Let C be a city, Ω(C) its spatial domain, |Ω(C)| its area, v(C) the average traffic speed in C and λ the average number of trips per hour with both endpoints in Ω(C). Figure 1(a) shows that the computed curve of shareability against λ for New York City16 closely resembles a “fast” saturation process, with a quick increase from lowest density, where shareability is minimal, to saturation where all trips can be shared. Our first main finding is that three other cities – San Francisco, Singapore, and Vienna (see Methods and Supplementary Information: Table S1a for datasets description and algorithms17) – show strikingly similar shareability curves (Fig. 1(b)–(d)). Such a similarity is remarkable, given that the shareability curves are obtained from data sets of real taxi trips, using a methodology that includes the hour-by-hour variability in traffic congestion (see Methods).

Figure 1: Shareability curves. The shareability curves for (a) New York, (b) San Francisco, (c) Singapore, and (d) Vienna. The curves were computed using a shareability network algorithm16 applied to data collected from over 156 million taxi trips in the four cities. See Methods for details about the datasets and algorithm. Full size image

Each curve in Fig. 1 saturates rapidly as a function of λ. Their rapid saturation distinguishes them from other saturation phenomena observed in urban/geographical processes, such as the growth of retail locations18 and the spreading of innovations19, which are instead characterized by an initial “slow start” phase with a sigmoidal shape. Fast saturation of shareability is a plausible explanation for the great success of innovative ride and vehicle sharing apps such as UberPoolTM, ZipCarTM, and Car2GoTM.

The similarity we observe between cities actually goes beyond the resemblance of their shareability curves: a single linear rescaling of the λ-axis makes all the curves nearly coincident (Fig. 2; see also Methods and Supplementary Information: Table S2), suggesting that a common mechanism governs shareability in those four cities. The data collapse is achieved by replotting the computed shareability S versus the dimensionless quantity

Figure 2: Shareability law. When replotted as functions of L, the computed shareability curves for New York, San Francisco, Singapore, and Vienna nearly coincide with each other and with the theoretical prediction given by Eq. (2). This rescaling involves no adjustable parameters. Full size image

The greater the L, the greater the shareability.

The quality of the data collapse indicates that a few urban level parameters, when combined into the dimensionless group L, suffice to accurately model a complex quantity like the fraction of trips that can be shared in a city. This result is all the more surprising when one considers that L is defined in terms of the average daily traffic speed in the city, while the shareability curves have been derived using hourly, street-level traffic speed estimations. Evidently, the variability of congestion occurring at different times of day has a limited effect on our predictions. We do not ignore that variability; on the contrary, it is captured by the data we use to derive the shareability curves. Yet the fact that our simple model accounts so well for those shareability curves demonstrates that the variability is not a dominant effect. The universal curve can be explained by using a single, average value of the traffic velocity v.

The particular combination of urban parameters in Eq. (1) can be rationalized by dimensional analysis. Intuitively, L represents a ratio between two timescales: the sharing delay ∆ and the characteristic waiting time t wait for a trip to be generated in a user’s vicinity. To see this, imagine that you are looking for a cab. Since λ is the average rate at which taxi trips are generated, 1/λ is the characteristic time for a new trip to be generated, somewhere in the city. But the city as a whole is not what concerns you. What matters more is how long you can expect to wait for a new trip to be generated in your vicinity. The characteristic linear scale of a vicinity is vΔ, the distance a cab moving at speed v would travel in the delay time Δ that another passenger could tolerate. Since the city has a total area |Ω| and each vicinity has area (vΔ)2, there are about |Ω|/(vΔ)2 vicinities in total. Assuming that trips are generated uniformly in space, you would expect a trip to be generated in your vicinity every time units. Hence the ratio of the tolerable delay time Δ to the expected waiting time is Δ/t wait = λv2Δ3/|Ω| = L.

At a more refined level, the influence of urban parameters on shareability can be approached mathematically as follows. Intuitively, one expects that shareability should be positively related to ∆, v(C), and λ. Indeed, as ∆ and v(C) increase, people become more tolerant about sharing delay and a larger urban space can be covered without exceeding the delay16. The effect of increasing trip density is more complex to assess since it simultaneously introduces new rides and new ride-sharing opportunities. However, the additional trips are drawn from the same distribution as the original ones, so they possess similar spatiotemporal properties, which on average results in an increase of shareability as a function of λ.

Assuming that rides are generated independently in the city according to a given spatiotemporal distribution, we wish to compute the probability that a ride can be shared as a function of ∆, v(C), and λ. Tackling this problem directly is very difficult, since the probability of actually sharing a ride depends not only on the spatiotemporal availability of candidate trips to share, but also on how potentially shareable trips are paired together, which in turn depends on complex structural properties of the underlying shareability network. Nevertheless, the spatial dimension of the problem, coupled with the observed fast saturation of the shareability curve, suggest analogies with geometric random graphs20 and percolation theory21. A common trait of these theories is that complex network structural properties such as connectivity can be closely approximated by much simpler properties, such as the existence of isolated nodes. This turns out to be the case also for shareability networks; we find that shareability S is highly correlated with the number of isolated nodes in the shareability network (Methods).

Based on the above discussion, we can model shareability by fixing an arbitrary trip T and estimating the probability that there exists at least one other trip T′ shareable with T. More specifically, an arbitrary trip T starting at time t 0 and going from origin o to destination d defines a trajectory in space and time. For fixed ∆ and average traffic speed v(C), we define the notion of the shareability shadow s(T) surrounding T and confining the region of sharing opportunities (Supplementary Information: Figs S1 and S2). For another trip T′ to be shareable with T, its trajectory needs to overlap (i.e., to take place at the same time, at least partially) and to be “aligned” (i.e., not deviate too much direction-wise) with s(T). Those two conditions simply translate our upper bound ∆ on delays into a geometric condition stating that shareable trips should be close enough in terms of trajectories, where close enough is quantified through the volume of s(T) chosen depending on v(C) and ∆. Analytically, the expected shareability becomes the probability that a compatible trip will be generated in the shareability shadow (see Supplementary Information: “Supplementary Equations”). To compute that quantity, the previously mentioned spatiotemporal distribution of trips has to be determined. Among the different options we considered, the following one gave the best compromise between accuracy and tractability: origin point o chosen uniformly in Ω, and destination point d chosen uniformly in a disk centered on o of radius R (ignoring boundary effects for the sake of simplicity). The geometry of the city plays a minimal part in the definition, which allows us to derive analytical formulas for the shareability. For R large enough, we find that S becomes independent of R, and the city’s influence on the shareability only appears through the quantity L. We prove that (see Supplementary Information: “Supplementary Equations”)

We tested our model predictions on the four cities mentioned above and found a strong agreement with the respective shareability curves (Supplementary Information: Fig. S3), with R2 values ranging from 0.91 to 0.98.