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REVIEW 2 major objections 5 minor 68 references

Crossing angle, not how we weight who pedestrians see, organizes how crowds adapt during encounters.

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2026-07-13 04:35 UTC pith:RX4XGWTO

load-bearing objection Solid empirical re-analysis of public crossing-flow data: geometry dominates FOV weighting of crowdedness, and dynamic phase-space diagrams show clear history dependence. the 2 major comments →

arxiv 2607.09221 v1 pith:RX4XGWTO submitted 2026-07-10 physics.soc-ph physics.comp-ph

I see you, do you see me? Perception-based crowdedness and behavioral responses in pedestrian dynamics

Igor Ho{\l}owacz , Pratik Mullick This is my paper
classification physics.soc-ph physics.comp-ph PACS 89.40.-a45.70.Vn89.75.Fb
keywords pedestrian dynamicscrossing flowsperceived crowdednessperceptual anisotropydynamic fundamental diagramsdirectional deviationinteraction geometry
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper asks whether the usual way of linking local crowding to walking speed still holds once we account for the fact that people notice those ahead of them more than those behind. Using high-frequency trajectories from controlled experiments in which two groups cross at angles from head-on to parallel, the authors build a distance-weighted crowdedness index and then systematically discount people outside a pedestrian’s field of view. The numerical value of crowdedness shrinks when the rear hemisphere is discounted, yet the time course of the interaction, the ordering by crossing angle, and the shape of the velocity–crowdedness curves stay essentially the same. Pedestrians do leave their intended group headings, but they do so through many small successive turns rather than sharp swerves; initial slowing depends strongly on geometry while the later recovery does not. Because the same instantaneous crowdedness can appear both while the streams are locking and while they are unlocking, the authors argue that only time-resolved “dynamic fundamental diagrams” capture the full process.

Core claim

For bidirectional crossing flows spanning 0°–180°, changing how strongly pedestrians outside the field of view contribute to a local crowdedness measure merely rescales that measure; the qualitative temporal dynamics, angle ordering, and behavioral relationships with speed, directional deviation and acceleration remain intact. Interaction geometry therefore dominates the organization of these flows, and non-retracing phase-space trajectories show that instantaneous state variables alone cannot distinguish build-up from recovery.

What carries the argument

A distance-weighted perceived-crowdedness index ρ_c whose rear-hemisphere weight c is varied continuously from fully isotropic (c=1) to fully anisotropic (c=0); trajectories are then examined in dynamic fundamental diagrams that track how ρ_c co-evolves with velocity, directional deviation and acceleration over a rescaled interaction interval.

Load-bearing premise

The entire temporal comparison rests on interaction start and end times taken from an earlier edge-cutting procedure; if those times mis-identify the true interaction window, every rescaled curve and phase-space loop shifts.

What would settle it

Re-analyze the same trajectories with an independent, objective definition of interaction onset and offset (for example first and last times any pair of opposing pedestrians come within a fixed distance) and check whether the claimed invariance to perceptual weighting and the non-retracing loops survive.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Dynamic fundamental diagrams that retain history should replace single-valued density–speed curves when describing transient crossing or merging flows.
  • Models that treat density as isotropic can still recover the correct qualitative crossing dynamics if the relative angle of the streams is correctly specified.
  • Smooth incremental heading corrections, not large discrete turns, are the microscopic signature of successful self-organization into stripes or lanes.
  • Recovery accelerations after a crossing may be largely geometry-independent once the streams begin to separate, simplifying post-conflict flow forecasts.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same robustness test applied to bottleneck or restricted-visibility settings could reverse the conclusion and make field-of-view weighting essential rather than cosmetic.
  • If incremental turning remains small while group-heading deviation grows, stripe formation may be diagnosable in real time from the ratio of the two deviation measures alone.
  • The observed hysteresis-like loops imply that short-term prediction of pedestrian velocity from instantaneous density will systematically err unless the recent interaction phase is also known.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The manuscript reanalyzes a public experimental dataset of bidirectional pedestrian crossing flows (angles 0°–180°) with a distance-weighted local crowdedness index ρ_c that incorporates perceptual anisotropy by down-weighting agents outside a 210° field of view by a factor c ∈ {0, 0.5, 1}. It reports that changing c primarily rescales the numerical values of crowdedness while leaving temporal evolution, velocity–crowdedness diagrams, directional deviations (δ1 from group heading, δ2 incremental turning), acceleration dynamics, and non-retracing phase-space trajectories qualitatively intact and ordered by crossing angle. The authors conclude that interaction geometry dominates the organization of these flows relative to the precise FOV weighting, that pedestrians adapt via smooth incremental corrections rather than abrupt turns, that deceleration is more geometry-sensitive than recovery, and that dynamic fundamental diagrams better capture history-dependent transient interactions than static instantaneous relations.

Significance. If the reported robustness and geometry dominance hold, the work usefully clarifies that, for open crossing geometries, the precise anisotropic weighting of a local interaction measure is secondary to relative flow directions, and it strengthens the case for dynamic (path-dependent) fundamental diagrams over conventional static ones when characterizing transient encounters. Concrete strengths include use of a fully public trajectory dataset, explicit formulas (Eqs. 1–3), open analysis code, systematic multi-angle coverage, and complementary behavioral observables (velocity, two weakly correlated deviation measures, acceleration). These make the qualitative claims reproducible and falsifiable against other flow configurations.

major comments (2)
  1. Section 2 (after Eq. 3) and all subsequent temporal/phase-space figures: the global time rescaling t' = (t − Ti)/(Tf − Ti) rests entirely on the edge-cutting algorithm of the prior study for defining interaction start/end. While the authors correctly treat α = 0° as a non-interacting baseline and exclude boundary intervals, no sensitivity check is provided (e.g., shifting Ti/Tf by a few seconds or using an alternative overlap criterion). Because every averaged curve and loop structure is conditioned on this common temporal frame, a brief robustness test would strengthen the claim that geometry, not the particular interaction window, organizes the dynamics.
  2. Figures 2–4, 6–9 and Appendices B–C: all reported curves are averages or medians without error bands, standard errors, or bootstrap intervals. The central qualitative claim—that shapes, orderings, and non-retracing loops are preserved across c—is visually clear, yet the absence of uncertainty quantification leaves open whether small angle-to-angle differences (e.g., the 3°–6° spread in |δ1|) or the precise location of acceleration zero-crossings are statistically distinguishable. Adding simple variability measures would make the robustness statement fully quantitative without altering the manuscript’s scope.
minor comments (5)
  1. Eq. (1) and surrounding text: the inverse-square distance weighting is presented without discussion of alternatives (e.g., exponential decay common in social-force models). A one-sentence justification or reference would help readers assess sensitivity of the rescaling result.
  2. Figure 1 caption and text: the FOV angle ϕ = 210° is taken from clinical perimetry literature; a brief note that results are expected to be robust to modest changes in ϕ (as with c) would be useful.
  3. Section 3: the weak correlation r ≲ 0.3 between δ1 and δ2 is reported with p-values; stating the exact sample size (or degrees of freedom) used for the correlation would improve transparency.
  4. Appendix A: the signed-deviation plots (Fig. 10) are helpful for interpreting the 30° early peaks; cross-referencing them more explicitly in the main-text discussion of α = 30° would aid readers who skip the appendix.
  5. Data/code availability: the Zenodo and GitHub links are welcome; ensuring the repository contains a short README that reproduces at least one main figure would further raise reproducibility standards.

Circularity Check

0 steps flagged

No significant circularity: empirical re-analysis of a public trajectory dataset with a literature-derived crowdedness index; qualitative robustness conclusions are not forced by construction of the measure or by self-citation.

full rationale

The paper defines a distance-weighted crowdedness index ho_c (Eq. 1) that incorporates a tunable FOV weight c otin {0, 0.5, 1} taken from the social-force literature and a fixed 210° FOV angle from clinical perimetry citations. It then applies this index, together with two directional-deviation angles and acceleration, to an independent public motion-capture dataset of crossing flows (Zenodo DOI given). All reported relationships (temporal profiles of ho, v– ho diagrams, ho– heta diagrams, a– ho diagrams, and non-retracing phase-space loops) are direct empirical averages over the trajectories; no free parameter is fitted to one subset of the data and then presented as a prediction of a related quantity. The interaction-window endpoints T_i, T_f used for the global time rescaling t' are inherited from the edge-cutting algorithm of the prior public release, but they are applied uniformly and the ho=0° baseline is handled separately; the rescaling does not algebraically force the observed geometry-versus-anisotropy ordering or the existence of the loops. Self-citations to the co-author’s earlier analyses of the same dataset supply context and the data themselves, yet the central claims (robustness of qualitative dynamics to c, smooth incremental turning, disruption–recovery asymmetry, superiority of dynamic fundamental diagrams) rest on the new multi-panel measurements rather than on an unverified uniqueness theorem or definitional identity. Consequently the derivation chain contains no self-definitional step, no fitted-input-as-prediction, and no load-bearing self-citation that collapses the result to its inputs.

Axiom & Free-Parameter Ledger

4 free parameters · 3 axioms · 1 invented entities

The central claim rests on a small set of modeling choices (inverse-square kernel, fixed 210° FOV, constant out-of-view weight c, edge-cut interaction windows) and on the representativeness of one laboratory data set. No free parameters are fitted to produce the robustness conclusion; the three c values are discrete probes. Invented entities are limited to the operational definition of perceived crowdedness itself.

free parameters (4)
  • FOV half-angle ϕ/2 = 210° total
    Fixed at 105° (total 210°) from cited perimetry literature; not varied.
  • out-of-FOV weight c = 0, 0.5, 1
    Probed at three discrete values 0, 0.5, 1; not fitted to data.
  • turning window W = 0.5 s
    Fixed at 0.5 s for δ2; not optimized.
  • crowdedness bin width = 0.2 m^{-2}
    0.2 m^{-2} for median curves; chosen for visualization.
axioms (3)
  • ad hoc to paper Local interaction intensity is adequately captured by a sum of inverse-square distances weighted by a binary FOV indicator scaled by constant c.
    Definition Eq. (1)–(3); no derivation from first principles of vision or collision avoidance.
  • domain assumption The edge-cutting algorithm of Mullick et al. (2022) correctly identifies the temporal bounds of each crossing interaction.
    Used without re-validation to define T_i, T_f and the scaled time t' (Section 2).
  • domain assumption Human visual field relevant to locomotion is 210°.
    Cited from Traquair (1927) and Strasburger (2020); not re-measured.
invented entities (1)
  • perceived crowdedness ρ_c no independent evidence
    purpose: Operational index of local interaction intensity that incorporates distance and FOV anisotropy.
    Defined by Eq. (1); treated as an index rather than a physical density; no independent measurement outside the paper.

pith-pipeline@v1.1.0-grok45 · 22279 in / 2455 out tokens · 43852 ms · 2026-07-13T04:35:07.129365+00:00 · methodology

0 comments
read the original abstract

Pedestrian traffic is commonly characterized using local density, yet the interactions experienced by individuals depend on the relative positions and perceptual relevance of surrounding pedestrians. This raises the question of whether behavioral relationships inferred from local crowdedness are robust to the representation of perceptual anisotropy, and how interaction geometry shapes pedestrian adaptation over time. We analyze experimental pedestrian crossing flows over angles from 0 to 180 degrees using a distance-weighted measure of local crowdedness. Perceptual anisotropy is varied by reducing the contribution of pedestrians outside the focal pedestrian's field of view. We examine the temporal evolution of crowdedness and its relationships with velocity, directional deviation, and acceleration. Anisotropy primarily changes the numerical scale of crowdedness, while the qualitative dynamics, temporal progression, and crossing-angle dependence remain largely preserved. Pedestrians deviate appreciably from their expected group directions, but changes between successive walking directions remain small, indicating adaptation through smooth, incremental corrections rather than abrupt turns. Acceleration dynamics reveal an asymmetry between disruption and recovery: initial deceleration varies strongly with crossing geometry, whereas recovery accelerations are more similar across angles. Non-retracing trajectories in the behavioral phase spaces show that similar instantaneous conditions can correspond to different phases of the interaction. Overall, interaction geometry has a stronger influence on the organization of crossing flows than the perceptual weighting used to quantify local crowdedness. More broadly, dynamic fundamental diagrams provide a more complete characterization of transient pedestrian interactions than conventional relationships based on instantaneous state variables alone.

Figures

Figures reproduced from arXiv: 2607.09221 by Igor Ho{\l}owacz, Pratik Mullick.

Figure 1
Figure 1. Figure 1: Schematic illustration of the field of view of a focal pedestrian. The blue pedestrian represents the focal agent, with the arrow indicating its direction of motion, while the dashed lines delimit its 210◦ field of view shown by the arc. Pedestrians within the field of view (green) are assigned a weight wi j = 1, whereas those outside it (gray) are assigned a weight wi j = c, where c ∈ [0, 1] controls the … view at source ↗
Figure 2
Figure 2. Figure 2: Variation of average values of indices of crowdedness (a) ρ1, (b) ρ0.5 and (c) ρ0 as functions of the scaled time t ′ for different values of the crossing angle. The averages are estimated over all the agents and all the experimental trials for a particular crossing angle. The analysis presented in the paper was restricted to the part of each trial associated with the crossing interaction. For each trial, … view at source ↗
Figure 3
Figure 3. Figure 3: Velocity-crowdedness fundamental diagrams for each crossing angle α in our data set. The points in the plot are median values of observed velocity for a crowdedness bin of width 0.2 m−2 [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Dynamic velocity–crowdedness fundamental diagrams for different crossing angles α. Each point corresponds to the average velocity ⟨v⟩ and crowdedness ⟨ρc⟩ of pedestrians at a given scaled time t ′ during a crossing-flow interaction. The color gradient represents the temporal evolution of the velocity-crowdedness relationship averaged across all trials. ● δ1 dexp dcur ● ● t − 2W ● t − W t d1 d2 δ2 (a) (b) … view at source ↗
Figure 5
Figure 5. Figure 5: Schematic representation of the directional deviation measures used in this work. (a) The angle δ1 denotes the deviation of a pedestrian’s current direction of motion, dcur, from the expected direction of motion of its group, dexp. (b) The angle δ2 represents the turning angle between two con￾secutive displacement vectors, d1 and d2, evaluated over time windows of duration W = 0.5 s [PITH_FULL_IMAGE:figur… view at source ↗
Figure 6
Figure 6. Figure 6: Temporal evolution of the mean absolute directional deviations (a) ⟨|δ1|⟩ and (b) ⟨|δ2|⟩ for different crossing angles α. The quantity δ1 measures the instantaneous deviation from the expected direction of motion of the pedestrian’s group, while δ2 quantifies the change in walking direction be￾tween successive displacement windows of duration W = 0.5 s. The averages are computed over all pedestrians and al… view at source ↗
Figure 7
Figure 7. Figure 7: Deviations |δ1| (top row) and |δ2| (bottom row) as functions of crowdedness for different crossing angles α. The three columns correspond to the indices of crowdedness ρ1, ρ0.5, and ρ0, respec￾tively, depending on the levels of perceptual anisotropy. The points in the plot are median values of observed deviation for a crowdedness bin of width 0.2 m−2 . do not perform abrupt turns while traversing the cross… view at source ↗
Figure 8
Figure 8. Figure 8: Temporal evolution of (a) mean acceleration ⟨a⟩ and (b) mean velocity ⟨v⟩ as functions of the scaled time t ′ for different crossing angles α. The averages are computed over all pedestrians and all trials at each scaled time. The plot of time dependence for mean velocity was also shown in [62]. The magnitude and timing of these changes also depend on the crossing angle. The α = 90◦ and 120◦ cases exhibit s… view at source ↗
Figure 9
Figure 9. Figure 9: Acceleration-crowdedness relationships for different crossing angles α and indices of crowd￾edness (a) ρ1, (b) ρ0.5, and (c) ρ0, depending on three levels of perceptual anisotropy. Each point corresponds to the median acceleration for a crowdedness bin of width 0.2 m−2 . sent a unique state relation between acceleration and crowdedness; rather, crowdedness acts partly as a proxy for the stage of the crossi… view at source ↗
Figure 10
Figure 10. Figure 10: Temporal evolution of the mean signed directional deviations ⟨δ1⟩ (left) and ⟨δ2⟩ (right) for different crossing angles α. The averages are computed over all pedestrians and all trials at each scaled time t ′ . crossing flow. In this regime, pedestrians may initially move in a manner that retains some features of the α = 0 ◦ case, before the weak transverse component of the interaction becomes sufficientl… view at source ↗
Figure 11
Figure 11. Figure 11: presents the temporal evolution of the velocity-deviation phase space for the two deviation measures, ⟨|δ1|⟩ and ⟨|δ2|⟩. For most crossing angles, the trajectories form curved or loop-like paths rather than collapsing onto a single relationship between velocity and directional deviation. Thus, the same mean velocity may be associated with different levels of deviation depending on the stage of the interac… view at source ↗
Figure 12
Figure 12. Figure 12: Temporal evolution of mean |δ1| and crowdedness for different crossing angles α. Each point corresponds to the average deviation ⟨|δ1|⟩ and crowdedness ⟨ρ⟩ of pedestrians at a given scaled time t ′ during a crossing-flow interaction. The color gradient represents the temporal evolution of the deviation-crowdedness relationship averaged across all trials. generally span a larger range of angular deviations… view at source ↗
Figure 13
Figure 13. Figure 13: Temporal evolution of mean |δ2| and crowdedness for different crossing angles α. Each point corresponds to the average deviation ⟨|δ2|⟩ and crowdedness ⟨ρ⟩ of pedestrians at a given scaled time t ′ during a crossing-flow interaction. The color gradient represents the temporal evolution of the deviation-crowdedness relationship averaged across all trials. move horizontally with crowdedness, while ⟨δ1⟩ chan… view at source ↗

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Reference graph

Works this paper leans on

68 extracted references

  1. [1]

    Helbing, Reviews of modern physics73, 1067 (2001)

    D. Helbing, Reviews of modern physics73, 1067 (2001)

    2001

  2. [2]

    Schreckenberg, S.D

    M. Schreckenberg, S.D. Sharma,Pedestrian and evacuation dynamics, V ol. 1 (Springer, 2002)

    2002

  3. [3]

    Gibson, British journal of psychology49, 182 (1958)

    J.J. Gibson, British journal of psychology49, 182 (1958)

    1958

  4. [4]

    Batty, Nature388, 19 (1997)

    M. Batty, Nature388, 19 (1997)

    1997

  5. [5]

    Turner, A

    A. Turner, A. Penn, Environment and planning B: Planning and Design29, 473 (2002)

    2002

  6. [6]

    Gulikers, J

    L. Gulikers, J. Evers, A. Muntean, A. Lyulin, Journal of Statistical Mechanics: Theory and Experiment2013, P04025 (2013)

    2013

  7. [7]

    Wirth, G.C

    T.D. Wirth, G.C. Dachner, K.W. Rio, W.H. Warren, PNAS nexus2, pgad118 (2023)

    2023

  8. [8]

    Schrater, D.C

    P.R. Schrater, D.C. Knill, E.P. Simoncelli, Nature neuroscience3, 64 (2000)

    2000

  9. [9]

    Hopkins, A

    B. Hopkins, A. Churchill, S. V ogt, L. Rönnqvist, Journal of motor behavior36, 3 (2004)

    2004

  10. [10]

    Huber, Y .H

    M. Huber, Y .H. Su, M. Krüger, K. Faschian, S. Glasauer, J. Hermsdörfer, PloS one9, e89589 (2014)

    2014

  11. [11]

    Helbing,Social self-organization: Agent-based simulations and experiments to study emergent social behavior(Springer, 2012)

    D. Helbing,Social self-organization: Agent-based simulations and experiments to study emergent social behavior(Springer, 2012)

    2012

  12. [12]

    Moussaid, E.G

    M. Moussaid, E.G. Guillot, M. Moreau, J. Fehrenbach, O. Chabiron, S. Lemercier, J. Pettré, C. Appert-Rolland, P. Degond, G. Theraulaz, PLoS computational biology8, e1002442 (2012)

    2012

  13. [13]

    F. Gu, B. Guiselin, N. Bain, I. Zuriguel, D. Bartolo, Nature638, 112 (2025)

    2025

  14. [14]

    Helbing, L

    D. Helbing, L. Buzna, A. Johansson, T. Werner, Transportation science39, 1 (2005)

    2005

  15. [15]

    Feliciani, K

    C. Feliciani, K. Nishinari, Physical Review E94, 032304 (2016)

    2016

  16. [16]

    Murakami, C

    H. Murakami, C. Feliciani, Y . Nishiyama, K. Nishinari, Science Advances7, eabe7758 (2021)

    2021

  17. [17]

    Khelfa, R

    B. Khelfa, R. Korbmacher, A. Schadschneider, A. Tordeux, Scientific reports12, 4768 (2022)

    2022

  18. [18]

    Bacik, B.S

    K.A. Bacik, B.S. Bacik, T. Rogers, Science379, 923 (2023)

    2023

  19. [19]

    Mullick, S

    P. Mullick, S. Fontaine, C. Appert-Rolland, A.H. Olivier, W.H. Warren, J. Pettré, PLoS computational biology18, e1010210 (2022)

    2022

  20. [20]

    Zanlungo, C

    F. Zanlungo, C. Feliciani, Z. Yücel, K. Nishinari, T. Kanda, Safety science158, 105953 (2023)

    2023

  21. [21]

    Zanlungo, C

    F. Zanlungo, C. Feliciani, Z. Yücel, K. Nishinari, T. Kanda, Safety science158, 105969 (2023)

    2023

  22. [22]

    Hoogendoorn, W

    S.P. Hoogendoorn, W. Daamen, Transportation science39, 147 (2005)

    2005

  23. [23]

    Seyfried, O

    A. Seyfried, O. Passon, B. Steffen, M. Boltes, T. Rupprecht, W. Klingsch, Transporta- tion science43, 395 (2009)

    2009

  24. [24]

    Zhang, W

    J. Zhang, W. Klingsch, A. Schadschneider, A. Seyfried, inTraffic and Granular Flow’11 (Springer, 2013), pp. 241–249

    2013

  25. [25]

    Moussaïd, D

    M. Moussaïd, D. Helbing, G. Theraulaz, Proceedings of the National Academy of Sci- ences108, 6884 (2011)

    2011

  26. [26]

    Dachner, T.D

    G.C. Dachner, T.D. Wirth, E. Richmond, W.H. Warren, Proceedings of the Royal Soci- ety B: Biological Sciences289, 20212089 (2022)

    2022

  27. [27]

    Corbetta, F

    A. Corbetta, F. Toschi, Annual Review of Condensed Matter Physics14, 311 (2023)

    2023

  28. [28]

    Dogbe, Journal of Mathematical Analysis and Applications387, 512 (2012)

    C. Dogbe, Journal of Mathematical Analysis and Applications387, 512 (2012)

    2012

  29. [29]

    Duives, W

    D.C. Duives, W. Daamen, S.P. Hoogendoorn, Transportation Research Part C: Emerging Technologies37, 193 (2013)

    2013

  30. [30]

    Van Toll, J

    W. Van Toll, J. Pettré,Algorithms for microscopic crowd simulation: Advancements in the 2010s, inComputer Graphics Forum(Wiley Online Library, 2021), V ol. 40, pp. 731–754

    2021

  31. [31]

    Korbmacher, A

    R. Korbmacher, A. Nicolas, A. Tordeux, C. Totzeck, Crowd Dynamics, V olume 4: An- alytics and Human Factors in Crowd Modeling pp. 55–80 (2023)

    2023

  32. [32]

    Helbing, P

    D. Helbing, P. Molnar, Physical review E51, 4282 (1995)

    1995

  33. [33]

    Burstedde, K

    C. Burstedde, K. Klauck, A. Schadschneider, J. Zittartz, Physica A: Statistical Mechan- ics and its Applications295, 507 (2001)

    2001

  34. [34]

    Vicsek, A

    T. Vicsek, A. Czirók, E. Ben-Jacob, I. Cohen, O. Shochet, Physical review letters75, 1226 (1995)

    1995

  35. [35]

    Kim, S.J

    S. Kim, S.J. Guy, K. Hillesland, B. Zafar, A.A.A. Gutub, D. Manocha, The Visual Computer31, 541 (2015)

    2015

  36. [36]

    Q. Xu, M. Chraibi, A. Seyfried, Transportation research part C: emerging technologies 133, 103464 (2021)

    2021

  37. [37]

    Ond ˇrej, J

    J. Ond ˇrej, J. Pettré, A.H. Olivier, S. Donikian, ACM Transactions on Graphics (TOG) 29, 1 (2010)

    2010

  38. [38]

    Hoogendoorn, F.L

    S.P. Hoogendoorn, F.L. van Wageningen-Kessels, W. Daamen, D.C. Duives, Physica A: Statistical Mechanics and its Applications416, 684 (2014)

    2014

  39. [39]

    W.L. Koh, S. Zhou, ACM Transactions on Modeling and Computer Simulation (TOMACS)21, 1 (2011)

    2011

  40. [40]

    Dietrich, G

    F. Dietrich, G. Köster, Physical Review E89, 062801 (2014)

    2014

  41. [41]

    Y . Xiao, Z. Gao, Y . Qu, X. Li, Transportation research part C: emerging technologies 68, 566 (2016)

    2016

  42. [42]

    Helbing, I.J

    D. Helbing, I.J. Farkas, T. Vicsek, inThe science of disasters: Climate disruptions, heart attacks, and market crashes(Springer, 2002), pp. 330–350

    2002

  43. [43]

    Helbing, A

    D. Helbing, A. Johansson, H.Z. Al-Abideen, Physical Review E—Statistical, Nonlinear, and Soft Matter Physics75, 046109 (2007)

    2007

  44. [44]

    Helbing, P

    D. Helbing, P. Mukerji, EPJ Data Science1, 7 (2012)

    2012

  45. [45]

    Haghani, M

    M. Haghani, M. Sarvi, Transportation Research Part B: Methodological107, 253 (2018)

    2018

  46. [46]

    Feliciani, K

    C. Feliciani, K. Shimura, K. Nishinari,Introduction to crowd management: Managing crowds in the digital era: Theory and Practice(Springer Nature, 2022)

    2022

  47. [47]

    Feliciani, A

    C. Feliciani, A. Corbetta, M. Haghani, K. Nishinari, Safety science164, 106174 (2023)

    2023

  48. [48]

    Steffen, A

    B. Steffen, A. Seyfried, Physica A: Statistical Mechanics and its Applications389, 1902 (2010)

    1902

  49. [49]

    Schauer, M

    L. Schauer, M. Werner, P. Marcus,Estimating crowd densities and pedestrian flows us- ing wi-fi and bluetooth, inProceedings of the 11th International Conference on Mobile and Ubiquitous Systems: Computing, Networking and Services(2014), pp. 171–177

    2014

  50. [50]

    M. Wirz, T. Franke, D. Roggen, E. Mitleton-Kelly, P. Lukowicz, G. Tröster, EPJ Data Science2, 5 (2013)

    2013

  51. [51]

    Duives, W

    D.C. Duives, W. Daamen, S.P. Hoogendoorn, Physica A: Statistical Mechanics and its Applications427, 162 (2015)

    2015

  52. [52]

    Tordeux, J

    A. Tordeux, J. Zhang, B. Steffen, A. Seyfried, Journal of Statistical Mechanics: Theory and Experiment p. P06030 (2015)

    2015

  53. [53]

    A.S. Rao, J. Gubbi, S. Marusic, M. Palaniswami, The Visual Computer31, 1533 (2015)

    2015

  54. [54]

    Nagao, D

    K. Nagao, D. Yanagisawa, K. Nishinari, Physica A: Statistical Mechanics and its Ap- plications510, 145 (2018)

    2018

  55. [55]

    X. Ding, F. He, Z. Lin, Y . Wang, H. Guo, Y . Huang, IEEE Transactions on Intelligent Transportation Systems22, 4776 (2020)

    2020

  56. [56]

    J. Wang, W. Lv, H. Jiang, Z. Fang, J. Ma, Transportation Research Part C: Emerging Technologies157, 104400 (2023)

    2023

  57. [57]

    Mullick, C

    P. Mullick, C. Appert-Rolland, W.H. Warren, J. Pettré, Physica A: Statistical Mechanics and its Applications657, 130251 (2025)

    2025

  58. [58]

    Seyfried, B

    A. Seyfried, B. Steffen, W. Klingsch, M. Boltes, Journal of Statistical Mechanics: The- ory and Experiment p. P10002 (2005)

    2005

  59. [59]

    Geroliminis, C.F

    N. Geroliminis, C.F. Daganzo, Transportation Research Part B: Methodological42, 759 (2008)

    2008

  60. [60]

    Keyvan-Ekbatani, A

    M. Keyvan-Ekbatani, A. Kouvelas, I. Papamichail, M. Papageorgiou, Transportation Research Part B: Methodological46, 1393 (2012)

    2012

  61. [61]

    Paetzke, M

    S. Paetzke, M. Boltes, A. Seyfried, Physica A: Statistical Mechanics and its Applica- tions595, 127077 (2022)

    2022

  62. [62]

    Mullick, Transportation Science59, 990 (2025)

    P. Mullick, Transportation Science59, 990 (2025)

    2025

  63. [63]

    Cordes, A

    J. Cordes, A. Nicolas, A. Schadschneider, Collective Dynamics9, 1 (2024)

    2024

  64. [64]

    Cordes, A

    J. Cordes, A. Schadschneider, A. Nicolas, PNAS nexus3, pgae120 (2024)

    2024

  65. [65]

    García, D

    A. García, D. Hernández-Delfin, D.J. Lee, M. Ellero, Physica A: Statistical Mechanics and its Applications612, 128461 (2023)

    2023

  66. [66]

    Appert-Rolland, J

    C. Appert-Rolland, J. Pettré, A.H. Olivier, W. Warren, A. Duigou-Majumdar, E. Pin- sard, A. Nicolas, Collective Dynamics5, 1–8 (2020)

    2020

  67. [67]

    Traquair,An introduction to clinical perimetry(Kimpton, 1927)

    H.M. Traquair,An introduction to clinical perimetry(Kimpton, 1927)

    1927

  68. [68]

    H. Strasburger, i-Perception11, 2041669520913052 (2020) A Appendix: Interpretation of the initial deviation peaks forα=30 ◦ A distinct early-time behavior is observed for the shallow crossing angleα=30 ◦. In the plots of the mean absolute deviations (shown in Figure 5 of the main text), both⟨|δ 1|⟩and ⟨|δ2|⟩exhibit pronounced peaks during the initial part...

    2020