking facilities. Therefore, (185 km at qB = 0.425). The LCC for synthetic Milan is to be conservative in our estimations, here we consider the whole network. the strongest possible effect of new bicycle infrastructure on streets apart from complete replacement. We assume that along the affected road sections the effec- ligibly from E 0.11 to E 0.10 (red curve in tive speed limit for cars will be reduced by a factor of loc ≈ loc ≈ 5, i.e. to walking speed. In practice this assumption Fig. 4(h)). Growing the bicycle network has no effect would be consistent with, for example, all infrastruc- on the length and coverage of the automobile network, ture being built as a child-friendly “fietsstraat” or liv- given that cars can still access all points on the street ing street, i.e. as a shared traffic space where cyclists network, albeit in a longer time than they would with- and pedestrians have priority and cars are tolerated to out the bicycle infrastructure. We find almost identical pass through in walking speed [11]. behavior for the closeness and random growth strate- Given this strong constraint, we find the metric that gies (Supplementary Figure 3). is most affected is directness. Note that for calculating One of the effects of modifying the street infrastruc- directness, we modify the affected road section lengths ture is the redistribution of load on the street inter- by a factor of 5, so this accounts for not just spatial, but sections, measured by the betweenness centrality. It spatio-temporal directness [11]. It decreases approxi- has been shown that while the global distribution of mately linearly with the bicycle network’s growth, from the betweenness centrality remains unchanged due to D 0.8 to D 0.6, as the network is grown using the change in density of streets, the spatial distribution betweenness≈ strategy≈ (red curve in Fig. 4(c)). In other and clustering of the high betweenness nodes tend to words, in the absence of any bicycle infrastructure, car- change, thus redistributing areas of higher traffic [38]. routes deviate by around 25% from the Euclidean dis- Two measures to quantify this effect are the spatial tance between any origin-destination, whereas once the clustering and the anisotropy of the high betweenness full bicycle infrastructure is established, this increases nodes (see Methods). We find a slight increase (around to around 66%. At around the 0.4 betweenness quan- 5%) in spatial clustering and anisotropy for nodes in tile, the directness of the bicycle network exceeds that the 90th percentile of betweenness values but the ef- of the car network. The global efficiency decreases fect is marginal (Supplementary Figure 3). Thus bicy- from around Eglob 0.75 to Eglob 0.6, (red curve cle networks do not substantially affect the congestion in Fig. 4(d)), while≈ the local efficiency≈ decreases neg- properties of existing street networks.

6 Comparing synthetic with existing net- a b work metrics qB = 0.05 Although the growth processes described here are somewhat artificial, given the lack of accounting for practical limitations of bicycle network design—street Existing, not grown Grown and existing width, incline, or political feasibility for instance—it is Grown, not existing nevertheless prudent to compare the synthetic network c with existing bicycle networks to gauge their general qB = 0.20 correspondence. To have a fair comparison in terms d of length (which is a proxy for cost), we first select all cities that have a protected bicycle network with shorter length Lreal than the fully grown synthetic network Lsyn (42 out of 62 cities), and for each of them we fix the growth quantile where the synthetic length is equal to the real length, Lsyn = Lreal. Given this set of bicycle network pairs – real versus synthetic at same length – we then measure the ratio Msyn/Mreal between the synthetic quality metric Msyn and the quality metric of the existing infrastructure Mreal. The results for the metrics of coverage, local efficiency, global effi- Figure 7: First stages of synthetic growth recreate ciency, and length of LCC are reported in Fig. 6 (a). existing networks. Shown are results for rail station We find that synthetic networks have on average 5 POIs averaged over all cities. Same legend as Fig. 4. a times larger LCCs, 3 times the global efficiency, and Growth by betweenness starts with high, then decreasing higher local efficiency. Existing networks only tend to overlap with existing protected bicycle infrastructure. In- have better coverage because they are more scattered, set: The effect is especially strong in cities with well de- as illustrated in Fig. 6 (b) for Milan which has 230 veloped on-street bicycle networks such as Copenhagen. disconnected components. Milan’s scattered network Here the growth algorithm starts with over 80% overlap. provides an important lesson: Mere measures of to- b Map of this high overlap in Copenhagen at the quantiles tal length or coverage are misleading when it comes qB = 0.05 and c qB = 0.20. d The overlap with bike- to an efficient and safe infrastructure if the network able infrastructure has a notable effect only for growth by closeness due to traffic-calmed city centers: With increasing is not well connected. Instead, if city planners were distance from the city center, overlap falls. to develop and implement bicycle networks holistically, considering a city-wide rather than piece-wise local approach, much higher quality infrastructure could be de- proximate reality, we measure the percent overlap of rived to the benefit of the residents. synthetic infrastructure with existing bicycle infras- For completeness we also compare our results to the tructure. Figure 7(a) and (b) report for rail station closeness and random growth approaches, Supplemen- POIs the average overlaps for protected cycle tracks tary Figure 1(c) and (e). For closeness we find almost and for bikeable infrastructure respectively, where the same result as for betweenness, only with notably bikeable infrastructure is defined as the union of pro- worse coverage which is to be expected given how close- tected tracks and streets with speed limits 30 km/h ness grows the covered area as slowly as possible. For (see Methods). Results are qualitatively similar≤ for the random growth approach we find the same cover- grid POIs, see Supplementary Figures 7–9. age as existing infrastructure, and around 2 times the For protected infrastructure, Fig. 7(a), we find that global efficiency and length of LCC. At first blush, this growth by betweenness (solid line) starts with around implies that even a naive random growth strategy can 6% overlap on average, then decreases fast reaching perform better than existing ones. However, this could 3.5%. We find a similar behavior for random growth be due to a number of reasons: For instance, in the (dotted line) but with a smaller effect. We find no clear random growth process described here, segments are effect for growth by closeness (dashed line). These ob- added over at least 1.7 km in each step, whereas in real servations suggest that cities take into account flow cities, segments are added in a more scattered fashion (betweenness) when building their cycling infrastruc- and in varying lengths. Further, cities can have non- ture, and that rail stations play some role – otherwise negligible off-street bicycle tracks, for example through there would be no effect in the random growth. The be- parks, a feature not considered in our analysis. tweenness overlap effect is especially strong in Copen- hagen, Fig. 7(a) inset, which has a well-developed, co- Comparing synthetic with existing net- hesive on-street bicycle network. Figure 7(b) shows the work overlaps synthetic growth stage at the second step, qB = 0.05. Remarkably, at this step over 80% of the links sug- To gain a better understanding into how the synthet- gested by the synthetic network model do already ex- ically grown parts compare to existing infrastructure, ist in reality. Even at qB = 0.20 there is almost 70% and thus the extent to which the growth models ap- overlap, Fig. 7(c). These values are far higher than ex-

7 pected by chance: Given the length of Copenhagen’s By no means do they suggest concrete recommenda- on-street cycle tracks we would expect only at most tions for new bicycle facilities, as a vast array of local 24% overlap in a random link placement. idiosyncrasies (second order effects) would need to be The overlap for bikeable infrastructure looks differ- accounted for [11, 37], including: road category, speed ent, Fig. 7(d). Here, there is no clear effect for be- limit, volume of motorized traffic, or aspects of com- tweenness and random growth, but a very clear effect fort [40]. Despite the importance of these aspects, a for closeness, which starts with high values (above 20% transport network’s geometry is its most fundamen- on average), falling slowly to 16%. This observation is tal limitation [41] which is the reason we explored it consistent with cities preferentially installing low speed first. Although our approach here is not yet aiming limit areas in their city centers. to provide concrete urban design solutions, it could be useful for planning purposes for easily generating an initial vision of a cohesive bicycle network – to be re- Discussion fined subsequently. By publishing all our code as open source we facilitate such future refinements. Our min- We grew synthetic bicycle networks in 62 cities fol- imal requirements on data are a deliberate limitation lowing three different growth strategies all aiming to we impose for our framework to be applicable to data- generate a cohesive network, i.e. a network that is well scarce environments and thus to a large part of the connected and covers a large fraction of the city area. planet [42]: no lane widths, inclines, traffic flows, etc. Studying the resulting networks we found a consistent are needed to optimize network topology. critical threshold affecting directness in all cities and Starting from rail station POIs seems a reasonable global efficiency in some of them, for the two most re- approach, however care has to be taken to not am- alistic strategies of growth by betweenness and random plify existing biases that are well-documented in the growth. This sudden network consolidation has a fun- transport planning profession [43–45]: For example, damental policy implication: To be successful, cities planning bicycle infrastructure only along metro sta- must invest into bicycle networks persistently, to sur- tions that were built following elitist or racist biases pass short-term deficiencies until a critical mass of bi- would reinforce them, neglecting under-served regions cycle infrastructure has been built up. Further, from a and their inhabitants even further. The strength of topological perspective, cities should avoid traditional our POI approach lies in its arbitrariness that can by- “random growth-like” strategies that follow local, step- pass such issues: Grid POIs implement equal coverage wise refinement. Such strategies substantially shift the and could be a starting point, to be refined carefully critical threshold, thereby hold up the development with e.g. population density or traffic demand mod- of a functional cycling infrastructure which could fuel els [21, 46]. The biggest limitation of our approach is adversarial objections to bicycle network expansions the sole focus on retrofitting street networks for safe along the lines of “We already built many bike tracks cycling. This approach has some issues because it only but nobody is using them, so why build more?” As we considers on-street but no off-street bicycle infrastruc- have shown, it is not a network’s length that matters ture. We discuss the technical details of this limitation, but how you grow it. mostly relevant for concrete bicycle network planning By comparing metrics and overlaps of synthetic with in low urban density, in Supplementary Note 1, con- existing networks, we gained an insight into the re- cluding that future research on bicycle network growth alism of our models. Our observations suggest that should consider off-street solutions wherever possible. growth processes of existing protected bicycle networks Finally we discuss the effect of growing bicycle net- contain a strong random ingredient and a detectable works on limiting street networks for car traffic. It is consideration for flow (betweenness) and rail stations. unclear whether to interpret a change from directness The random ingredient can be explained by the tra- D 0.8 to D 0.6 as substantial or inconsequen- ditionally slow-paced urban planning processes arising tial.≈ We deem it≈ more important to discuss whether a from political inertia [8, 16]. Unfortunately, the ran- small or a large change is desired. There are arguments dom strategy is also the slowest in terms of network for both sides: From the perspective of car-dependent consolidation: It needs at least three times the in- transport planning the change should be small to not vestments than the betweenness strategy to reach the disrupt the existing system too abruptly [7, 47]. From critical threshold. The rail effect can be explained by the perspective of sustainability, human-centric urban transit-oriented development efforts, where bicycle fa- planning, and climate research, the change should be cilities are planned close to transit lines [20, 39]. The large to boost efficient bicycle transport, livable cities, remarkably high overlap with Copenhagen’s well devel- and to fight climate change effectively. Indeed, the oped network suggests that our models could also be CROW manual states that directness should be higher adapted to identify “missing links” in existing bicycle for cyclists than for cars [11]. On top of that it could networks. be argued that our Copenhagen-style model of building Only by being global and minimalistic, deliberately a relatively sparse sub-network for cyclists is not going ignoring second order effects, does our approach un- far enough. For example, it could be inverted into a cover fundamental topological limitations of bicycle Barcelona-style model where dense patches of living network growth independent of place. At the same streets – Superblocks – are built within a sparse sub- time our results must be treated as statistical solutions. network of automobile arterials [48]. In any case, resis-

8 tance to such ideas needs to be anticipated [5,47], and proxy for the NP-hard minimum weight triangulation reclaiming ineﬃciently used urban space for humans [32] with an approximation ratio of Θ(√N) [56]. The requires vigorous policy making following leading ex- greedy triangulation is fast and solvable for any set of amples such as the Netherlands [8,49]. The necessity of nodes. Computing a quadrangular grid, as suggested doing so is backed by ample evidence from sustainabil- by the CROW manual [11], or a quadrangulation, is ity science, and if implemented persistently it will facil- in general not possible for arbitrary sets of nodes and itate the transition to car-free cities to counteract cli- also computationally less feasible [57]. mate change, providing extraordinary beneﬁts to public health and urban livability on the side [2,13,50,51]. Network metrics

Methods Cohesion. The CROW manual [11] describes qualitatively what it means for a network to be cohesive: Data acquisition and processing a “combination of grid size and interconnection”. It states that this is the most elementary requirement for Infrastructure networks. We downloaded existing a bicycle network but without a rigorous definition. We street and bicycle networks for 62 cities from Open- interpret this concept as having both high connected- StreetMap (OSM) on 2021-02-26 using OSMnx [31]. ness (few disconnected components) and coverage, see For each city, three networks were downloaded: Street below. A cohesive network should also be resilient, see network, protected bicycle network, bikeable network. below, which excludes pathological cases like the min- Each node is an intersection, each link is a connection imum spanning tree. between two intersections. A protected bicycle network is the union of all OSM data structures that encode protected bicycle infrastructure, both on-street and off-street. Following the cycling safety literature, we Coverage. We measure spatial coverage of the net- consider only protected bicycle networks in our main work as the union of the ε-neighborhoods of all network analysis because safe cycling in general conditions is elements, i.e. a buffer of ε m around all links and nodes. only ensured through physical separation from vehicu- Here we set ε = 500 m together with the grid POI dis- lar traffic [6, 11, 24, 37, 49]. However, we also consider tance, as this implies a theoretical coverage of 100% of for additional analysis the “bikeable” network, which is the city area for a grid triangulation and an average the union of a protected bicycle network and all streets distance to the network of 167 m, see Supplementary with speed limits 30 km/h or 20 m/h (including Note 2. In general, a cohesive bicycle network should living streets). In≤ special conditions≤ such street seg- cover the majority of the city area. ments can be considered safe for cycling, but not in general [11,24]. OSM data has been generally found to be of high quality and completeness [52, 53], but mul- POI coverage. This metric refers to the number of ticity studies using bicycle infrastructure data such as POIs that have been covered by network elements (by ours could potentially suffer from some labeling incon- the coverage defined above). sistencies, especially for less common types of bicycle infrastructure [54]. Resilience. By resilience we mean a general level of fault-tolerance [58]: A resilient bicycle network should Points of interest (POI). Rail station POIs con- provide an acceptable level of service in the face of sist of all railway and metro stations. A few of the faults and challenges to normal operation, for example considered cities do not have rail stations. For creat- interruptions due to road works. The removal of a ing grid POIs, we created grid points at a distance of small fraction of links should not have a substantial 1707 m, ensuring a tolerable average distance of 167 m impact on network metrics. (2 minutes walking) over the whole city to the trian- gulated network in the worst case, see Supplementary Note 2. We then rotated this grid to align it with the Directness. The ratio of euclidean distances d (i, j) city’s most common street bearing [55], and snapped E and shortest path distances d (i, j) on the network’s the grid points to the closest street network intersec- G largest connected component (LCC): tions within a 500 m tolerance. The rotation is mostly important for US cities that have a grid-like street network, e.g. Manhattan, for creating straight routes. i=j LCC dE(i, j) D = 6 ∈ (1) i=j LCC dG(i, j) P 6 ∈ Greedy triangulation P Directness is naturally defined only for the LCC be- The greedy triangulation orders all pairs of nodes by cause shortest path distances between nodes in discon- route distance and connects them stepwise as the crow nected components are by definition infinite. A similar, flies. A link is added only if it does not cross an existing but network-wide, measure is given by the global effi- link. This triangulation is a O(N log N) computable ciency, see below.

9 Local and Global Efficiency. A network’s global Closeness centrality. Measures the total length of efficiency is defined as [34]: the shortest paths from a node i to all other nodes in the network [60], 1 dG(i,j) i=j N 1 Eglob = 6 (2) P 1 CC (i) = − (7) dE (i,j) d(i, j) i=j 6 i=j P P6 A network’s local efficiency is defined as the average of global efficiencies Eglob(i) over each node i and its Code availability neighbors, 1 N All code used in the research is open-sourced, available E = E (i) (3) loc N glob at: https://github.com/mszell/bikenwgrowth i=1 X Spatial clustering and Anisotropy. We first spec- Data availability ify a threshold θ and identify the Nθ nodes with high betweenness above the θ-th percentile. Then, we com- All data used and generated in the research are pub- pute their spread about their center of mass licly available at zenodo [61]. Interactively growing

N networks, plots and video visualizations for all 62 cities 1 θ x = x can be explored and downloaded at the accompanying cm N i visualization platform: http://growbike.net θ i=1 X where xi speciﬁes their coordinates, normalizing for comparison across networks of diﬀerent sizes via Acknowledgments

N 1 θ We thank Cecilia Laura Kolding Andersen and Morten C = x x , (4) θ N X k i − cmk Lynghede for developing and implementing the visu- θ i=1 h i X alization platform. We are grateful to Anders Hart- where man, Anastassia Vybornova, and Laura Alessandretti 1 N for helpful discussions. We thank the ITU High- X = x x h i N k i − cmk Performance Computing cluster for computing re- i=1 X sources and support. is the average distance of all nodes in the network to the center of mass of the high betweenness cluster. Transition between the topological and spatial Author Contributions regimes is quantified by the increasingly isotropic lay- out of the high betweenness nodes with increasing edge- M.S. designed the study with input from R.S. and G.G. density. The anisotropy factor is defined by the ratio M.S. wrote the manuscript with input from all authors. M.S. acquired and pre-processed the data with input λ1 from S.M. and T.P., and performed the simulations. Aθ = (5) λ2 M.S. directed the project, R.S. and G.G. helped su- pervise the project. M.S. measured the results, aided where λ1 λ2 are the positive eigenvalues of the co- variance matrix≤ of the spatial position of the nodes by S.M. T.P. performed the analysis on grid size and with betweenness above the threshold θ [38]. network coverage. All authors discussed the results. For the largest 15 cities we calculated these values only at the 0, 0.5, and 1 quantiles of the growth strategies due to computational feasibility. Therefore, Sup- References plementary Figure 3 reports average values over the 47 [1] Banister, D. Unsustainable transport: city trans- smallest cities. port in the new century (Routledge, 2005). Growth strategies [2] Nieuwenhuijsen, M. J. & Khreis, H. Car free cities: Pathway to healthy urban living. Environment Betweenness centrality. This is a path-based mea- international 94, 251–262 (2016). sure that computes the fraction of paths passing through a given node i [59], [3] Alessandretti, L., Aslak, U. & Lehmann, S. The scales of human mobility. Nature 587, 402–407 1 σst(i) CB(i) = (6) (2020). N σst s=t X6 [4] Gössling,S., Choi, A., Dekker, K. & Metzler, D. where σst is the number of shortest paths going from The social cost of automobility, cycling and walk- nodes s to t and σst(i) is the number of these paths ing in the European Union. Ecological Economics that go through i. 158, 65–74 (2019).

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12 Growing Urban Bicycle Networks Supplementary Information

Michael Szella,b, Sayat Mimarc, Tyler Perlmanc, Gourab Ghoshalc, and Roberta Sinatraa,b

This is the supplementary information for the manuscript containing supplementary notes, ﬁgures, and tables.

List of Figures

1 Square and triangle for network coverage calculations...... 3 2 Synthetic versus real bicycle networks...... 6 3 Streets: Change of network metrics with growing bicycle networks...... 7 4 Change of network metrics with betweenness growth on rail station POIs...... 8 5 Change of network metrics with closeness growth on rail station POIs...... 9 6 Change of network metrics with random growth on rail station POIs...... 10 7 Change of network metrics with betweenness growth on grid POIs...... 11 8 Change of network metrics with closeness growth on grid POIs...... 12 9 Change of network metrics with random growth on grid POIs...... 13

List of Tables

1 Cities and their urban parameters...... 5 arXiv:2107.02185v2 [physics.soc-ph] 13 Jul 2021

1 Supplementary Note 1: Limitations and reﬁnements for planning

Since we publish all our code as open source, arbitrary refinements are possible, expected, and encouraged in future work, especially towards better applicability of our approach for concrete bicycle network planning. For example, the CROW manual recommends a hierarchical network approach with a top-level bicycle highway connecting the major neighborhoods, a medium-level “main cycle network”, and a low-level “basic infrastructure” [1]. In our process we only considered the medium level so far, but it is straightforward to place custom links before or after the greedy triangulation which allows to create any desired hierarchy. To operational- ize CROW’s “adapted grid method” [1] even closer, a quadrangulation [2] could be implemented for grid-like POIs and streets, especially for US cities [3], as well as the missing harmonization with other transport lay- ers [4]. However, computational difficulties must be expected with quadrangulations since in general they are not guaranteed to have a solution and are computationally less feasible [2]. Further, our approach does not consider bicycle network planning outside of built-up urban areas, which is becoming increasingly pressing with the rise of affordable e-bike technologies. This line of research is also becoming more important due to large-scale bicycle planning initiatives such as Denmark’s infrastructure plan for 2035, where 2-3 billion DKK are to be invested into national bicycle infrastructure, primarily into cycle superhighways [5]. The development of inter-urban, regional networks demands different considerations outside the scope of our paper, from changed parameters or topology such as “ladder structures” [1] to accounting for different regional stakeholders and investors. The biggest limitation of our approach is the sole focus on retrofitting street networks for safe cycling. This approach has some issues because it only considers on-street but no off-street bicycle infrastructure – in particular, our algorithms ignore existing bicycle-only infrastructure such as Copenhagen’s Cykelslangen or similar bicycle-only infrastructure in many parts of the Netherlands – but this issue is only relevant in such well developed cycling environments and negligible in the vast majority of cities on the planet. The first main problem is about low-density regions. Such regions, like most parts of the Netherlands, rightly prioritize the development of off-street networks to keep cycle routes as far as possible from cars, avoiding intersecting traffic, thus making them attractive low-stress environments [1]. Therefore, although our approach is the only feasible for high-density cities, in low-density urban environments a Dutch style off-street approach can be argued to be more desirable than on-street networks. For example, although Copenhagen has a very well developed on- street bicycle network with a high 62% modal share for cycling [6], it is questionable 1) to which extent such a cycling environment is low-stress, inclusive, and child-friendly [7], and 2) whether such on-street tracks are literally cementing the “arrogant” car-centric distribution of mobility space that has indeed been reported in Copenhagen [6, 8, 9]. Thus, a focus on building on-street networks possibly represents a lock-in into a path- dependent city evolution that could complicate the transition to car-free cities, therefore being unsustainable in the long term [10]. Indeed, there is preliminary evidence that automobile ownership and mobility is on the rise in Copenhagen [11]. Second, an exclusion of off-street infrastructure also could exclude improved geometric solutions such as Steiner trees/points [2,12,13]. Third, from a communication perspective, this approach comes with the risk of anchoring automobile infrastructure as the default when in reality automobile transport is one of the most unsustainable and possessive modes (in terms of public space demands) for cities [14, 15] posing the biggest threat to the lives of other road users [16]. For all these reasons, future research on bicycle network growth should consider and prioritize off-street solutions where possible. Our minimal requirements on data, making use of only the street network, could be seen as another limitation. However, we follow this approach on purpose for our framework to be applicable to data-scarce environments, and thus to a large part of the planet [17]: no lane widths, inclines, traffic flows, etc. are needed to optimize geometry. Further, even if we had for example cyclist flows available, how would we ensure they are a ground truth for optimal infrastructure? Such flow data could have been influenced by inappropriately or suboptimally developed cycling infrastructure and other biases [18]. Thus, our research approach places itself on the right end of the spectrum: Single city with complex/multiple data sets — Multiple cities with simple/few data sets. Given that the struggle to implement more bicycle infrastructure is mostly a political one, it was recently proposed that there might already be enough theoretical knowledge generated by cycling research [19]. While we agree on the political bottleneck and that it is most important to “provide more road space for cyclists at the cost of motorized traffic”, the questions of How to do that and of Why that is happening so slowly cannot be neglected. Our study shows that cycling research has barely scratched the surface when it comes to understanding the underlying network effects, given that network percolation and its fundamental implication on network growth is one of the most basic findings of graph theory dating back to the 1950s [20] but is not common knowledge in transport network planning – otherwise it would be acknowledged in leading planning manuals such as CROW [1]. Last but not least, research and politics are not independent: If cycling research is becoming “mainstream” and is accompanied by high-quality visualizations, open source tools, and a compelling vision, it can create increased potential to reach the general public and induce political change [15, 21–23].

2 Supplementary Note 2: Grid size and network coverage

The grid points in the grid POI network triangulation were chosen at a distance of a = 1707 m ensuring full 4 coverage of the city when rmax = 500 m is assumed as a reasonable maximum distance (around 5 minutes of walking). The proof for this coverage comes from the fact that the right triangle with sides of lengths a , a and 4 4 a +a √2a √ 4 4− 4 2a is the triangle that emerges in the triangulation, together with its inradius equation rmax = 2 . Solving4 for a yields a = 2 r 3.41r . Comparing this argumentation to a square grid, which is the 2 √2 max max 4 4 − ≈ network geometry suggested by the CROW manual, a maximum distance of rmax = 500 m would correspond to a grid length of a = 2rmax = 1000 m. The grid size standard given by CROW for built-up areas is 300-500 m. If a similarly tightly covering mesh is desired, our grid triangulation would need to be run for a grid with half to third of its grid length. Instead of the maximum distance rmax, it is more useful for planning purposes to consider the average distance ravg of a random point to the network. As we show in the calculations below, the average distance for our choice a = 1707 m corresponds to ravg = rmax/3 167 m. For a square CROW grid of length 500 m the 4 ≈ average distance would be ravg 83 m. These numbers are only crow flies distances and do not account for less direct distances routed on concrete≈ street or pedestrian networks – to account for street grid effects all distance values could be multiplied by up to √2. In other words, the effective average distance to the network could be closer to √2 167 236 m than to 167 m. · ≈ Calculation of average distance to the nearest side of a square What is of interest to us is to calculate the average distance from a randomly selected interior point to the side of the square closest to it. By subdividing the square into triangles by drawing the line segments connecting each corner to the center, all points for which a given sides is the closest will be in the same sub triangle. From there, we can integrate the distances to the closest side over each of these triangles, sum these values, and divide by the total area of the square to get our desired result. a a For a square with side length a, if we were to align two of the with the axes, our center would be at ( 2 , 2 ). The lines that connect the corners to the center (the diagonals) would be given as follows, see Supplementary Figure 1 (left):

f(x) = x g(x) = x + a . − First, by noting that all of our sub-triangles are congruent, isosceles triangles, we can further divide our triangles along the altitudes from each side of the square to the center. This results in a new count of 8 congruent, isosceles triangles for which all interior points of each triangle are closest to the same side of the square. For the triangles with sides on the x-axis, the distance from an interior point to the closest edge is the point’s distance to the x-axis. As all of our triangles are congruent, we need to calculate one integral:

a /2 f(x) I = ydy dx Z0 Z0 !

Supplementary Figure 1: Square and triangle for network coverage calculations.

3 Evaluating this integrals yields: a3 I = . 48 2 3 a a 8 a Thus, the average distance from an interior point of the square to the nearest edge will be: I = = . ÷ 8 48 a2 6 As such: 1 r = a avg 6 Calculation of average distance to the nearest side of an isosceles triangle What is of interest to us is to calculate the average distance from a randomly selected interior point to the side of the triangle closest to it. By ﬁnding the incenter of the circle (the point equidistant from all sides) and subdividing the triangle by drawing the line segments connecting each corner to the incenter, all points for which a given sides is the closest will be in the same triangle. From there, we can integrate the distances to the closest side over each of these triangles, sum these values, and divide by the total area of the triangle to get our desired result. √ 2 √2 For a right isosceles triangle with side lengths a , a , and a 2, the inradius is r = −2 a . As such, if we were to align the sides that intersect at a right4 angle4 with the4 axes, our incenter would be4 at (r, r). The lines that connect the corners to the incenter (and thus bisect each of the angles) would be given as follows, see Supplementary Figure 1 (right):

f(x) = x g(x) = (1 + √2)x + a − 4 h(x) = (1 √2)x (1 √2)a . − − − 4 First, we can simplify the problem: by noting that as the triangle is isosceles, we can cut the triangle in half along the line y = x and calculate our average distance over one of these halves of the triangle. We then take the part of the sub-triangle for which the interior points are closest to hypotenuse, and rotating our coordinates so that the hypotenuse is on the x-axis. By doing so, for both this triangle as well as the other sub-triangle, the distance from an interior point to the closest edge is the point’s distance to the x-axis. We will also subdivide the second triangle, but breaking it up along the altitude from the incenter to the edge. As such, our integrals of interest are:

r f(x) I1 = ydy dx Z0 Z0 ! a h(x) 4 I2 = ydy dx Zr Z0 ! a √2 (√2 1)x 4 2 − I3 = ydy dx Z0 Z0 ! where y = (√2 1)x is the line connecting (0, 0) to (a √2 , r). It is worth noting that the sub-triangles cor- − 2 responding to I2 and I3 are congruent, and as such, these integrals will be equal. Evaluating these integrals yields:

3 1 3 I1 = 2 √2 a 48 − 4 1 3 I2 = 3√2 4 a 24 − 4 1 3 I3 = 3√2 4 a . 24 − 4 1 √ 3 Thus, the sum of these integrals will be: I1 + I2 + I3 = 24 2 2 a . Now, we divide by half of the area of − 4 a2 4 the triangle, 4 . This yields: 2 √2 ravg = − a 6 4

4 2 2 City Lcar [km] Lbike [km] Γbike Population Area [km ] Continent Density [Pop/km ] Tokyo 26,293 339 504 37,977,000 8,547 Asia 4,443 Jakarta 11,797 4 15 34,540,000 3,225 Asia 10,710 Mumbai 3,983 2 1 23,355,000 546 Asia 42,775 Sao Paulo 17,029 250 127 22,046,000 2,707 S. America 8,144 Mexico City 15,056 247 63 20,996,000 2,072 N. America 10,133 Moscow 5,507 286 229 17,125,000 4,662 Europe 3,673 Buenos Aires 3,399 257 57 16,157,000 2,681 S. America 6,026 Karachi 20,272 0 1 14,835,000 945 Asia 15,698 Los Angeles 12,286 292 276 12,750,807 6,299 N. America 2,024 Paris 1,536 292 314 11,020,000 2,845 Europe 3,873 London 15,810 1,236 4,196 10,979,000 1,738 Europe 6,317 Bogota 6,788 471 179 9,464,000 492 S. America 19,236 Chicago 7,003 163 160 8,604,203 6,856 N. America 1,255 Luanda 1,878 0 0 8,417,000 894 Africa 9,415 Hong Kong 3,692 272 329 7,347,000 275 Asia 26,716 Singapore 4,502 470 249 5,745,000 518 Asia 11,091 Philadelphia 4,576 109 70 5,649,300 5,131 N. America 1,101 Houston 13,120 287 97 5,464,251 4,644 N. America 1,177 Toronto 6,275 299 183 5,429,524 2,287 N. America 2,374 Boston 1,744 124 86 4,688,346 5,325 N. America 880 Barcelona 1,412 217 79 4,588,000 1,075 Europe 4,268 Phoenix 9,537 341 107 4,219,697 3,196 N. America 1,320 Berlin 6,181 1,334 803 3,644,826 1,347 Europe 2,706 San Francisco 1,847 76 115 3,592,294 2,797 N. America 1,284 Montreal 5,687 362 165 3,519,595 1,546 N. America 2,277 Detroit 5,108 75 44 3,506,126 3,463 N. America 1,012 Birmingham 2,774 237 265 2,897,303 598 Europe 4,845 Rome 7,282 214 190 2,872,800 1,114 Europe 2,579 Greater Manchester 9,836 668 797 2,705,000 1,277 Europe 2,118 Tashkent 3,633 4 9 2,424,100 334 Asia 7,258 Leeds 3,290 247 425 1,901,934 487 Europe 3,905 Hamburg 4,349 858 885 1,841,179 755 Europe 2,439 Vienna 3,023 573 478 1,840,573 414 Europe 4,446 Warsaw 3,649 630 217 1,790,658 517 Europe 3,464 Budapest 4,507 210 206 1,752,286 525 Europe 3,338 Manhattan 972 152 77 1,628,706 87 N. America 18,721 Rabat 1,152 4 7 1,628,000 117 Africa 13,915 Munich 2,785 1,080 466 1,471,508 310 Europe 4,747 Ulaanbaatar 6,180 24 14 1,396,288 4,704 Asia 297 Milan 1,964 185 230 1,351,562 181 Europe 7,467 Cologne 2,720 764 427 1,085,664 405 Europe 2,681 Copenhagen 1,159 454 221 1,085,000 292 Europe 3,716 Amsterdam 1,770 718 288 1,031,000 219 Europe 4,708 Glasgow 2,088 147 189 985,290 142 Europe 6,939 Kathmandu 808 3 21 975,453 49 Asia 19,907 Marrakesh 1,860 2 3 928,850 143 Africa 6,495 Turin 1,643 171 100 870,952 130 Europe 6,700 Oslo 1,737 294 555 693,494 480 Europe 1,445 Sheﬃeld 2,017 103 233 685,368 368 Europe 1,862 Helsinki 1,572 1,299 506 642,045 715 Europe 898 Stuttgart 1,526 161 334 634,830 207 Europe 3,067 Shah Alam 3,079 57 15 584,340 290 Asia 2,015 Bradford 1,985 59 102 540,000 370 Europe 1,459 Lyon 653 62 116 516,092 48 Europe 10,752 Edinburgh 1,623 162 199 488,050 264 Europe 1,849 Tel Aviv 813 163 195 451,523 52 Asia 8,683 Zurich 764 64 330 434,008 87 Europe 4,989 Malmo 919 428 142 321,845 332 Europe 969 Santiago Centro 399 48 9 200,792 22 S. America 9,126 Bern 408 11 89 133,798 51 Europe 2,623 Delft 285 111 145 103,000 24 Europe 4,292 Bath 272 17 32 88,859 29 Europe 3,064

Supplementary Table 1: Cities and their urban parameters.

5 Synthetic versus real networks Synthetic versus real networks

Coverage grid | betweenness Coverage railwaystation | betweenness

Local Efficiency Local Efficiency

Global Efficiency Global Efficiency

Length of LCC Length of LCC

0.5 1 2 5 10 20 0.5 1 2 5 10 20

Msyn/Mreal | Lsyn = Lreal Msyn/Mreal | Lsyn = Lreal

Synthetic versus real networks Synthetic versus real networks

Coverage grid | closeness Coverage railwaystation | closeness

Local Efficiency Local Efficiency

Global Efficiency Global Efficiency

Length of LCC Length of LCC

0.5 1 2 5 10 20 0.5 1 2 5 10 20

Msyn/Mreal | Lsyn = Lreal Msyn/Mreal | Lsyn = Lreal

Synthetic versus real networks Synthetic versus real networks

Coverage grid | random Coverage railwaystation | random

Local Efficiency Local Efficiency

Global Efficiency Global Efficiency

Length of LCC Length of LCC

0.5 1 2 5 10 20 0.5 1 2 5 10 20

Msyn/Mreal | Lsyn = Lreal Msyn/Mreal | Lsyn = Lreal

Supplementary Figure 2: Synthetic versus real bicycle networks. Shown are all 6 combinations of growth strategies (betweenness, closeness, random) with POI types (grid, rail).

6 Global Efficiency Local Efficiency Directness of LCC Spatial Clustering Anisotropy 0.80 0.115 0.80 0.80 0.56 0.75 0.110 0.75 Top 10% 0.75 0.54 0.70 0.70 0.105 0.70 grid 0.52 Top 5% 0.65 0.65 0.100 betweenness 0.65 0.50 0.60 Top 3% 0.60 0.095 0.60 0.48

0.80 0.115 0.80 0.80 0.56 0.75 0.110 0.75 0.75 0.54 0.70 0.70 0.105 0.70 grid 0.52 0.65 closeness 0.65 0.100 0.65 0.50 0.60 0.60 0.095 0.60 0.48

0.80 0.115 0.80 0.80 0.56 0.75 0.110 0.75 0.75 0.54 0.70 0.70 0.105 0.70 grid 0.52

random 0.65 0.65 0.100 0.65 0.50 0.60 0.60 0.095 0.60 0.48

0.80 0.115 0.80 0.80 0.56 0.75 0.110 0.75 0.75 0.54 0.70 0.70 0.105 0.70 0.52 0.65 0.65 0.100 betweenness railwaystation 0.65 0.50 0.60 0.60 0.095 0.60 0.48

0.80 0.115 0.80 0.80 0.56 0.75 0.110 0.75 0.75 0.54 0.70 0.70 0.105 0.70 0.52 0.65 closeness 0.65 0.100 railwaystation 0.65 0.50 0.60 0.60 0.095 0.60 0.48

0.80 0.115 0.80 0.80 0.56 0.75 0.110 0.75 0.75 0.54 0.70 0.70 0.105 0.70 0.52

random 0.65 0.65 0.100 railwaystation 0.65 0.50 0.60 0.60 0.095 0.60 0.48 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Growth strategy quantile Growth strategy quantile Growth strategy quantile Growth strategy quantile Growth strategy quantile

Supplementary Figure 3: Streets: Change of network metrics with growing bicycle networks.

7 Supplementary Figure 4: Change of network metrics with betweenness growth on rail station POIs. Shown are the selected cities Copenhagen, Boston, Mumbai, Tokyo. See Section Data availability in the main text for how to access these plots for all cities.

8 Supplementary Figure 5: Change of network metrics with closeness growth on rail station POIs. Shown are the selected cities Copenhagen, Boston, Mumbai, Tokyo. See Section Data availability in the main text for how to access these plots for all cities.

9 Supplementary Figure 6: Change of network metrics with random growth on rail station POIs. Shown are the selected cities Copenhagen, Boston, Mumbai, Tokyo. See Section Data availability in the main text for how to access these plots for all cities.

10 Supplementary Figure 7: Change of network metrics with betweenness growth on grid POIs. Shown are the selected cities Copenhagen, Boston, Mumbai, Tokyo. See Section Data availability in the main text for how to access these plots for all cities.

11 Supplementary Figure 8: Change of network metrics with closeness growth on grid POIs. Shown are the selected cities Copenhagen, Boston, Mumbai, Tokyo. See Section Data availability in the main text for how to access these plots for all cities.

12 Supplementary Figure 9: Change of network metrics with random growth on grid POIs. Shown are the selected cities Copenhagen, Boston, Mumbai, Tokyo. See Section Data availability in the main text for how to access these plots for all cities.

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