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Quantitative analysis of pedestrian counterflow in a cellular automaton model

Stefan Nowak, Andreas Schadschneider

arXiv:1206.3084v1physics.soc-phnlin.CG

TL;DR

Quantitative accounts of pedestrian lane formation are scarce, despite the phenomenon’s familiarity and the limited empirical evidence on gridlock. This study uses a floor-field cellular automaton and an order parameter adapted from colloidal suspensions to characterize counterflow states. Its results distinguish conditions associated with disorder, lanes, and jamming, and report that anticipation reduces collision tendency.

Abstract

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Pedestrian dynamics exhibits various collective phenomena. Here we study bidirectional pedestrian flow in a floor field cellular automaton model. Under certain conditions, lane formation is observed. Although it has often been studied qualitatively, e.g., as a test for the realism of a model, there are almost no quantitative results, neither empirically nor theoretically. As basis for a quantitative analysis we introduce an order parameter which is adopted from the analysis of colloidal suspensions. This allows to determine a phase diagram for the system where four different states (free flow, disorder, lanes, gridlock) can be distinguished. Although the number of lanes formed is fluctuating, lanes are characterized by a typical density. It is found that the basic floor field model overestimates the tendency towards a gridlock compared to experimental bounds. Therefore an anticipation mechanism is introduced which reduces the jamming probability.

I. INTRODUCTION

The introduction frames pedestrian counterflow as a setting where lane formation is familiar but its mechanism and quantitative description remain limited. It then outlines the floor-field cellular automaton used to study the phenomenon.

I. INTRODUCTION

  • Lane formation is spontaneous segregation by walking direction, but its mechanism is not fully understood and does not require a left-right preference.Following others is given as one assumption under which symmetry can break spontaneously.Source: §I · p. 1
  • Quantitative descriptions of lane formation and comparisons with empirical data are rare, while empirical evidence for a gridlock transition is absent in the cited discussion.Recent experiments cited there place the lower density limit for that transition above 3.5 persons/m².Source: §I · p. 1

A. Floor Field Cellular Automaton Model

  • The FFCA represents pedestrians on a square lattice with at most one pedestrian per cell and synchronous moves to unoccupied neighboring cells.The model may use four von Neumann neighbors or eight Moore neighbors.Source: A. Floor Field Cellular Automaton Model · p. 1, Figure 1 · p. 2
  • In the studied variant, transition probabilities use static, dynamic, and anticipation floor fields, with the anticipation field distinguishing it from the standard FFCA.The standard model corresponds to kA = 0.Source: A. Floor Field Cellular Automaton Model · p. 2

1. Static Floor Field

The model combines a static field encoding desired direction with a dynamic trace that diffuses and decays over time.

1. Static Floor Field

  • The static floor field encodes preferred motion direction; in counterflow, particles of types A and B are affected by different static fields.The field typically represents distance to a destination, and the counterflow expression is simplified.Source: §1 · p. 2

2. Dynamic Floor Field

  • Moving particles leave a dynamic floor-field trace that attracts others and encourages following, while temporarily discounting each particle’s own last trace.Separate dynamic fields interact with the corresponding particle types.Source: §2 · p. 2
  • The dynamic field is updated through diffusion and decay, with its diffusion term expressed as a discretized Laplace operator.The corresponding diffusion equation has constant β = α(1 − δ)/4 and an additional decay term.Source: §2 · p. 2

3. Anticipation Floor Field

The anticipation floor field represents expected near-future occupancy so pedestrians can avoid the routes of opposing walkers.

3. Anticipation Floor Field

  • The anticipation floor field encodes cells likely to be traversed soon by each particle type, making those cells repulsive to the opposing type.The range is controlled by λ; the distance measure counts the minimum forward steps between cells, and the field is defined analogously for type B.Source: §3 · p. 3

4. Update Rules

Each timestep computes anticipation, selects destinations, resolves conflicts, moves particles, and updates the dynamic field. The lane order parameter quantifies directional segregation while correcting for a random-distribution baseline.

4. Update Rules

  • The update sequence computes anticipation, selects destinations by transition probability, resolves destination conflicts randomly, moves particles, then diffuses and decays the dynamic field.The simplest conflict rule gives each of n particles choosing one destination an equal probability 1/n.Source: §4 · p. 3
  • The study uses the simplest conflict rule, noting no indication that conflict handling qualitatively affects lane formation.Alternative formulations include friction or coupling to evolutionary games.Source: §4 · p. 3

B. Order Parameter for Lane Formation

  • The lane order parameter averages per-particle contrasts between same-type and opposite-type pedestrians in the same row.Its underlying measure is adapted from colloidal lane analysis, and the row-based calculation reflects the lattice’s spatial discreteness.Source: B. Order Parameter for Lane Formation · p. 3
  • The reduced order parameter subtracts the random-distribution baseline because the uncorrected value can be large at low densities without lanes.The order parameter can in principle become negative.Source: B. Order Parameter for Lane Formation · pp. 3–4

III. DETERMINATION OF STATES

The simulations distinguish four pedestrian states using density, motion, and the reduced order parameter. Unless noted otherwise, the two pedestrian types are equally represented, and the specified setup uses a 40 m × 4 m corridor.

III. DETERMINATION OF STATES

  • The default parameters set kS = 2.5, kD = kA = 0, α = 0.3, δ = 0.1, and λ = 0.8, on a 10 × 100 lattice.Source: §III · p. 4
  • The model uses a 40 m × 4 m corridor, periodic boundaries along the walking direction, open boundaries across it, and a four-neighbor von Neumann neighborhood.Source: §III · p. 4

A. Definition of Different States

B. Simulation Procedure

Randomly initialized systems were simulated across density and dynamic-field coupling values, with at least 100 runs per parameter pair. Runs stopped upon detected gridlock, established lanes, or reaching a density-dependent time limit.

B. Simulation Procedure

  • At least 100 randomly initialized runs were performed for each density–coupling pair, with ρ from 0.05 to 1.00 and kD from 0.0 to 7.5.Source: B. Simulation Procedure · p. 4
  • A run stopped as gridlock when recent average flow fell below 5×10^-4, and the fraction of runs stopped this way defined PJam.Source: B. Simulation Procedure · p. 4
  • Lane formation was detected when the order parameter’s maximum and minimum over the last 1000 timesteps met the lane criterion; the criterion imposed no absolute Φ threshold.Source: B. Simulation Procedure · p. 4
  • Runs exceeding Tmax = 20000√ρ typically represented disordered or free-flow states, and their velocity, flow, and order parameter were averaged over the last 1000 timesteps.Source: B. Simulation Procedure · p. 4

C. Results

Jamming becomes more likely at higher density and weaker coupling, while anticipation suppresses jams more effectively than the dynamic floor field. Among non-gridlocked runs, lane order strengthens with density and coupling, and the order parameter’s width dependence levels off near W = 30.

1. Jam Probability

  • Jam probability generally rises with density and falls as kD or kA increases; anticipation prevents jams more effectively than the dynamic floor field.Higher density means more opposing particles and less room to avoid collisions; anticipation keeps opposing particles from coming too close in the same row.Source: §1 · p. 5
  • For kA = 0, every simulation above ρ = 0.5 gridlocked, whereas kA > 3 made PJam vanish at almost every density.At ρ = 0.2, the DFF-only setting still had nonzero jam probability.Source: §1 · p. 5

2. Order Parameter

  • Among non-gridlocked runs, the reduced order parameter is near zero when kD = kA = 0, but increases with density and coupling.Its slight positive value at zero coupling is attributed to sidesteps after contact between different particle types, which destabilize homogeneous states.Source: §2 · p. 5
  • A minimum density is needed for an order parameter near 1, and the disorder-to-lane transition sharpens as density increases; above ρ = 0.2, low-coupling gridlocks leave few transition data points.At large density, the system lacks a stable disordered state and ends in gridlock or almost perfect lanes.Source: §2 · p. 5

3. Dependence on the System Size

  • The order parameter increases with corridor width and appears to stop changing beyond about W = 30 when length is fixed.Source: §3 · p. 5
  • Length scaling was not feasible because jam probability increased; its dependence on L/W is only a rough, small-system approximation, with non-monotonic behavior at large W.Source: §3 · p. 5

IV. PROPERTIES OF THE LANE STATE

In lane states, the row-density distribution depends on the dynamic floor field’s attraction between same-type particles. It also exhibits characteristic lane densities across tested coupling and global-density ranges, alongside additional peaks under some conditions.

A. Distribution of Densities

  • With DFF, the row-density distribution gains a peak at ρi = 0 and broader support as kD increases, indicating more empty rows, denser rows, and reduced average velocity.Without DFF, even with anticipation, rows remain concentrated near the global density.Source: A. Distribution of Densities · p. 5
  • Characteristic lane-density peaks occur at ρ*3 = 0.56 across tested kD = 1–7 and global densities ρ = 0.1–0.3, with each peak corresponding to a different lane count.ρ*j corresponds to j + 1 lanes; narrower lanes have higher density than wider lanes.Source: A. Distribution of Densities · p. 5
  • Increasing kD raises the tendency to form more than two lanes: kD = 1 yields only two-lane configurations, while kD = 5 also yields three- and four-lane configurations.Source: A. Distribution of Densities · p. 6
  • Additional density-dependent peaks can occur at low global density and high kD when particles of each type concentrate in one row.At ρ = 0.1 in a 100×10 system, each type has 50 particles and a single row’s maximum density is 0.5; larger global densities can also produce a peak beyond ρ*3.Source: A. Distribution of Densities · p. 6

B. Velocities

The effects of coupling on velocity differ between dynamic and anticipation floor fields. Lane lifetimes also depend on coupling strength and lane count, while destabilized configurations can return to lanes, disorder, or gridlock.

B. Velocities

  • Anticipation makes mean velocity increase monotonically with kA, whereas dynamic-floor-field velocity can decrease as same-type clustering raises row densities.With DFF, v(kD) need not be monotonic, and maximum velocity does not necessarily coincide with maximum order.Source: B. Velocities · p. 7, Figure 7 · p. 7
  • At ρ = 0.1, v(kD) has no maximum; at high densities, missing low-kD data can also leave a monotonically decreasing curve.At ρ = 0.1, local density does not exceed 0.5, so particles have on average at least one empty cell ahead.Source: B. Velocities · p. 7

C. Stability and Lifetime

  • Two-lane lifetimes rise roughly exponentially with kD, while three- and four-lane lifetimes level near 1000 and 250 timesteps, respectively.The three-lane curve levels off beyond kD ≈ 1.25; Fig. 9 shows the lifetime curves.Source: C. Stability and Lifetime · p. 8, Figure 9 · p. 8
  • With anticipation but no DFF, lane-count differences in lifetime shrink as kA increases, and all three configurations grow faster than exponentially.Two lanes still have the longest lifetimes.Source: C. Stability and Lifetime · p. 8, Figure 9 · p. 8
  • After losing stability, systems can become gridlocked, disordered, or form lanes again, depending on density and coupling.When more than two lanes merge, strong DFF attraction commonly leaves a two-lane configuration.Source: C. Stability and Lifetime · p. 8
  • At increasing kD, merging without a detectable Φ drop occurs in up to 50% of cases, so results above the lifetime-distribution gap are excluded.Runs capped at Tmax = 1.5 × 10^5 may distort states whose lifetimes are of that order.Source: C. Stability and Lifetime · p. 8

V. OPEN BOUNDARY CONDITIONS

The open-boundary simulations examine whether two opposing groups can pass and how their coupling choices shape collisions and lane persistence. Anticipation reduces observed gridlock, and combining it with the dynamic floor field yields persistent lanes after passage.

A. Two Groups Passing Each Other

  • With both fields off, every run gridlocks; at kD = 2 without anticipation, about 50% gridlock, while kA ≥ 2 yielded no observed gridlocks for any tested kD.Only runs that did not gridlock were included in subsequent analyses.Source: A. Two Groups Passing Each Other · p. 8
  • The collision index counts neighboring-cell pairs with a type-A particle on the left and a type-B particle on the right.The simulations begin with opposing groups at opposite ends and repeat 5000 times for each tested coupling pair.Source: A. Two Groups Passing Each Other · pp. 8–9
  • Without anticipation, opposing-particle collisions leave small jams that dissipate slowly, and lanes are less distinct than in the periodic system.The collision index fades slowly even after most particles have exited.Source: A. Two Groups Passing Each Other · p. 9
  • Anticipation alone produces many thin lanes during contact, but they disappear immediately after the groups pass.The order parameter also reflects this rapid destruction.Source: A. Two Groups Passing Each Other · p. 9
  • Combining anticipation and DFF reduces collisions further and usually leaves two or three lanes that persist after passage.These lanes can be nonparallel to the walking direction; the snapshots compare field settings at timestep 75.Source: A. Two Groups Passing Each Other · p. 9, Figure 10 · p. 9

B. Random Insertion of Particles

With open boundaries, anticipation reduces gridlock risk, while the dynamic floor field does not prevent jams. Lanes usually form before gridlock at sufficiently large coupling constants, and the conclusions report a quantitative order-parameter analysis of lane formation.

B. Random Insertion of Particles

  • Particles enter dynamically with probability β at boundary cells, while removal occurs with probability 1 and depends on particle type and boundary.Type A enters at the left and type B at the right; the left absorbs B and the right absorbs A.Source: B. Random Insertion of Particles · p. 9
  • The dynamic floor field does not help much to prevent jams and instead supports jam formation.Figure 12 presents jam probability at timestep 20000 for open boundary conditions.Source: Figure 12 · p. 9, B. Random Insertion of Particles · p. 9
  • With anticipation, gridlocks can be avoided at β = 0.12; β and ρ vary linearly, with proportionality 2.5–2.8 and a maximum gridlock-free density of ρ = 0.34.The proportionality constant depends slightly on the coupling constants.Source: B. Random Insertion of Particles · pp. 9–10
  • Before gridlocks form, lanes usually appear for sufficiently large coupling constants, with an order parameter lower than in the periodic case.The order parameter is comparable to findings in the previous section.Source: B. Random Insertion of Particles · p. 10
  • In open systems, continued particle inflow from behind strongly reduces the chances that a gridlock will resolve.This contrasts with periodic systems, where flow near gridlock becomes very small.Source: B. Random Insertion of Particles · p. 10

VI. CONCLUSIONS

  • The conclusions report the first quantitative results for lane formation, enabled by an order parameter adapted from colloidal-suspension analysis.The order parameter is reported as suitable for detecting lanes at least in a straight-corridor scenario.Source: §VI · p. 10
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