Source-linked AI summary
Scaling Law of Urban Ride Sharing
Remi Tachet, Oleguer Sagarra, Paolo Santi, Giovanni Resta, Michael Szell, Steven Strogatz, Carlo Ratti
TL;DR
The paper examines ride-share shareability and the difficulty of predicting it directly. It computes shareability curves across cities using shareability networks, then models their common rescaled form; the model’s fidelity supports shared mechanisms governing ride sharing in cities.
Problem
Shareability may help explain ride-sharing success, but directly addressing the probability that trips can actually be shared is difficult.
Method
The study builds shareability networks from taxi trips, computes maximum matching, compares cities, and rescales their shareability curves using a mathematical model.
Results
The model’s fidelity suggests that fundamental mechanisms governing ride sharing are shared across cities, including under low trip density with delays no longer than Δ = 5 minutes.
Takeaways & Limitations
The urban parameters and corresponding predictions provide information relevant to the recent upheaval that ride sharing has caused in cities worldwide.
Takeaways & Limitations
The analysis reports little improvement in model accuracy because the overwhelming majority of trips are affected by the relevant limitation.
Abstract
from arXiv · showhide
Sharing rides could drastically improve the efficiency of car and taxi transportation. Unleashing such potential, however, requires understanding how urban parameters affect the fraction of individual trips that can be shared, a quantity that we call shareability. Using data on millions of taxi trips in New York City, San Francisco, Singapore, and Vienna, we compute the shareability curves for each city, and find that a natural rescaling collapses them onto a single, universal curve. We explain this scaling law theoretically with a simple model that predicts the potential for ride sharing in any city, using a few basic urban quantities and no adjustable parameters. Accurate extrapolations of this type will help planners, transportation companies, and society at large to shape a sustainable path for urban growth.
Results
Across four cities, shareability rises rapidly with trip density and follows a common curve after rescaling density by the dimensionless group L. A simple model using urban parameters closely predicts these patterns, while congestion has limited influence.
- Four cities show strikingly similar shareability curves, with rapid saturation as trip density increases.The curves rise quickly from low density toward saturation, unlike the slow-start sigmoidal patterns reported for some other urban processes.
- A single linear rescaling of the λ-axis makes the cities’ shareability curves nearly coincident, suggesting a common governing mechanism.The collapse is obtained by plotting shareability against the dimensionless quantity L.
- Greater L corresponds to greater shareability, and the dimensionless group combines a few urban-level parameters to model the fraction of trips that can be shared.The model uses quantities including city area, average traffic speed, trip density, and sharing delay.
- The model assumes independently generated trips with uniformly distributed origins and destinations within a disk, while city geometry has a minimal role.For sufficiently large R, the predicted shareability becomes independent of R.
- Shareability is highly correlated with isolated nodes in the shareability network, allowing network structure to be approximated through a simpler quantity.The model estimates whether an arbitrary trip has at least one compatible trip within its shareability shadow.
- The model predicts shareability accurately across the four cities, with R^2 values ranging from 0.91 to 0.98.Its predictions agree strongly with the respective empirical shareability curves.
Discussion
The framework uses a simple mathematical model to characterize ride sharing despite real-world congestion and simplifying assumptions. Its parameter-free predictions quantify substantial sharing potential and support extensions for urban mobility planning.
- Discussion: The model accurately characterizes ride sharing in congested road networks using a relatively simple mathematical framework.It is built on simplifying assumptions including Euclidean geometry, straight-line trajectories, and basic shareability-shadow shapes.
- Discussion: Most information needed to determine shareability is contained in the dimensionless group L, the framework’s only required quantity.Because L is relatively easy to estimate, the otherwise parameter-free framework can be applied across cities.
- Discussion: The framework can incorporate second-order congestion effects when their influence on average speed is known.Ride sharing may reduce congestion and increase average travel speed, thereby increasing shareability.
- Discussion: The findings quantify ride sharing’s effects on the urban environment and provide guidance for designing more efficient mobility systems.Tables report urban parameters and corresponding ride-shareability predictions for several major world cities.
- Discussion: Even at low trip density, allowing delays no longer than Δ = 5 minutes yields massive sharing potential.The paper presents this result as a prediction for the cities considered in its tables.
Methods
The study combines taxi datasets from four cities with common preprocessing, traffic-aware travel-time estimation, shareability-network construction, and density resampling. Dynamic supersampling extends the observed trip distributions when higher densities are needed.
- Methods: Taxi data came from New York, San Francisco, Singapore, and Vienna, with datasets differing in duration, coverage, and tracked-fleet size.The New York data span more than a year; the other datasets span roughly a month and include records from single operators.
- Methods: All datasets were filtered to occupied taxi trips whose start and end positions lay within 200 meters of study-area intersections.The study areas included Manhattan, Singapore, San Francisco and Vienna, with selected airports included for three cities.
- Methods: Pickup and dropoff times estimate travel times between intersections for each hour, allowing traffic effects to enter shareability-network calculations.Coordinates were mapped to intersections obtained from OpenStreetMap and projected into Euclidean UTM coordinates.
- Methods: Saturation curves were generated by subsampling saturated datasets and supersampling Vienna to reach densities above those observed.The supersampling procedure extends an existing trip sample to higher densities.
- Methods: Dynamic supersampling allocates trips using stable origin-destination transition probabilities, empirical hourly generation patterns, and Poisson timing processes.The authors extended a prior method to account for temporal dynamics and made the code public.
Supplementary Text
The supplementary analysis defines shareability networks and curve similarity measures, then compares statistically optimized rescaling with rescaling by the dimensionless group L. It also links shareability to network structure.
- Supplementary Text: A shareability network represents trips as nodes and connects pairs that can be shared; maximum shareability is computed through maximum matching.For a fixed number of trips, random subnetworks are evaluated and their maximum-matching sizes averaged.
- Supplementary Text: Shareability is modeled as S = a(1 − iso) + b, with R2 = 0.991 for a = 1.06114 and b = −0.0703398 across four datasets.The correlation relates shareability to the fraction of isolated nodes in the shareability network.
- Supplementary Text: Curve similarity is assessed using a symmetric coefficient of determination computed from interpolated points on both curves.The supplementary analysis uses k = 1000 points and averages the variances of the two curves.
- Supplementary Text: Rescaling the independent variable by K transforms a curve as SK(λ) := S(λ / K).The authors compare least-squares-optimal rescaling with parameter-free rescaling using K = L(C).
1 General problem
The paper formulates urban ride sharing as the probability that a trip can be shared under spatial, temporal, and delay constraints. It studies how trip density, delay tolerance, speed, and city area shape that probability.
- 1 General problem: The paper seeks general laws governing urban ride sharing and introduces an analytical model capturing the essential features of sharing.The model is designed to predict the probability that an arbitrary ride can be shared from basic urban parameters.
- 1 General problem: A trip is characterized by its origin, destination, and starting time, which the model assumes uniquely determine its trajectory.Trips can be shared when a route satisfies spatial ordering and temporal delay constraints.
- 1 General problem: Trip density λ measures the average number of potentially shareable trips originating in the city per hour.The study varies density through subsampling and supersampling, while λf denotes the density of the full datasets.
- 1 General problem: Shareability increases rapidly with trip density before approaching saturation.This relationship is reported as a clear association between the number of available trips and the fraction that can be shared.
- 1 General problem: Higher delay tolerance and travel speed increase shareability, whereas greater city area decreases it when other parameters are fixed.Higher speed expands the area reachable within the delay bound, while a more spread-out city offers fewer sharing opportunities.
2 Trip sharing model: overview
The model introduces a shareability shadow: at each instant, a trip defines origin and destination regions for compatible trips. This framework approximates overall shareability from spatiotemporal trip distributions.
- Trip-generation assumptions: Trip start times follow a time-dependent Poisson process, while origins and destinations come from a four-dimensional distribution ρ(x, y).The baseline model assumes ρ is time-independent, although the framework can be generalized to time-varying distributions.
- Shareability shadow: A trip’s shareability shadow consists of origins close to its current position and destinations compatible with its destination.The destination constraint prevents the shared trip from deviating too much from its course.
- Shareability calculation: The model estimates the probability that a trip can be shared from the probability of another trip being generated within compatible origin and destination regions.For simplicity, these regions are assumed independent and to have simple shapes.
- Mathematical formulation: Overall shareability is expressed as a five-dimensional integral involving the exponential of another five-dimensional integral.Analytic evaluation requires selecting a specific spatiotemporal distribution ρ.
3 Fundamental properties and assumptions
The four cities have nearly identical shareability-curve shapes after rescaling, supporting a universal dimensionless description. The model uses simple geometric and trip-generation assumptions to explain this similarity.
- Empirical similarity: R^2 values exceeding 95.5% show that optimally rescaled shareability curves for New York, San Francisco, Singapore, and Vienna nearly overlap.The curves have extremely similar shapes across the four cities.
- Universal rescaling: L = λv^2Δ^3/|Ω| provides a universal rescaling without adjustable parameters.Although not quite optimal, L naturally accounts for relevant differences between cities.
- Scaling properties: Scaling a city’s linear dimensions by μ while scaling its area by μ^2 leaves L unchanged when average speed is scaled by the same constant.Under this transformation, the city’s shareability level should not change, aside from microscopic road-network effects.
- Spatial assumptions: Uniformly sampling trip origins and sampling destinations uniformly within a disk around each origin generates the required properties and fits measured curves well.This choice avoids requiring an empirically determined spatial trip-generation function.
- Trip representation: Trips are represented by origin, destination, and start time, with straight-line trajectories at system-wide average velocity v.Trip duration is determined by Euclidean distance divided by v, and Δ represents the maximum tolerable delay.
5 Probability that a random trip can be shared
The model computes whether a random trip can be shared with earlier or later trips by integrating over compatible trips in time, space, and trajectory. Its assumptions make these compatibility probabilities analytically tractable.
- Sharing cases: A trip can be backward-shareable, forward-shareable, or not shareable at all.Forward pairing considers trips generated during the interval [t_A, t_A + Δ_A].
- Probability derivation: Poisson conditioning yields exponential expressions for the probability that none of the generated candidate trips shares with A.The derivation averages (1 − p(x_A, y_A, t_A))^N over the Poisson-distributed number of candidate trips.
- Model ingredients: Trip duration is Δ_A = ||y_A − x_A||/v, so trip length, the spatial distribution ρ(x, y), and the probability p of a compatible trip determine shareability.The model assumes Poisson trip generation, a two-dimensional spatial distribution, and deterministic trajectories.
- Compatibility conditions: A compatible trip must begin near the existing vehicle and have an endpoint compatible with the existing trip’s trajectory.The compatible endpoint may lie before the existing destination or beyond it toward the city boundary.
- Geometric construction: The shareability shadow is formed by the compatible origin region together with disjoint compatible destination regions.These regions encode the geometric conditions under which another trip can share the existing trip.
- Backward sharing: Only trips starting within [t_A − Δ_B, t_A] can potentially share backward with trip A.Here Δ_B is the time needed to reach the city boundary in the reverse direction of A’s trajectory.
6 Uniform trip generation in time
Assuming uniform trip generation in time replaces the time-varying rate with a constant rate, simplifying the expected number of trips during a trip’s duration.
- Uniform temporal generation: Uniform trip generation assumes λ(t) = λ when the rate changes more slowly than a typical trip duration.Under this approximation, Λ(t_A, Δ_A) = λΔ_A.
7 Bounded uniform and isotropic spatial generation
The model assumes uniformly distributed trip origins and isotropic destinations within a bounded disk, then simplifies shareability regions geometrically to derive an analytic shareability expression. Border effects are omitted because most trips occur far from city boundaries and including them would add complexity with little accuracy improvement.
- Trips start uniformly in the city, with destinations chosen at random inside a centered disk and lengths bounded by R.
- The model approximates delay-feasible regions with disks and rectangles because exact geometric computation is computationally heavy and adds little accuracy.
- Border effects are disregarded because accounting for them would require cumbersome derivations while improving accuracy little, since most trips occur far from city boundaries.
- Combining forward and backward probabilities yields the model's shareability expression, S = 1 − 1/(2L^3)(1 − e^−L)(1 − (1 + 2L)e^−2L).
8 Interpolation for time generation
The paper addresses time-varying trip generation by interpolating hourly rates rather than using only a daily average. This refinement marginally improves accuracy because most trips occur during daytime, when rate variation is mild, supporting predictions even when exact hourly rates are unavailable.
- Trip-generation rates depend strongly on time of day, motivating an interpolation that divides the day into one-hour bins with piecewise constant rates.
- Shareability is computed as a weighted average of hourly shareability values, using the time-varying generation rates.
- Randomly subsampling trips models changes in daily trip volume by multiplying all hourly rates λ_i by a factor p in [0, 1].
- Interpolation improves model accuracy from 97.7 to 98.9% for New York, 95.1 to 97.7% for San Francisco, 94.9 to 95.0% for Singapore, and 91.2 to 91.4% for Vienna.
- The limited impact of interpolation reflects mild daytime variation and suggests accurate predictions where exact hourly rates are unknown.
9 Second-order effect on vehicle speed
The paper extends the model by coupling trip density to vehicle speed and accounting for feedback from shared rides to congestion. Shareability is defined as the unique fixed point of this feedback map, but empirical evaluation is limited by the unavailable speed–density function.
- Average vehicle speed decreases with traffic, while fewer cars can reduce congestion, increase speed, and thereby increase shareability.
- The extended framework introduces a speed function ṽ(λ), a willingness-to-share fraction μ, and a trip-generation rate for matched sharers limited to two people per ride.
- The actual city shareability is modeled through a map F whose unique fixed point s* represents the reached shareability.
- Existence and uniqueness follow because F is increasing, with F(0) > 0 and F(1) < 1.
- Empirical results cannot yet incorporate the feedback fully because the speed–density function ṽ(λ) is unavailable, although sensors may provide it in the future.