December 2018, 15(6): 1271-1290. doi: 10.3934/mbe.2018059

Modeling crowd dynamics through coarse-grained data analysis

1. 

School of Mathematical and Statistical Sciences, Arizona State University, Tempe, USA

2. 

Adaptive Behavior and Cognition Group, Max Planck Institut for Human Development, Berlin, Germany

3. 

Centre de Recherches sur la Cognition Animale, Centre de Biologie Intégrative (CBI), Centre National de la Recherche Scientifique (CNRS) & Université de Toulouse 3 Paul Sabatier, 31062 Toulouse, France

4. 

INRIA Rennes-Bretagne Atlantique, Campus de Beaulieu, Rennes, France

5. 

CNRS, Laboratoire de Physique Théorique, Orsay, France

6. 

Department of Mathematics, Imperial College London, London SW7 2AZ, UK

* Corresponding author: smotsch@asu.edu

This paper entilted "Modeling crowd dynamics through coarse-grained data analysis" is licensed under a Creative Commons Attribution 3.0 Unported License. See http://creativecommons.org/licenses/by/3.0/.

Received  June 03, 2017 Revised  April 25, 2018 Published  September 2018

Understanding and predicting the collective behaviour of crowds is essential to improve the efficiency of pedestrian flows in urban areas and minimize the risks of accidents at mass events. We advocate for the development of crowd traffic management systems, whereby observations of crowds can be coupled to fast and reliable models to produce rapid predictions of the crowd movement and eventually help crowd managers choose between tailored optimization strategies. Here, we propose a Bi-directional Macroscopic (BM) model as the core of such a system. Its key input is the fundamental diagram for bi-directional flows, i.e. the relation between the pedestrian fluxes and densities. We design and run a laboratory experiments involving a total of 119 participants walking in opposite directions in a circular corridor and show that the model is able to accurately capture the experimental data in a typical crowd forecasting situation. Finally, we propose a simple segregation strategy for enhancing the traffic efficiency, and use the BM model to determine the conditions under which this strategy would be beneficial. The BM model, therefore, could serve as a building block to develop on the fly prediction of crowd movements and help deploying real-time crowd optimization strategies.

Citation: Sebastien Motsch, Mehdi Moussaïd, Elsa G. Guillot, Mathieu Moreau, Julien Pettré, Guy Theraulaz, Cécile Appert-Rolland, Pierre Degond. Modeling crowd dynamics through coarse-grained data analysis. Mathematical Biosciences & Engineering, 2018, 15 (6) : 1271-1290. doi: 10.3934/mbe.2018059
References:
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S. A. AlGadhiH. S. Mahmassani and R. Herman, A speed-concentration relation for bi-directional crowd movements with strong interaction, Pedestrian and Evacuation Dynamics, (2002), 3-20.

[2]

S. Ali and M. Shah, Floor fields for tracking in high density crowd scenes, Computer Vision – ECCV 2008: 10th European Conference on Computer Vision, 2008, 1–14; Netw Heterog Media., 6 (2008), 401–423.

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C. Appert-RollandP. Degond and S. Motsch, Two-way multi-lane traffic model for pedestrians in corridors, Netw Heterog Media, 6 (2011), 351-381. doi: 10.3934/nhm.2011.6.351.

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A. Aw and M. Rascle, Resurrection of "second order" models of traffic flow, SIAM J Appl Math., 60 (2000), 916-938. doi: 10.1137/S0036139997332099.

[5]

N. Bellomo and L. Gibelli, Behavioral crowds: Modeling and Monte Carlo simulations toward validation, Comp. & Fluids, 141 (2016), 13-21. doi: 10.1016/j.compfluid.2016.04.022.

[6]

N. Bellomo, B. Piccoli and A. Tosin, Modeling crowd dynamics from a complex system viewpoint Math Mod Meth Appl S., 22 (2012), 1230004, 29pp. doi: 10.1142/S0218202512300049.

[7]

B. Benfold and I. Reid, 2011 Stable multi-target tracking in real-time surveillance video, Computer Vision and Pattern Recognition (CVPR), 2011 IEEE Conference on, 3457–3464.

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J. H. Bick and G. F. Newell, A continuum model for two-directional traffic flow, Q. Appl. Math., 1960.

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C. K. Birdsall and A. B. Langdon, Plasma physics via computer simulation, CRC Press, 2014.

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V. J. Blue and J. L. Adler, Cellular automata microsimulation of bidirectional pedestrian flows, Trans Res B., 1678 (1999), 135-141.

[11]

W. Daamen and S. P. Hoogendoorn, Controlled experiments to derive walking behaviour, Eur J of Trans Infrastruct Res., 3 (2003), 39-59.

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H. Fehske, R. Schneider and A. Weisse, Computational Many-Particle Physics, Springer, 2007.

[13]

C. Feliciani and K. Nishinari, Empirical analysis of the lane formation process in bidirectional pedestrian flow, Physical Review E., 94 (2016), 032304.

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G. Fltterd and G. Lmmel, Bidirectional pedestrian fundamental diagram, Transportation Research Part B: Methodological, 71 (2015), 194-212.

[15]

P. Goatin and M. Mimault, A mixed system modeling two-directional pedestrian flows, Math Biosci Eng., 12 (2015), 375-392.

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D. Helbing, Self-organized pedestrian crowd dynamics: Experiments, simulations, and design solutions, Transport Sci., 39 (2005), 1-24.

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D. Helbing, Saving human lives: What complexity science and information systems can contribute, J. Stat Phys., 158 (2015), 735-781. doi: 10.1007/s10955-014-1024-9.

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D. Helbing and P. Mukerji, Crowd disasters as systemic failures: Analysis of the Love Parade disaster, EPJ Data Science, 1 (2012), 1-40.

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D. Helbing and P. Molnar, Social force model for pedestrian dynamics, Physical Review E., 51 (1995), 4282-4286.

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D. Helbing, Traffic and related self-driven many-particle systems, Rev Mod Phys., 73 (2001), 1067.

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D. Helbing, A. Johansson and H. Z. Al-Abideen, Dynamics of crowd disasters: An empirical study, Phys Rev E., 75 (2007), 046109.

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L. F. Henderson, The statistics of crowd fluids, Nature, 229 (1971), 381-383.

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S. P. Hoogendoorn and P. H. Bovy, Pedestrian route-choice and activity scheduling theory and models, Transport Res B-Meth., 38 (2004), 169-190.

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R. L. Hughes, A continuum theory for the flow of pedestrians, Transport Res B-Meth., 36 (2002), 507-535.

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L. JianY. Lizhong and Z. Daoliang, Simulation of bi-direction pedestrian movement in corridor, Physica A., 354 (2005), 619-628.

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A. Johansson, From crowd dynamics to crowd safety: A video-based analysis, Adv Complex Syst., 11 (2008), 497-527.

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A. Kirchner, K. Nishinari and A. Schadschneider, Friction effects and clogging in a cellular automaton model for pedestrian dynamics, Phys Rev E., 67 (2003), 056122.

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A. Klar, Multivalued fundamental diagrams and stop and go waves for continuum traffic flow equations, SIAM J Appl Math., 64 (2004), 468-483. doi: 10.1137/S0036139902404700.

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L. Kratz and K. Nishino, Tracking pedestrians using local spatio-temporal motion patterns in extremely crowded scenes, IEEE Trans. on Pattern Analysis and Machine Intelligence, 34 (2012), 987-1002.

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T. Kretz, et. al., Experimental study of pedestrian counterflow in a corridor, J Stat Mech-Theory E., 2006 (2006), P10001.

[32]

A. Kurganov and E. Tadmor, New high-resolution central schemes for nonlinear conservation laws and convection-diffusion equations, J Comput Phys., 160 (2000), 241-282. doi: 10.1006/jcph.2000.6459.

[33]

W. H. Lam, A generalised function for modeling bi-directional flow effects on indoor walkways in Hong Kong, Transport Res A-Pol, 37 (2003), 789-810.

[34]

S. Lemercier, Reconstructing motion capture data for human crowd study, Motion in Games, (2011), 365-376.

[35]

S. Lemercier, Realistic following behaviors for crowd simulation, Comput Graph Forum, 31 (2012), 489-498.

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R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, Cambridge, 2002. doi: 10.1017/CBO9780511791253.

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R. J. LeVeque, Numerical Methods for Conservation Laws, Birkhäuser, 1992. doi: 10.1007/978-3-0348-8629-1.

[38]

M. J. Lighthill and G. B. Whitham, On kinematic waves. Ⅱ. A theory of traffic flow on long crowded roads, P Roy Soc Lond A Mat., 229 (1955), 317-345. doi: 10.1098/rspa.1955.0089.

[39]

A. N. Marana, Real-time crowd density estimation using images, Lect Notes Comput Sc., (2005), 355-362.

[40]

G. Martine and A. Marshall, State of world population 2007: Unleashing the potential of urban growth, UNFPA, 2007.

[41]

M. Moussaid, How simple rules determine pedestrian behavior and crowd disasters, Proc Natl Acad Sci., 108 (2011), 6884-6888.

[42]

M. Moussaid, et al., Traffic instabilities in self-organized pedestrian crowds, Plos Comput Biol., 8 (2012), e1002442.

[43]

J. Ondrej, et al., A synthetic-vision based steering approach for crowd simulation, ACM Transactions on Graphics, 29 2010.

[44]

M. PapageorgiouC. DiakakiV. DinopoulouA. Kotsialos and Y. Wang, Review of road traffic control strategies, Proceedings of the IEEE, 91 (2003), 2043-2067.

[45]

H J. Payne, FREFLO: A macroscopic simulation model of freeway traffic, Transp Res Record, 1979.

[46]

M. PlaueM. ChenG. Brwolff and H. Schwandt, Trajectory extraction and density analysis of intersecting pedestrian flows from video recordings, Photogrammetric Image Analysis, Springer, (2011), 285-296.

[47]

C. W. Reynolds, Flocks, herds and schools: A distributed behavioral model, ACM SIGGRAPH Computer Graphics, 21 (1987), 25-34.

[48]

P. Rietveld, Non-motorised modes in transport systems: A multimodal chain perspective for The Netherlands, Transportation Research Part D: Transport and Environment, 5 (2000), 31-36.

[49]

E. RonchiF. N. UrizX. Criel and P. Reilly, Modelling large-scale evacuation of music festivals, Case Studies in Fire Safety, 5 (2016), 11-19.

[50]

M. Rosini, Macroscopic Models for Vehicular Flows and Crowd Dynamics: Theory and Applications, Springer, 2013. doi: 10.1007/978-3-319-00155-5.

[51]

M. SaberiK. Aghabayk and A. Sobhani, Spatial fluctuations of pedestrian velocities in bidirectional streams: Exploring the effects of self-organization, Physica A: Statistical Mechanics and its Applications, 434 (2016), 120-128.

[52]

N. Shiwakoti and M. Sarvi, Enhancing the panic escape of crowd through architectural design, Transportation Research Part C: Emerging Technologies, 37 (2013), 260-267.

[53]

G. K. Still, Crowd Dynamics, PhD Thesis, University of Warwick, 2000.

[54]

E. Tory, et. al., An adaptive finite-volume method for a model of two-phase pedestrian flow, 2011.

[55]

W. G. Weng, A behavior-based model for pedestrian counter flow, Physica A, 375 (2007), 668-678.

[56]

N. WijermansC. ConradoM. van SteenC. Martella and J. Li, A landscape of crowd-management support: An integrative approach, Safety Science, 86 (2016), 142-164.

[57]

M. Wirz, Probing crowd density through smartphones in city-scale mass gatherings, EPJ Data Science, 2 (2013), 1-24.

[58]

S. Yaseen, Real-time crowd density mapping using a novel sensory fusion model of infrared and visual systems, Safety Sci, 57 (2013), 313-325.

[59]

J. Zhang, W. Klingsch, A. Schadschneider and A. Seyfried, Ordering in bidirectional pedestrian flows and its influence on the fundamental diagram, J. Stat. Mech. Theory Exp., 2 (2012), P02002.

[60]

B. ZhouF. Zhang and L. Peng, Higher-order SVD analysis for crowd density estimation, Comput Vis Image Und., 116 (2012), 1014-1021.

show all references

References:
[1]

S. A. AlGadhiH. S. Mahmassani and R. Herman, A speed-concentration relation for bi-directional crowd movements with strong interaction, Pedestrian and Evacuation Dynamics, (2002), 3-20.

[2]

S. Ali and M. Shah, Floor fields for tracking in high density crowd scenes, Computer Vision – ECCV 2008: 10th European Conference on Computer Vision, 2008, 1–14; Netw Heterog Media., 6 (2008), 401–423.

[3]

C. Appert-RollandP. Degond and S. Motsch, Two-way multi-lane traffic model for pedestrians in corridors, Netw Heterog Media, 6 (2011), 351-381. doi: 10.3934/nhm.2011.6.351.

[4]

A. Aw and M. Rascle, Resurrection of "second order" models of traffic flow, SIAM J Appl Math., 60 (2000), 916-938. doi: 10.1137/S0036139997332099.

[5]

N. Bellomo and L. Gibelli, Behavioral crowds: Modeling and Monte Carlo simulations toward validation, Comp. & Fluids, 141 (2016), 13-21. doi: 10.1016/j.compfluid.2016.04.022.

[6]

N. Bellomo, B. Piccoli and A. Tosin, Modeling crowd dynamics from a complex system viewpoint Math Mod Meth Appl S., 22 (2012), 1230004, 29pp. doi: 10.1142/S0218202512300049.

[7]

B. Benfold and I. Reid, 2011 Stable multi-target tracking in real-time surveillance video, Computer Vision and Pattern Recognition (CVPR), 2011 IEEE Conference on, 3457–3464.

[8]

J. H. Bick and G. F. Newell, A continuum model for two-directional traffic flow, Q. Appl. Math., 1960.

[9]

C. K. Birdsall and A. B. Langdon, Plasma physics via computer simulation, CRC Press, 2014.

[10]

V. J. Blue and J. L. Adler, Cellular automata microsimulation of bidirectional pedestrian flows, Trans Res B., 1678 (1999), 135-141.

[11]

W. Daamen and S. P. Hoogendoorn, Controlled experiments to derive walking behaviour, Eur J of Trans Infrastruct Res., 3 (2003), 39-59.

[12]

H. Fehske, R. Schneider and A. Weisse, Computational Many-Particle Physics, Springer, 2007.

[13]

C. Feliciani and K. Nishinari, Empirical analysis of the lane formation process in bidirectional pedestrian flow, Physical Review E., 94 (2016), 032304.

[14]

G. Fltterd and G. Lmmel, Bidirectional pedestrian fundamental diagram, Transportation Research Part B: Methodological, 71 (2015), 194-212.

[15]

P. Goatin and M. Mimault, A mixed system modeling two-directional pedestrian flows, Math Biosci Eng., 12 (2015), 375-392.

[16]

D. Helbing, Self-organized pedestrian crowd dynamics: Experiments, simulations, and design solutions, Transport Sci., 39 (2005), 1-24.

[17]

D. Helbing, Saving human lives: What complexity science and information systems can contribute, J. Stat Phys., 158 (2015), 735-781. doi: 10.1007/s10955-014-1024-9.

[18]

D. Helbing and P. Mukerji, Crowd disasters as systemic failures: Analysis of the Love Parade disaster, EPJ Data Science, 1 (2012), 1-40.

[19]

D. Helbing and P. Molnar, Social force model for pedestrian dynamics, Physical Review E., 51 (1995), 4282-4286.

[20]

D. Helbing, Traffic and related self-driven many-particle systems, Rev Mod Phys., 73 (2001), 1067.

[21]

D. Helbing, A. Johansson and H. Z. Al-Abideen, Dynamics of crowd disasters: An empirical study, Phys Rev E., 75 (2007), 046109.

[22]

L. F. Henderson, The statistics of crowd fluids, Nature, 229 (1971), 381-383.

[23]

S. P. Hoogendoorn and P. H. Bovy, Pedestrian route-choice and activity scheduling theory and models, Transport Res B-Meth., 38 (2004), 169-190.

[24]

R. L. Hughes, A continuum theory for the flow of pedestrians, Transport Res B-Meth., 36 (2002), 507-535.

[25]

L. JianY. Lizhong and Z. Daoliang, Simulation of bi-direction pedestrian movement in corridor, Physica A., 354 (2005), 619-628.

[26]

A. Johansson, From crowd dynamics to crowd safety: A video-based analysis, Adv Complex Syst., 11 (2008), 497-527.

[27]

B. S. Kerner, The physics of traffic: Empirical freeway pattern features, engineering applications, and theory, Springer Verlag, 2004.

[28]

A. Kirchner, K. Nishinari and A. Schadschneider, Friction effects and clogging in a cellular automaton model for pedestrian dynamics, Phys Rev E., 67 (2003), 056122.

[29]

A. Klar, Multivalued fundamental diagrams and stop and go waves for continuum traffic flow equations, SIAM J Appl Math., 64 (2004), 468-483. doi: 10.1137/S0036139902404700.

[30]

L. Kratz and K. Nishino, Tracking pedestrians using local spatio-temporal motion patterns in extremely crowded scenes, IEEE Trans. on Pattern Analysis and Machine Intelligence, 34 (2012), 987-1002.

[31]

T. Kretz, et. al., Experimental study of pedestrian counterflow in a corridor, J Stat Mech-Theory E., 2006 (2006), P10001.

[32]

A. Kurganov and E. Tadmor, New high-resolution central schemes for nonlinear conservation laws and convection-diffusion equations, J Comput Phys., 160 (2000), 241-282. doi: 10.1006/jcph.2000.6459.

[33]

W. H. Lam, A generalised function for modeling bi-directional flow effects on indoor walkways in Hong Kong, Transport Res A-Pol, 37 (2003), 789-810.

[34]

S. Lemercier, Reconstructing motion capture data for human crowd study, Motion in Games, (2011), 365-376.

[35]

S. Lemercier, Realistic following behaviors for crowd simulation, Comput Graph Forum, 31 (2012), 489-498.

[36]

R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, Cambridge, 2002. doi: 10.1017/CBO9780511791253.

[37]

R. J. LeVeque, Numerical Methods for Conservation Laws, Birkhäuser, 1992. doi: 10.1007/978-3-0348-8629-1.

[38]

M. J. Lighthill and G. B. Whitham, On kinematic waves. Ⅱ. A theory of traffic flow on long crowded roads, P Roy Soc Lond A Mat., 229 (1955), 317-345. doi: 10.1098/rspa.1955.0089.

[39]

A. N. Marana, Real-time crowd density estimation using images, Lect Notes Comput Sc., (2005), 355-362.

[40]

G. Martine and A. Marshall, State of world population 2007: Unleashing the potential of urban growth, UNFPA, 2007.

[41]

M. Moussaid, How simple rules determine pedestrian behavior and crowd disasters, Proc Natl Acad Sci., 108 (2011), 6884-6888.

[42]

M. Moussaid, et al., Traffic instabilities in self-organized pedestrian crowds, Plos Comput Biol., 8 (2012), e1002442.

[43]

J. Ondrej, et al., A synthetic-vision based steering approach for crowd simulation, ACM Transactions on Graphics, 29 2010.

[44]

M. PapageorgiouC. DiakakiV. DinopoulouA. Kotsialos and Y. Wang, Review of road traffic control strategies, Proceedings of the IEEE, 91 (2003), 2043-2067.

[45]

H J. Payne, FREFLO: A macroscopic simulation model of freeway traffic, Transp Res Record, 1979.

[46]

M. PlaueM. ChenG. Brwolff and H. Schwandt, Trajectory extraction and density analysis of intersecting pedestrian flows from video recordings, Photogrammetric Image Analysis, Springer, (2011), 285-296.

[47]

C. W. Reynolds, Flocks, herds and schools: A distributed behavioral model, ACM SIGGRAPH Computer Graphics, 21 (1987), 25-34.

[48]

P. Rietveld, Non-motorised modes in transport systems: A multimodal chain perspective for The Netherlands, Transportation Research Part D: Transport and Environment, 5 (2000), 31-36.

[49]

E. RonchiF. N. UrizX. Criel and P. Reilly, Modelling large-scale evacuation of music festivals, Case Studies in Fire Safety, 5 (2016), 11-19.

[50]

M. Rosini, Macroscopic Models for Vehicular Flows and Crowd Dynamics: Theory and Applications, Springer, 2013. doi: 10.1007/978-3-319-00155-5.

[51]

M. SaberiK. Aghabayk and A. Sobhani, Spatial fluctuations of pedestrian velocities in bidirectional streams: Exploring the effects of self-organization, Physica A: Statistical Mechanics and its Applications, 434 (2016), 120-128.

[52]

N. Shiwakoti and M. Sarvi, Enhancing the panic escape of crowd through architectural design, Transportation Research Part C: Emerging Technologies, 37 (2013), 260-267.

[53]

G. K. Still, Crowd Dynamics, PhD Thesis, University of Warwick, 2000.

[54]

E. Tory, et. al., An adaptive finite-volume method for a model of two-phase pedestrian flow, 2011.

[55]

W. G. Weng, A behavior-based model for pedestrian counter flow, Physica A, 375 (2007), 668-678.

[56]

N. WijermansC. ConradoM. van SteenC. Martella and J. Li, A landscape of crowd-management support: An integrative approach, Safety Science, 86 (2016), 142-164.

[57]

M. Wirz, Probing crowd density through smartphones in city-scale mass gatherings, EPJ Data Science, 2 (2013), 1-24.

[58]

S. Yaseen, Real-time crowd density mapping using a novel sensory fusion model of infrared and visual systems, Safety Sci, 57 (2013), 313-325.

[59]

J. Zhang, W. Klingsch, A. Schadschneider and A. Seyfried, Ordering in bidirectional pedestrian flows and its influence on the fundamental diagram, J. Stat. Mech. Theory Exp., 2 (2012), P02002.

[60]

B. ZhouF. Zhang and L. Peng, Higher-order SVD analysis for crowd density estimation, Comput Vis Image Und., 116 (2012), 1014-1021.

Figure 1.  Experiments and data acquisition. (a) A typical experiment where bi-directional circulation is analyzed. (b) Participants equipped with reflexive markers were tracked by means of an optoelectronic motion capture system (VICON MX-40, Oxford Metrics, UK). (c) The area-weighting assignment procedure: The particle $i$ (red dot) is located in the cell ${\mathcal C}^n(i) = [\theta_k, \theta_{k+1}]$ and is assigned to the two nodes $\theta_k$ and $\theta_{k+1}$ in proportion to the area enclosing the opposite node. For instance, on the picture, the assignment to the node $\theta_k$ is in proportion to the ratio of the shaded area to the area of ${\mathcal C}^n(i)$
Figure 2.  Characteristic speeds in the BM model. (a) Perspective view of the BFD $f(\rho_+, \rho_-)$ as a function of $\rho_+$ and $\rho_-$. (b) Sketchy fundamental diagram (in red) of a one-way flow $f(\rho)$ as a function of the single density $\rho$. The function $f(\rho)$ has the same monotony as cuts of the BFD $f(\rho_+, \rho_-)$ along lines $\rho_- = \mbox{Constant}$. The average velocity $u(\rho) = f(\rho)/\rho$ and the cluster velocity $\lambda(\rho) = f'(\rho)$ are respectively the slopes of the secant (in red) and tangent (in green) lines to this curve
Figure 3.  Bi-directional Fundamental Diagram (BFD). (a) Estimated BFD expressing the flux $f$ in one direction as a function of the density of pedestrians moving in the same direction $\rho_+$ and in the opposite direction $\rho_-$. (b) Parametrized BFD used in the model. The values of $f$, in $(m.s)^{-1}$, are color coded according to the color bar
Figure 4.  Model results and comparisons with experimental data. (a) Setting for the model: the density at the entry of the corridor ($x = 0$) is taken from the experiment and the model is used to predict the occupancy inside the corridor. (b-c) Clockwise and anti-clockwise (resp. b and c) pedestrian densities as functions of position (horizontal axis) and time (vertical axis running downwards), for one of the replications with $60$ pedestrians corresponding to balanced fluxes ($50 \%$ of pedestrians walking in each direction). Left picture of each panel: experiment; right picture: BM model, run with initial and boundary data and BFD estimated from the experimental data. The density is color-coded according to the lateral scale (in m$^{-2}$)
Figure 5.  Model results and comparisons with experiment data with the $75 \%\! -\! 25\%$ flux balance: $\%75$ are moving to the left ($\rho_-$) and $\%25$ are moving to the right ($\rho_+$). See Fig. 4b, c for balanced flux
Figure 6.  Pedestrian and cluster velocities. Comparison between the pedestrian velocity $u$ (red curve) and cluster velocity $\lambda$ (green curve) as functions of the local density of co-moving pedestrians $\rho$ from the experiments (circles) and from the BM model (solid line) with balanced flux ($50\%\! -\! 50\%$) on the left figure and with $75\%\! -\! 25\%$ flux balance on the right figure. Since $\lambda <u$, information is propagating upstream as predicted by the BM model
Figure 7.  Efficiency segregation strategy. Estimation of the relative gain using the segregation strategy (in %) as a function of the densities $\rho_+, \, \rho_-$ (in m$^{-2}$) of the two types of pedestrians (level curve representation). The strategy is efficient when both densities $\rho_+$ and $\rho_-$ are roughly balanced and large
Figure 8.  Cluster and cluster velocity. (a) Illustration of a cluster defined from a density distribution $\rho(x, t^n)$ for a given threshold $h$. The edges of the cluster (i.e. $X^+$ and $X^-$) described the level curves of $\rho(x, t)$. (b) Graphical representation of the cluster velocity using the level curves of $\rho$. The picture depicts the edges (in yellow) of the cluster (in red). The cluster edge velocity i.e. the slope of the cluster edge in the position-time plane is given by $f'(h)$ and is illustrated by the green segment
Figure 9.  Level curves of the density. Level curves $X(t)$ for various levels $h$ for the replication displayed in Fig. 4 and clockwise pedestrian density $\rho_+$. Horizontal axis is space and vertical axis is time, running downwards. The color code corresponds to the level height $h$, from blue (lower levels) to red (higher levels). (a): with no cutoff and no filtering. (b): after cutoff (only level curves with life-time greater than $3$ seconds are kept) but before filtering. (c): after cutoff and filtering
Table 1.  Parametric estimation of the bi-directional fundamental diagram. The coefficients $a$, $b$ and $c$ refer to Eq. 9. To measure the accuracy of each regression, we estimate the coefficient of determination $R^2$ in the last column (the closer $R^2$ is to $1$, the better the estimation is)
Sample setabc $R^2$
$50\%-50\%$ $1.218$ $0.273$ $0.181$ 0.944
$75\%-25\%$ $1.216$ $0.087$ $0.203$ 0.972
$100\%-0\%$ $1.269$ $0.077$ $0$ 0.982
Sample setabc $R^2$
$50\%-50\%$ $1.218$ $0.273$ $0.181$ 0.944
$75\%-25\%$ $1.216$ $0.087$ $0.203$ 0.972
$100\%-0\%$ $1.269$ $0.077$ $0$ 0.982
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