September  2015, 10(3): 579-608. doi: 10.3934/nhm.2015.10.579

Time-delayed follow-the-leader model for pedestrians walking in line

1. 

Université de Toulouse; UPS, INSA, UT1, UTM, Institut de Mathématiques de Toulouse; F-31062 Toulouse, France, France

2. 

Donghua University, No. 2999 North Renmin Road, Songjiang, Shanghai 201620, China

3. 

INRIA Rennes - Bretagne Atlantique, Campus de Beaulieu, 35042 Rennes, France

4. 

Istituto Sistemi Complessi, Consiglio Nazionale delle Ricerche, UOS Sapienza, 00185 Rome, Italy

5. 

Laboratoire de Physique Théorique, Université Paris Sud, btiment 210, 91405 Orsay cedex

6. 

Golaem S.A.S., Bâtiment Germanium, 80 avenue des Buttes de Coësmes, 35 700 Rennes, France

7. 

Imperial College London, South Kensington Campus, London SW7 2AZ

Received  October 2014 Revised  January 2015 Published  July 2015

We use the results of a pedestrian tracking experiment to identify a follow-the-leader model for pedestrians walking-in-line. We demonstrate the existence of a time-delay between a subject's response and the predecessor's corresponding behavior. This time-delay induces an instability which can be damped out by a suitable relaxation. By comparisons with the experimental data, we show that the model reproduces well the emergence of large-scale structures such as congestions waves. The resulting model can be used either for modeling pedestrian queuing behavior or can be incorporated into bi-dimensional models of pedestrian traffic.
Citation: Jérôme Fehrenbach, Jacek Narski, Jiale Hua, Samuel Lemercier, Asja Jelić, Cécile Appert-Rolland, Stéphane Donikian, Julien Pettré, Pierre Degond. Time-delayed follow-the-leader model for pedestrians walking in line. Networks & Heterogeneous Media, 2015, 10 (3) : 579-608. doi: 10.3934/nhm.2015.10.579
References:
[1]

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

[2]

A. Aw, A. Klar, T. Materne and M. Rascle, Derivation of continuum traffic flow models from microscopic follow-the-leader models,, SIAM J. Appl. Math., 63 (2002), 259. doi: 10.1137/S0036139900380955. Google Scholar

[3]

R. Bellman and K. Cooke, Differential-Difference Equations,, Academic Press, (1963). Google Scholar

[4]

N. Bellomo and C. Dogbé, On the modelling crowd dynamics from scaling to hyperbolic macroscopic models,, Math. Models Methods Appl. Sci., 18 (2008), 1317. doi: 10.1142/S0218202508003054. Google Scholar

[5]

N. Bellomo and C. Dogbé, On the modeling of traffic and crowds: A survey of models, speculations and perspectives,, SIAM Review, 53 (2011), 409. doi: 10.1137/090746677. Google Scholar

[6]

S. Berres, R. Ruiz-Baier, H. Schwandt and E. M. Tory, An adaptive finite-volume method for a model of two-phase pedestrian flow,, Netw. Heter. Media., 6 (2011), 401. doi: 10.3934/nhm.2011.6.401. Google Scholar

[7]

C. Burstedde, K. Klauck , A. Schadschneider and J. Zittartz, Simulation of pedestrian dynamics using a two-dimensional cellular automaton,, Physica A, 295 (2001), 507. doi: 10.1016/S0378-4371(01)00141-8. Google Scholar

[8]

R. E. Chandler, R. Herman and E. W. Montroll, Traffic dynamics: Studies in car following,, Operations Res., 6 (1958), 165. doi: 10.1287/opre.6.2.165. Google Scholar

[9]

M. Chraibi, A. Seyfried and A. Schadschneider, Generalized centrifugal-force model for pedestrian dynamics,, Phys. Rev. E, 82 (2010). doi: 10.1103/PhysRevE.82.046111. Google Scholar

[10]

R. M. Colombo and M. D. Rosini, Pedestrian flows and nonclassical shocks,, Math. Methods Appl. Sci., 28 (2005), 1553. doi: 10.1002/mma.624. Google Scholar

[11]

V. Coscia and C. Canavesio, First-order macroscopic modelling of human crowd dynamics,, Math. Models Methods Appl. Sci., 18 (2008), 1217. doi: 10.1142/S0218202508003017. Google Scholar

[12]

D. C. Gazis, R. Herman and R. Rothery, Nonlinear follow-the-leader models of traffic flow,, Operations Res., 9 (1961), 545. doi: 10.1287/opre.9.4.545. Google Scholar

[13]

S. J. Guy, J. Chhugani, C. Kim, N. Satish, M. C. Lin, D. Manocha and P. Dubey, Clearpath: Highly parallel collision avoidance for multi-agent simulation,, in ACM SIGGRAPH/Eurographics Symposium on Computer Animation, (2009), 177. doi: 10.1145/1599470.1599494. Google Scholar

[14]

P. Degond, C. Appert-Rolland, M. Moussaid, J. Pettre and G. Theraulaz, A hierarchy of heuristic-based models of crowd dynamics,, J. Stat. Phys., 152 (2013), 1033. doi: 10.1007/s10955-013-0805-x. Google Scholar

[15]

P. Degond, C. Appert-Rolland, J. Pettre and G. Theraulaz, Vision-based macroscopic pedestrian models,, Kinet. Relat. Models, 6 (2013), 809. doi: 10.3934/krm.2013.6.809. Google Scholar

[16]

P. Degond and J. Hua, Self-Organized Hydrodynamics with congestion and path formation in crowds,, J. Comput. Phys., 237 (2013), 299. doi: 10.1016/j.jcp.2012.11.033. Google Scholar

[17]

M. Di Francesco, P. A. Markowich, J.-F. Pietschmann and M.-T. Wolfram, On the Hughes' model for pedestrian flow: The one-dimensional case,, J. Diff. Eq., 250 (2011), 1334. doi: 10.1016/j.jde.2010.10.015. Google Scholar

[18]

D. Helbing, A mathematical model for the behavior of pedestrians,, Behavioral Science, 36 (1991), 298. doi: 10.1002/bs.3830360405. Google Scholar

[19]

D. Helbing, A fluid dynamic model for the movement of pedestrians,, Complex Systems, 6 (1992), 391. Google Scholar

[20]

D. Helbing and P. Molnàr, Social force model for pedestrian dynamics,, Phys. Rev. E, 51 (1995), 4282. doi: 10.1103/PhysRevE.51.4282. Google Scholar

[21]

D. Helbing and P. Molnàr, Self-organization phenomena in pedestrian crowds,, in Self-Organization of Complex Structures: From Individual to Collective Dynamics (ed. F. Schweitzer), (1997), 569. Google Scholar

[22]

L. F. Henderson, On the fluid mechanics of human crowd motion,, Transp. Res., 8 (1974), 509. doi: 10.1016/0041-1647(74)90027-6. Google Scholar

[23]

S. Hoogendoorn and P. H. L. Bovy, Simulation of pedestrian flows by optimal control and differential games,, Optimal Control Appl. Methods, 24 (2003), 153. doi: 10.1002/oca.727. Google Scholar

[24]

W. H. Huang, B. R. Fajen, J. R. Fink and W. H. Warren, Visual navigation and obstacle avoidance using a steering potential function,, Robotic and Autonomous Systems, 54 (2006), 288. doi: 10.1016/j.robot.2005.11.004. Google Scholar

[25]

L. Huang, S. C. Wong, M. Zhang, C.-W. Shu and W. H. K. Lam, Revisiting Hughes' dynamic continuum model for pedestrian flow and the development of an efficient solution algorithm,, Transp. Res. B, 43 (2009), 127. doi: 10.1016/j.trb.2008.06.003. Google Scholar

[26]

R. L. Hughes, A continuum theory for the flow of pedestrians,, Transp. Res. B, 36 (2002), 507. doi: 10.1016/S0191-2615(01)00015-7. Google Scholar

[27]

R. L. Hughes, The flow of human crowds,, Ann. Rev. Fluid Mech., 35 (2003), 169. doi: 10.1146/annurev.fluid.35.101101.161136. Google Scholar

[28]

A. Jelić, C. Appert-Rolland, S. Lemercier and J. Pettré, Properties of pedestrians walking in line - Fundamental diagrams,, Phys. Rev. E, 85 (2012). Google Scholar

[29]

A. Jelić, C. Appert-Rolland, S. Lemercier and J. Pettré, Properties of pedestrians walking in line. II. stepping behavior,, Phys. Rev. E, 86 (2012). Google Scholar

[30]

D. Jezbera, D. Kordek, J. Kříž, Petr Šeba and P. Šroll, Walkers on the circle,, J. Stat. Mech. Theory Exp., 2010 (2010). doi: 10.1088/1742-5468/2010/01/L01001. Google Scholar

[31]

Y.-q. Jiang, P. Zhang, S. C. Wong and R.-x. Liu, A higher-order macroscopic model for pedestrian flows,, Physica A, 389 (2010), 4623. doi: 10.1016/j.physa.2010.05.003. Google Scholar

[32]

A. Johansson, Constant-net-time headway as a key mechanism behind pedestrian flow dynamics,, Phys. Rev. E, 80 (2009). doi: 10.1103/PhysRevE.80.026120. Google Scholar

[33]

S. Lemercier, A. Jelić, R. Kulpa, J. Hua, J. Fehrenbach, P. Degond, C. Appert-Rolland, S. Donikian and J. Pettré, Realistic following behaviors for crowd simulation,, Computer Graphics Forum, 31 (2012), 489. doi: 10.1111/j.1467-8659.2012.03028.x. Google Scholar

[34]

S. Lemercier, M. Moreau, M. Moussaïd, G. Theraulaz, S. Donikian and J. Pettré, Reconstructing motion capture data for human crowd study,, in Motion in Games, (7060), 365. doi: 10.1007/978-3-642-25090-3_31. Google Scholar

[35]

B. Maury, A. Roudneff-Chupin, F. Santambrogio and J. Venel, Handling congestion in crowd motion models,, Netw. Heterog. Media, 6 (2011), 485. doi: 10.3934/nhm.2011.6.485. Google Scholar

[36]

M. Moussaïd, E. G. Guillot, M. Moreau, J. Fehrenbach, O. Chabiron, S. Lemercier, J. Pettré, C. Appert-Rolland, P. Degond and G. Theraulaz, Traffic Instabilities in Self-organized Pedestrian Crowds,, PLoS Comput. Biol., 8 (2012). Google Scholar

[37]

M. Moussaïd, D. Helbing and G. Theraulaz, How simple rules determine pedestrian behavior and crowd disasters,, Proc. Nat. Acad. Sci., 108 (2011), 6884. Google Scholar

[38]

K. Nishinari, A. Kirchner, A. Namazi and A. Schadschneider, Extended floor field CA model for evacuation dynamics,, IEICE Transp. Inf. & Syst., E87-D (2004), 726. Google Scholar

[39]

J. Ondrej, J. Pettré, A. H. Olivier and S. Donikian, A Synthetic-vision based steering approach for crowd simulation,, in SIGGRAPH'10, 29 (2010). doi: 10.1145/1833349.1778860. Google Scholar

[40]

S. Paris, J. Pettré and S. Donikian, Pedestrian reactive navigation for crowd simulation: A predictive approach,, Eurographics, 26 (2007), 665. doi: 10.1111/j.1467-8659.2007.01090.x. Google Scholar

[41]

J. Pettré, J. Ondřej, A.-H. Olivier, A. Cretual and S. Donikian, Experiment-based modeling, simulation and validation of interactions between virtual walkers,, in SCA '09: Proceedings of the 2009 ACM SIGGRAPH/Eurographics Symposium on Computer Animation, (2009), 189. Google Scholar

[42]

B. Piccoli and A. Tosin, Pedestrian flows in bounded domains with obstacles,, Contin. Mech. Thermodyn., 21 (2009), 85. doi: 10.1007/s00161-009-0100-x. Google Scholar

[43]

L. Pontrjagin, On the zeros of some elementary transcendental functions,, Amer. Math. Soc. Transl. Ser. 2, 1 (1955), 95. Google Scholar

[44]

C. W. Reynolds, Steering behaviors for autonomous characters,, in Proceedings of Game Developers Conference, (1999), 763. Google Scholar

[45]

A. Seyfried, B. Steffen, W. Klingsch and M. Boltes, The fundamental diagram of pedestrian movement revisited,, J. Stat. Mech. Theory Exp., 2005 (2005). doi: 10.1088/1742-5468/2005/10/P10002. Google Scholar

[46]

A. Seyfried, B. Steffen and T. Lippert, Basics of modelling the pedestrian flow,, Phys. A, 368 (2006), 232. doi: 10.1016/j.physa.2005.11.052. Google Scholar

[47]

J. van den Berg and H. Overmars, Planning time-minimal safe paths amidst unpredictably moving obstacles,, Int. Journal on Robotics Research, 27 (2008), 1274. Google Scholar

[48]

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., 2012 (2012). doi: 10.1088/1742-5468/2012/02/P02002. Google Scholar

show all references

References:
[1]

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

[2]

A. Aw, A. Klar, T. Materne and M. Rascle, Derivation of continuum traffic flow models from microscopic follow-the-leader models,, SIAM J. Appl. Math., 63 (2002), 259. doi: 10.1137/S0036139900380955. Google Scholar

[3]

R. Bellman and K. Cooke, Differential-Difference Equations,, Academic Press, (1963). Google Scholar

[4]

N. Bellomo and C. Dogbé, On the modelling crowd dynamics from scaling to hyperbolic macroscopic models,, Math. Models Methods Appl. Sci., 18 (2008), 1317. doi: 10.1142/S0218202508003054. Google Scholar

[5]

N. Bellomo and C. Dogbé, On the modeling of traffic and crowds: A survey of models, speculations and perspectives,, SIAM Review, 53 (2011), 409. doi: 10.1137/090746677. Google Scholar

[6]

S. Berres, R. Ruiz-Baier, H. Schwandt and E. M. Tory, An adaptive finite-volume method for a model of two-phase pedestrian flow,, Netw. Heter. Media., 6 (2011), 401. doi: 10.3934/nhm.2011.6.401. Google Scholar

[7]

C. Burstedde, K. Klauck , A. Schadschneider and J. Zittartz, Simulation of pedestrian dynamics using a two-dimensional cellular automaton,, Physica A, 295 (2001), 507. doi: 10.1016/S0378-4371(01)00141-8. Google Scholar

[8]

R. E. Chandler, R. Herman and E. W. Montroll, Traffic dynamics: Studies in car following,, Operations Res., 6 (1958), 165. doi: 10.1287/opre.6.2.165. Google Scholar

[9]

M. Chraibi, A. Seyfried and A. Schadschneider, Generalized centrifugal-force model for pedestrian dynamics,, Phys. Rev. E, 82 (2010). doi: 10.1103/PhysRevE.82.046111. Google Scholar

[10]

R. M. Colombo and M. D. Rosini, Pedestrian flows and nonclassical shocks,, Math. Methods Appl. Sci., 28 (2005), 1553. doi: 10.1002/mma.624. Google Scholar

[11]

V. Coscia and C. Canavesio, First-order macroscopic modelling of human crowd dynamics,, Math. Models Methods Appl. Sci., 18 (2008), 1217. doi: 10.1142/S0218202508003017. Google Scholar

[12]

D. C. Gazis, R. Herman and R. Rothery, Nonlinear follow-the-leader models of traffic flow,, Operations Res., 9 (1961), 545. doi: 10.1287/opre.9.4.545. Google Scholar

[13]

S. J. Guy, J. Chhugani, C. Kim, N. Satish, M. C. Lin, D. Manocha and P. Dubey, Clearpath: Highly parallel collision avoidance for multi-agent simulation,, in ACM SIGGRAPH/Eurographics Symposium on Computer Animation, (2009), 177. doi: 10.1145/1599470.1599494. Google Scholar

[14]

P. Degond, C. Appert-Rolland, M. Moussaid, J. Pettre and G. Theraulaz, A hierarchy of heuristic-based models of crowd dynamics,, J. Stat. Phys., 152 (2013), 1033. doi: 10.1007/s10955-013-0805-x. Google Scholar

[15]

P. Degond, C. Appert-Rolland, J. Pettre and G. Theraulaz, Vision-based macroscopic pedestrian models,, Kinet. Relat. Models, 6 (2013), 809. doi: 10.3934/krm.2013.6.809. Google Scholar

[16]

P. Degond and J. Hua, Self-Organized Hydrodynamics with congestion and path formation in crowds,, J. Comput. Phys., 237 (2013), 299. doi: 10.1016/j.jcp.2012.11.033. Google Scholar

[17]

M. Di Francesco, P. A. Markowich, J.-F. Pietschmann and M.-T. Wolfram, On the Hughes' model for pedestrian flow: The one-dimensional case,, J. Diff. Eq., 250 (2011), 1334. doi: 10.1016/j.jde.2010.10.015. Google Scholar

[18]

D. Helbing, A mathematical model for the behavior of pedestrians,, Behavioral Science, 36 (1991), 298. doi: 10.1002/bs.3830360405. Google Scholar

[19]

D. Helbing, A fluid dynamic model for the movement of pedestrians,, Complex Systems, 6 (1992), 391. Google Scholar

[20]

D. Helbing and P. Molnàr, Social force model for pedestrian dynamics,, Phys. Rev. E, 51 (1995), 4282. doi: 10.1103/PhysRevE.51.4282. Google Scholar

[21]

D. Helbing and P. Molnàr, Self-organization phenomena in pedestrian crowds,, in Self-Organization of Complex Structures: From Individual to Collective Dynamics (ed. F. Schweitzer), (1997), 569. Google Scholar

[22]

L. F. Henderson, On the fluid mechanics of human crowd motion,, Transp. Res., 8 (1974), 509. doi: 10.1016/0041-1647(74)90027-6. Google Scholar

[23]

S. Hoogendoorn and P. H. L. Bovy, Simulation of pedestrian flows by optimal control and differential games,, Optimal Control Appl. Methods, 24 (2003), 153. doi: 10.1002/oca.727. Google Scholar

[24]

W. H. Huang, B. R. Fajen, J. R. Fink and W. H. Warren, Visual navigation and obstacle avoidance using a steering potential function,, Robotic and Autonomous Systems, 54 (2006), 288. doi: 10.1016/j.robot.2005.11.004. Google Scholar

[25]

L. Huang, S. C. Wong, M. Zhang, C.-W. Shu and W. H. K. Lam, Revisiting Hughes' dynamic continuum model for pedestrian flow and the development of an efficient solution algorithm,, Transp. Res. B, 43 (2009), 127. doi: 10.1016/j.trb.2008.06.003. Google Scholar

[26]

R. L. Hughes, A continuum theory for the flow of pedestrians,, Transp. Res. B, 36 (2002), 507. doi: 10.1016/S0191-2615(01)00015-7. Google Scholar

[27]

R. L. Hughes, The flow of human crowds,, Ann. Rev. Fluid Mech., 35 (2003), 169. doi: 10.1146/annurev.fluid.35.101101.161136. Google Scholar

[28]

A. Jelić, C. Appert-Rolland, S. Lemercier and J. Pettré, Properties of pedestrians walking in line - Fundamental diagrams,, Phys. Rev. E, 85 (2012). Google Scholar

[29]

A. Jelić, C. Appert-Rolland, S. Lemercier and J. Pettré, Properties of pedestrians walking in line. II. stepping behavior,, Phys. Rev. E, 86 (2012). Google Scholar

[30]

D. Jezbera, D. Kordek, J. Kříž, Petr Šeba and P. Šroll, Walkers on the circle,, J. Stat. Mech. Theory Exp., 2010 (2010). doi: 10.1088/1742-5468/2010/01/L01001. Google Scholar

[31]

Y.-q. Jiang, P. Zhang, S. C. Wong and R.-x. Liu, A higher-order macroscopic model for pedestrian flows,, Physica A, 389 (2010), 4623. doi: 10.1016/j.physa.2010.05.003. Google Scholar

[32]

A. Johansson, Constant-net-time headway as a key mechanism behind pedestrian flow dynamics,, Phys. Rev. E, 80 (2009). doi: 10.1103/PhysRevE.80.026120. Google Scholar

[33]

S. Lemercier, A. Jelić, R. Kulpa, J. Hua, J. Fehrenbach, P. Degond, C. Appert-Rolland, S. Donikian and J. Pettré, Realistic following behaviors for crowd simulation,, Computer Graphics Forum, 31 (2012), 489. doi: 10.1111/j.1467-8659.2012.03028.x. Google Scholar

[34]

S. Lemercier, M. Moreau, M. Moussaïd, G. Theraulaz, S. Donikian and J. Pettré, Reconstructing motion capture data for human crowd study,, in Motion in Games, (7060), 365. doi: 10.1007/978-3-642-25090-3_31. Google Scholar

[35]

B. Maury, A. Roudneff-Chupin, F. Santambrogio and J. Venel, Handling congestion in crowd motion models,, Netw. Heterog. Media, 6 (2011), 485. doi: 10.3934/nhm.2011.6.485. Google Scholar

[36]

M. Moussaïd, E. G. Guillot, M. Moreau, J. Fehrenbach, O. Chabiron, S. Lemercier, J. Pettré, C. Appert-Rolland, P. Degond and G. Theraulaz, Traffic Instabilities in Self-organized Pedestrian Crowds,, PLoS Comput. Biol., 8 (2012). Google Scholar

[37]

M. Moussaïd, D. Helbing and G. Theraulaz, How simple rules determine pedestrian behavior and crowd disasters,, Proc. Nat. Acad. Sci., 108 (2011), 6884. Google Scholar

[38]

K. Nishinari, A. Kirchner, A. Namazi and A. Schadschneider, Extended floor field CA model for evacuation dynamics,, IEICE Transp. Inf. & Syst., E87-D (2004), 726. Google Scholar

[39]

J. Ondrej, J. Pettré, A. H. Olivier and S. Donikian, A Synthetic-vision based steering approach for crowd simulation,, in SIGGRAPH'10, 29 (2010). doi: 10.1145/1833349.1778860. Google Scholar

[40]

S. Paris, J. Pettré and S. Donikian, Pedestrian reactive navigation for crowd simulation: A predictive approach,, Eurographics, 26 (2007), 665. doi: 10.1111/j.1467-8659.2007.01090.x. Google Scholar

[41]

J. Pettré, J. Ondřej, A.-H. Olivier, A. Cretual and S. Donikian, Experiment-based modeling, simulation and validation of interactions between virtual walkers,, in SCA '09: Proceedings of the 2009 ACM SIGGRAPH/Eurographics Symposium on Computer Animation, (2009), 189. Google Scholar

[42]

B. Piccoli and A. Tosin, Pedestrian flows in bounded domains with obstacles,, Contin. Mech. Thermodyn., 21 (2009), 85. doi: 10.1007/s00161-009-0100-x. Google Scholar

[43]

L. Pontrjagin, On the zeros of some elementary transcendental functions,, Amer. Math. Soc. Transl. Ser. 2, 1 (1955), 95. Google Scholar

[44]

C. W. Reynolds, Steering behaviors for autonomous characters,, in Proceedings of Game Developers Conference, (1999), 763. Google Scholar

[45]

A. Seyfried, B. Steffen, W. Klingsch and M. Boltes, The fundamental diagram of pedestrian movement revisited,, J. Stat. Mech. Theory Exp., 2005 (2005). doi: 10.1088/1742-5468/2005/10/P10002. Google Scholar

[46]

A. Seyfried, B. Steffen and T. Lippert, Basics of modelling the pedestrian flow,, Phys. A, 368 (2006), 232. doi: 10.1016/j.physa.2005.11.052. Google Scholar

[47]

J. van den Berg and H. Overmars, Planning time-minimal safe paths amidst unpredictably moving obstacles,, Int. Journal on Robotics Research, 27 (2008), 1274. Google Scholar

[48]

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., 2012 (2012). doi: 10.1088/1742-5468/2012/02/P02002. Google Scholar

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Simone Göttlich, Camill Harter. A weakly coupled model of differential equations for thief tracking. Networks & Heterogeneous Media, 2016, 11 (3) : 447-469. doi: 10.3934/nhm.2016004

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