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Universal power law governing pedestrian interactions

Ioannis Karamouzas, Brian Skinner, Stephen J. Guy

arXiv:1412.1082v1physics.soc-phcond-mat.stat-mechphysics.data-an

TL;DR

The paper addresses how to quantitatively describe pedestrian interactions when spatial separation alone is insufficient. It uses pair-distribution analysis to identify projected time to collision as a sufficient descriptor, finding an inverse-quadratic interaction-energy law that reproduces observed crowd behavior in simulations.

  • Problem

    Spatial coordinates alone cannot accurately quantify pedestrian interactions, leaving the interaction descriptor unresolved.

  • Method

    The paper compares pair distribution functions based on spatial separation and projected collision time, using g(x)=P(x)/P_NI(x) to infer interaction energy.

  • Results

    An exponent of 2 models the interaction energy E(τ) in both Outdoor and Bottleneck datasets, while simulations reproduce empirical g(τ) and several crowd phenomena.

  • Takeaways & Limitations

    Projected time to collision provides a sufficient, anticipatory descriptor for pedestrian interactions across varying approach rates and simulation conditions.

Abstract

from arXiv · show

Human crowds often bear a striking resemblance to interacting particle systems, and this has prompted many researchers to describe pedestrian dynamics in terms of interaction forces and potential energies. The correct quantitative form of this interaction, however, has remained an open question. Here, we introduce a novel statistical-mechanical approach to directly measure the interaction energy between pedestrians. This analysis, when applied to a large collection of human motion data, reveals a simple power law interaction that is based not on the physical separation between pedestrians but on their projected time to a potential future collision, and is therefore fundamentally anticipatory in nature. Remarkably, this simple law is able to describe human interactions across a wide variety of situations, speeds and densities. We further show, through simulations, that the interaction law we identify is sufficient to reproduce many known crowd phenomena.

Supplemental material for: Universal power law governing pedestrian interactions

The supplemental material identifies the paper, its authors, affiliations, and date.

  • The authors are Ioannis Karamouzas, Brian Skinner, and Stephen J. Guy.
  • The authors are affiliated with the University of Minnesota and Argonne National Laboratory.
  • The document is dated December 4, 2014, and identified as arXiv:1412.1082v1.

EXPERIMENTAL DATASETS

The study combines motion datasets from outdoor and controlled laboratory environments, then processes pedestrian trajectory data for analysis.

  • EXPERIMENTAL DATASETS: Six datasets comprise combined Outdoor and Bottleneck datasets representing sparse-to-moderate outdoor settings and dense crowds passing through bottlenecks.
  • EXPERIMENTAL DATASETS: Trajectory data were recorded using motion-capture or computer-vision techniques across outdoor environments and controlled laboratory settings.
  • EXPERIMENTAL DATASETS: The datasets include sparse outdoor bidirectional flows, moderately dense multidirectional flow, and controlled dense crowds moving through a narrow constriction.
  • EXPERIMENTAL DATASETS: The table reports dataset characteristics, including average density computed with Edie’s generalized definition.

DETAILED DESCRIPTION OF THE PAIR-DISTRIBUTION FUNCTION

The pair distribution function compares observed pedestrian separations with a non-interacting reference distribution, estimated using time-scrambled trajectories.

  • The pair distribution function g(x) measures how interactions alter the likelihood of pedestrian separation x.
  • g(x) is defined as the observed separation density P(x) divided by the hypothetical non-interacting density PNI(x).
  • The non-interacting reference is approximated by randomly permuting time information between pedestrians while preserving instantaneous positions and velocities.

STATISTICAL METHODS

Statistical tests evaluate whether distance- or time-to-collision-based pair distributions depend on approach rate, while power-law fitting estimates the interaction-energy exponent.

  • STATISTICAL METHODS: g(r) significantly depends on approach rate, [F(2, 576) = 27.811, P < 0.001], whereas g(τ) does not, [F(2, 510) = 0.143, P = 0.866].
  • STATISTICAL METHODS: The absence of approach-rate dependence in g(τ) indicates that τ is a sufficient descriptor of pedestrian interactions.
  • STATISTICAL METHODS: The interaction energy is estimated over finite τ intervals because tracking errors and statistical noise limit its well-defined range.
  • STATISTICAL METHODS: 2.05 ± 0.123 and 2.017 ± 0.192 are the fitted exponents for the Outdoor and Bottleneck datasets, respectively.
  • STATISTICAL METHODS: An exponent of 2 provides an adequate interaction-energy model for both datasets.

PAIR DISTRIBUTION FUNCTION IN 2D SPATIAL COORDINATES

Relative spatial coordinates alone do not adequately describe pedestrian interactions: the pair distribution depends on approach rate even when displacement direction is fixed. This supports using a time-based descriptor rather than position alone.

  • PAIR DISTRIBUTION FUNCTION IN 2D SPATIAL COORDINATES: The full displacement vector r cannot parameterize pedestrian interactions because g depends on the pedestrian rate of approach v.This inconsistency rules out interaction forms based only on relative spatial coordinates.
  • PAIR DISTRIBUTION FUNCTION IN 2D SPATIAL COORDINATES: The dependence of g(r) on approach rate contradicts the expectation that spatial distance alone would determine the pair distribution.
  • PAIR DISTRIBUTION FUNCTION IN 2D SPATIAL COORDINATES: At fixed orientation θ = 0, g(r, θ = 0) significantly depends on the rate v at which pedestrians approach each other.
  • PAIR DISTRIBUTION FUNCTION IN 2D SPATIAL COORDINATES: For fixed θ, g(r, θ) varies with approach rate v, indicating that the displacement vector alone cannot accurately quantify interactions.

ORIENTATION-INDEPENDENCE OF g(τ)

The time-to-collision variable τ provides an orientation-independent description of pedestrian interactions. Pair-distribution curves for different relative orientations collapse onto one another, extending its demonstrated velocity independence.

  • ORIENTATION-INDEPENDENCE OF g(τ): g(τ) is independent of both approach rate v and relative orientation between pedestrians.
  • ORIENTATION-INDEPENDENCE OF g(τ): The pair-distribution function g(τ) is independent of pedestrians’ relative orientation, as curves for different orientations collapse onto each other.

ABSENCE OF INTERACTION FOR UNDEFINED τ

Pedestrian pairs with undefined τ show no finite interaction beyond a short-range exclusion. In both datasets, the relevant boundary is approximately 0.4 m, although the Outdoor data also show a positive correlation near 0.6 m.

  • ABSENCE OF INTERACTION FOR UNDEFINED τ: Pairs not on a collision course show no evidence of finite interaction beyond a short-ranged exclusion.
  • ABSENCE OF INTERACTION FOR UNDEFINED τ: For Bottleneck pairs with undefined τ, g(r) ≈ 1 at all r ≳ 0.4 m, indicating no interaction beyond that distance.
  • ABSENCE OF INTERACTION FOR UNDEFINED τ: For Outdoor pairs with undefined τ, finite repulsion appears only at r ≲ 0.4 m.
  • ABSENCE OF INTERACTION FOR UNDEFINED τ: The Outdoor dataset has a peak in g(r) near r ≈ 0.6 m, suggesting positive correlation among some non-colliding pedestrians.

ANALYTICAL EXPRESSION FOR THE SIMULATED INTERACTION FORCE

The simulation computes pedestrian interaction forces from the time-to-collision τ, which is estimated by linearly extrapolating current trajectories. The force is combined with obstacle-repulsion and driving forces to produce the complete simulation.

  • ANALYTICAL EXPRESSION FOR THE SIMULATED INTERACTION FORCE: Equation (2) defines the interaction energy for pedestrian pairs with finite time-to-collision τ and relates it to the force Fij in force-based simulations.
  • ANALYTICAL EXPRESSION FOR THE SIMULATED INTERACTION FORCE: At each simulation step, τ is estimated by linearly extrapolating pedestrians’ trajectories using their current velocities.
  • ANALYTICAL EXPRESSION FOR THE SIMULATED INTERACTION FORCE: A collision occurs at τ > 0 when extrapolated pedestrian discs of radii Ri and Rj intersect; otherwise, the interaction force Fij is 0.
  • ANALYTICAL EXPRESSION FOR THE SIMULATED INTERACTION FORCE: The collision time is computed from relative displacement xij, relative velocity vij, pedestrian radii, and the discriminant d = b^2 − ac.
  • ANALYTICAL EXPRESSION FOR THE SIMULATED INTERACTION FORCE: The complete simulation combines pedestrian interaction and static-obstacle repulsion forces with a driving force.

SIMULATION RESULTS

Simulations using the derived anticipatory force law reproduce diverse collective crowd phenomena, empirical interaction behavior, and realistic density-dependent speed and flow relations.

  • SIMULATION RESULTS: The simulations use normally distributed preferred walking speeds averaging 1.3 ± 0.3 m/s, with default interaction parameters k = 1.5 and τ0 = 3 s.These parameters were used to approximate typical human behavior in the simulations.
  • SIMULATION RESULTS: The derived force law reproduces a wide variety of collective phenomena across evacuation, hallway, bottleneck, crossing, and collective-motion simulations.These include clogging, lane formation, zipping, diagonal group patterns, and spontaneous vortex formation.
  • SIMULATION RESULTS: The simulated fundamental diagram shows good agreement with the measured human relation among speed, flow, and density.Density and speed were measured using Edie’s generalized definitions over 0.5 m × 0.5 m cells and 4 s intervals.
  • SIMULATION RESULTS: The force model reproduces the observed dependence of the spatial pair distribution function g(r) on pedestrians’ rate of approach v.Distance-dependent-force simulations show weaker dependence on v and fail to show a strong dependence of g on time-to-collision.
  • SIMULATION RESULTS: The inferred interaction energy E(τ) closely follows the inverse quadratic power law in hallway, bottleneck, and evacuation simulations.Existing anticipatory models do not consistently capture this law.
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