A stochastic model of randomly accelerated walkers for human mobility

Travel-times

• Car travel times have an exponential tail, and their distributions become universal across cities when time is rescaled by the city’s average.The average travel time ranges from 9 to 18 minutes and grows with city population, with ⟨t⟩∝P^µ and µ = 0.07 ± 0.02.Source: Travel-times · p. 3
• The average travel time t contains all the information needed to describe car travel times in a particular city.The figure caption describes an exponential distribution whose differences between cities are encoded by the average travel time.Source: Travel-times · p. 3, Figure 1 · p. 3

Rest times

• Rest-time distributions show little variation among cities for stops within 24 hours and are close to a power law for shorter pauses.A lognormal tail is also plausible, while a stretched exponential fits the curve from 5 minutes to 30 days but lacks a theoretical explanation.Source: Rest times · p. 3, Figure 2 · p. 4

• Uncorrelated acceleration kicks produce a Gaussian-like velocity distribution in the randomly accelerated walk.Source: Dependence of velocities over… · p. 4
• Combining the velocity model with exponentially distributed travel times yields a large-displacement stretched exponential that reasonably fits data with one parameter, versus a three-parameter truncated power law.The stretched-exponential fit uses γ = 1/4, δ = 1/2, and C = 0.63 km−0.5.Source: Dependence of velocities over… · pp. 4–5
• For long trips, empirical average velocities grow linearly; Fig. 3’s constant-acceleration fit uses v0 = 17.9 km/h and a = 16.7 km/h2.Source: Figure 3 · p. 4, Dependence of velocities over… · p. 5
• The simple random velocity model predicts ⟨|v|⟩∝t^1/2, which the empirical linear growth does not match.Source: Dependence of velocities over… · p. 5
• The authors attribute the linear growth to longer trips’ greater use of faster modes or roads within hierarchical transportation networks.Source: Dependence of velocities over… · p. 5

Randomly accelerated walkers

• Trips accelerate through speed layers by constant δv kicks, then decelerate through the same layers with kicks of −δv at rate p.The model assumes equal average durations for both phases and estimates peak speed at the trip midpoint.Source: Randomly accelerated walkers · p. 5, Figure 4 · p. 5
• The model predicts mean speed grows linearly with travel time before saturation from the finite number of speed layers.The expected number of jumps by the midpoint is pt/2.Source: Randomly accelerated walkers · p. 5
• Poisson-distributed jump counts, with mean pt/2, yield the model’s conditional average-speed distribution P(v|t).The Gamma function provides an analytic continuation of the factorial in the jump-count distribution.Source: Randomly accelerated walkers · pp. 5–6
• p′ = 1.06 jumps/h and δv′ = 20.9 km/h fit empirical P(v|t) distributions across travel times of 5–180 minutes.The fit corresponds to velocity gaps of about 40 km/h, consistent with common Italian speed-limit steps.Source: Figure 5 · p. 6, Randomly accelerated walkers · p. 6

Displacement distribution

• The model computes P(∆r) by superimposing Poisson distributions associated with different trip durations.The figure describes each component as having λ = p′t, with its area decreasing as ∝e^−t/t.Source: Displacement distribution · p. 6, Figure 6 · p. 7
• The predicted distribution is not broad, unlike Lévy flights, which have divergent moments and are governed by large fluctuations.The authors therefore do not expect Lévy-flight-associated phenomena such as super-diffusion for randomly accelerated walks.Source: Displacement distribution · pp. 6–7
• The model’s predicted displacement distribution is consistent with the empirical data and compares favorably with a three-parameter truncated-power-law fit.The prediction uses the mean travel time and parameters estimated from the conditional speed distribution, rather than being an a posteriori fit to displacement data.Source: Figure 6 · p. 7, Displacement distribution · p. 7
• Different forms, including truncated power laws and a stretched exponential, can fit the displacement data, so macroscopic fits alone do not identify the underlying dynamics.The authors attribute their model to transitions between roads or transport modes with different velocities.Source: Displacement distribution · p. 7

Private Transportation Data

• The dataset contains 128,363,000 trips by 779,000 vehicles in Italy during May 2011, with GPS records at approximately one-second temporal resolution.It covers about 2% of registered vehicles; records include engine starts and stops, and travel times include time spent looking for parking.Source: Private Transportation Data · p. 8
• About 10% of data were discarded for systematic errors, and brief engine-off intervals under 30 seconds could be treated as continuation of the same trip.Redundant stopping-point information was used to correct 20% of the data affected by signal loss.Source: Private Transportation Data · p. 8
• Drivers were assigned to a city by where they spent most of their parking time, and their trips within and outside that urban area were included.The drivers’ city of residence was unavailable for privacy reasons.Source: Private Transportation Data · p. 8

Details of the model

• Trips shorter than 1 km or 5 minutes were excluded because very short trips may not reach the model’s assumed base speed of about 18 km/h.The model’s 130 km/h upper bound permits about three velocity-layer jumps; the authors expect significant prediction deviations for trips longer than four hours.Source: Details of the model · p. 8

Trip duration in public and private transportation

• London Oyster journeys show an exponential tail with b = 9.3 ± 0.3 minutes, reached after at least c = 36 ± 2 minutes.The data cover a 5% sample of bus, Tube, DLR, and London Overground journeys in one week in November 2009.Source: Figure S1 · p. 11
• Across Italian cities with at least 500 monitored drivers, average private-car travel times are approximately 9–18 minutes and correlate with population at r = 0.40.A power-law fit gives t ∝ population^γ with γ = 0.07 ± 0.02.Source: Figure S2 · p. 11

Dependence of velocities over time in public transportation.

• The public-transport estimates are not based on actual flows; they assume uniform travel demand across all origin–destination pairs and shortest time-respecting paths.The paths are estimated from a Great Britain multilayer transport-network dataset, integrating walking and connection waiting times.Source: Dependence of velocities over time in… · p. 12
• For public transport, average speed grows linearly for trips lasting 0.5–3.5 hours, with v0 = 7.4 ± 0.4 km/h and a = 12.0 ± 0.2 km/h².Trips under 30 minutes tend to be faster, likely because they lack time-consuming connections.Source: Dependence of velocities over time in… · p. 12, Figure S3 · p. 12
• For London urban trips under one hour from Charing Cross, the conditional speed distribution is universal and Gaussian-like.The authors suggest weighted edge speeds and waiting times may contribute to this shape through the central limit theorem.Source: Dependence of velocities over time in… · p. 12
• In the two-layer transportation model, average speed is the weighted average of time spent travelling on the base and fast layers.The figure reports that a numerical simulation confirms the two-layer result of Eq. (9).Source: Figure S4 · p. 13

Accelerated walker with a limited number of layers

• In a two-layer network, a traveler starts at speed v0 and jumps to the faster layer at speed v1 = v0 + dv with probability p per unit time.The jump is modeled as a Poisson process.Source: Accelerated walker with a limited… · p. 13
• Averaging position over the jump time gives the model’s average speed as v = x/t.The bar denotes averaging over different trips.Source: Accelerated walker with a limited… · p. 13
• For pt ≪ 1, average speed grows linearly from v0; for pt ≫ 1, it approaches v1.The model recovers linear speed growth at some durations and a limiting speed at longer durations.Source: Accelerated walker with a limited… · p. 13
• With equally probable jumps and a fixed speed gap between successive layers, constant acceleration extends to pt > 1, consistent with a hierarchical transport infrastructure explaining observed acceleration.The acceleration is expected to be proportional to jump probability p and speed gap δv.Source: Accelerated walker with a limited… · p. 13
• In the six largest Italian cities, the shared speed at t = 5 min is v = 16.5 ± 0.5 km/h, used as the prediction’s v0.Individual-city speed growth is less uniform than the aggregate pattern, possibly reflecting road-network and geographical differences.Source: Figure S5 · p. 14

Variability of P(∆r) among cities

• The model’s predicted displacement distributions are reported to match the empirical distributions across the six cities.The predictions are compared with a-posteriori truncated-power-law and stretched-exponential fits.Source: Variability of P(∆r) among cities · p. 14, Figure S6 · p. 15
• City fits use v0 = 16.5 km/h and estimate p′ and δv′ from P(v|t) between v0 and 130 km/h.The prediction also uses each city’s average travel time.Source: Variability of P(∆r) among cities · p. 14, Figure S6 · p. 15
• Individual-city speed growth is less uniform than the aggregate trend, with deviations attributed tentatively to road-network and regional geography differences.Source: Variability of P(∆r) among cities · p. 14
• Across cities, fitted values range from p′ = [0.97, 1.81] jumps/h, δv′ = [13.2, 18.5] km/h, and t = [0.30, 0.37] h.Source: Figure S6 · p. 15

Saddle-point analysis of P(∆r).

• At large r and t, the integral is governed by a saddle point defined by dF/dt = 0.Source: Saddle-point analysis of P(∆r) · p. 16
• For large times, the analysis obtains the scaling r ∼ t^2.Source: Saddle-point analysis of P(∆r) · p. 16
• The same large-time behavior is recovered with a Gaussian velocity distribution, as observed for public transportation.Source: Saddle-point analysis of P(∆r) · p. 16