- Email: [email protected]
Fractional Decision Making Model for Crowds of Pedestrians in Two-Alternative Choice Evacuation
Proceedings of the 20th World The International Federation of Congress Automatic Control Proceedings of the 20th9-14, World The International Federation of Congress Automatic Control Toulouse, France, July 2017 Available online at www.sciencedirect.com The International of Automatic Control Toulouse, France,Federation July 9-14, 2017 Toulouse, France, July 9-14, 2017
ScienceDirect
IFAC PapersOnLine 50-1 (2017) 11764–11769 Fractional Decision Making Model for Fractional Decision Making Model for Fractional Decision Making Model for Crowds of Pedestrians in Two-Alternative Crowds of Pedestrians in Two-Alternative Crowds of Pedestrians in Two-Alternative Choice Evacuation Choice Evacuation Choice∗∗ Evacuation ∗∗∗ ∗ ∗∗∗∗
Ke-cai Cao ∗ CaiBin Zeng ∗∗ YangQuan Chen ∗∗∗ Dong Yue ∗∗∗∗ Ke-cai Cao ∗∗ CaiBin Zeng ∗∗ YangQuan Chen ∗∗∗ Dong Yue ∗∗∗∗ Ke-cai∗ Cao CaiBin Zeng ∗∗ YangQuan Chen ∗∗∗ Dong Yue ∗∗∗∗ College of Automation, Nanjing University of Posts and ∗ ∗ College of Automation, Nanjing of Posts and Telecommunications, Nanjing, P.R.University China, 210023, (e-mail: ∗ College of Automation, Nanjing University of Posts and Telecommunications, Nanjing, P.R. China, 210023, (e-mail: [email protected]). Telecommunications, Nanjing, P.R. China, 210023, (e-mail: ∗∗ [email protected]). School of Sciences, South China University of Technology, ∗∗ [email protected]). ∗∗ School of Sciences, South(e-mail: China University of Technology, Guangzhou, 510640, China [email protected]) ∗∗ School of510640, Sciences, South(e-mail: China University of Technology, ∗∗∗Guangzhou, China [email protected]) Mechatronics Embedded Systems and Automation Lab, School of ∗∗∗Guangzhou, 510640, China (e-mail: [email protected]) ∗∗∗ Mechatronics Embedded Systems and Automation Lab, School of ∗∗∗Engineering, University of California, Merced, CA, USA 95343 MechatronicsUniversity EmbeddedofSystems and Merced, Automation Lab, Engineering, California, CA, USASchool 95343 of (e-mail: [email protected]) Engineering, University of California, Merced, CA, USA 95343 ∗∗∗∗ (e-mail: [email protected]) College of Automation, Nanjing University of Posts and ∗∗∗∗ (e-mail: [email protected]) ∗∗∗∗ College of Automation, Nanjing University of Posts and Telecommunications, Nanjing, P.R. China, 210023, (e-mail: ∗∗∗∗ College of Automation, Nanjing University of Posts and Telecommunications, Nanjing, P.R. China, 210023, (e-mail: [email protected]) Telecommunications, Nanjing, P.R. China, 210023, (e-mail: [email protected]) [email protected]) Abstract: Modeling of the two-alternative decision making process for evacuation of crowds Abstract: Modeling of the two-alternative decision making process for under evacuation of crowds of pedestrians from bounded room has been considered in this paper the framework Abstract: Modelingbounded of the two-alternative decision making process for under evacuation of crowds of pedestrians room has been considered in this paper the order framework of Calculus of from Fractional Order. Dynamic decision making models of fractional have of pedestrians from bounded room has been considered in this paper under the order framework of Calculus of Fractional Order. Dynamic decision making models of fractional have been presented under symmetric interactions and asymmetric interactions, respectively. Stability of Calculus of under Fractional Order.interactions Dynamic and decision makinginteractions, models of fractional order have been presented symmetric asymmetric respectively. Stability analysis under asymmetric interactions has been studied using linear approximation. Final value been presented under symmetric interactions and asymmetric interactions, respectively. Stability analysis under asymmetric interactions has been studied is using linear approximation. Final value of the collective opinion under asymmetric interactions dependent on the initial distribution analysis under asymmetric interactions has been studied is using linear approximation. Final value of the collective opinion dependent on the initial distribution of crowd’s opinions whileunder it is asymmetric not the caseinteractions for symmetric interactions. Future work based on of the collective opinion under asymmetric interactions is dependent on the initial distribution of crowd’s opinions while it is not the case for symmetric interactions. Future work based on the results of this paper are also discussed at the end of this paper. of whileare it also is not the caseatfor Future work based on thecrowd’s results opinions of this paper discussed thesymmetric end of thisinteractions. paper. the results of (International this paper are also discussed at the end ofHosting this paper. © 2017, IFAC Federation of Automatic Control) by Elsevier Ltd. All rights reserved. Keywords: Decision Making Model of Fractional order, Evacuation of Crowds, Keywords: Decision Making Model of Fractional order, Evacuation of Crowds, Two-Alternative Choice, Interactions. Keywords: Decision Making Model of Fractional order, Evacuation of Crowds, Two-Alternative Choice, Interactions. Two-Alternative Choice, Interactions. 1. INTRODUCTION has been proved to be a better choice for describing the has been proved to be amaking better model choice (DMM) for describing the 1. INTRODUCTION TAFC Tasks. Decision of integer 1. INTRODUCTION has been proved to be amaking better model choice (DMM) for describing the TAFC Tasks. Decision of integer for Two-Alternative Choice Tasks has also been The phenomenon of decision making has attracted a lot of order TAFC Tasks. Decision making model (DMM) of integer Two-Alternative Choice Tasks the hascooperative also been The phenomenon decision makingresearch has attracted a lot of order shown for in Turalska et al. (2009) to describe attention in recentofyears. Extensive has been done for Two-Alternative Choice Tasks the hascooperative also been The phenomenon ofyears. decision makingresearch has attracted a lot of order in Turalska et al. (2009) to describe attention in recentdynamic Extensive hasanimals been done interaction of individuals in network where the value of for this complex process from social to shown shown in Turalska et al. (2009) to describe thethe cooperative attention in recentdynamic years. Extensive research hasanimals been done interaction of individuals in network where of for this complex process from social to of interaction has played an importantvalue role in human beings that scattered from modeling in Dirk and strength of individuals in played network where the value of for this beings complex dynamic process from socialinanimals to interaction strength of interaction has an important role in human that scattered from modeling Dirk and collective behavior. Compared with research Peter (1995); Bellomo and Dogbe (2011); Bellomo et al. generating strength of interaction has played an important role in human beings that scattered from modeling in Dirk and generating collective behavior. Compared with research Peter Bellomo and et Dogbe (2011); Bellomo et al. on isolated decision making process, a lot of efforts of (2012),(1995); analysis in Helbing al. (2000); Cao et al. (2015, collective behavior.process, Compared with research Peter (1995); Bellomo and et Dogbe (2011); Bellomo et al. generating isolatedand decision making lotmodeling of efforts of (2012),and analysis in Helbing al. (2000); Cao et al. (2015, biologists engineers has been putaon and 2016) even control in Kachroo (2007); Kachroo et al. on on isolatedand decision making process, aon lotmodeling of effortsand of (2012), analysis in Helbing et al. (2000); Cao et al. (2015, biologists engineers has been put 2016) and even control in Kachroo (2007); Kachroo et al. analysis of the collective decision making process so that (2008); Ivancevic and Ivancevic (2012). biologists and engineers has been put on modeling and 2016) and even control in Kachroo (2007); Kachroo et al. analysis of the collective decision making process so that (2008); Ivancevic and Ivancevic (2012). we can understand and control the collective behavior of analysis of the collective decision making process so that (2008); Ivancevic and Ivancevic (2012). we can understand and control the collective behavior of animals, man-made agents and even that of human beings. 1.1 Dynamic Modeling for Decision Making we can understand and control the that collective behavior of animals, man-made agents and even of human beings. 1.1 Dynamic Modeling for Decision Making animals, man-made agentsMaking and even Model that of human beings. Networked Decision for TAFC: 1.1 Dynamic Modeling for Decision Making Isolated Decision Making Model for Two-Alternative Networked Making for onTAFC: Based on the Decision drift-diffusion modelModel for TAFC micro Isolated Decision Making Although Model fora Two-Alternative Decision Making Model for TAFC: Forced-Choice (TAFC): lot of modeling Networked Based on the drift-diffusion model for TAFC scale, coupled stochastic differential equations on has micro been Isolated Decision Making Model for Two-Alternative on the drift-diffusion model for TAFC on micro Forced-Choice (TAFC): Although a lot of of the modeling methods have been proposed for modeling Two- Based scale, coupled stochasticetdifferential hasthe been employed by Poulakakis al. (2010) equations in analyzing efForced-Choice (TAFC): Although a lot of modeling coupled stochasticetdifferential equations hasthe been methods have been problem, proposedthese for modeling of been the TwoAlternative Choices models have com- scale, employed by Poulakakis al. (2010) in analyzing effects of different communication topology on collective demethods have been proposed for modeling of the Twoemployed by Poulakakis et al. (2010) in analyzing the deefAlternative problem, these models have process been compared basedChoices on physics of decision making in fects ofmaking different communication topology on collective cision process for the TAFC task. Results obtained Alternative Choices problem, these models have process been compared based physics decision making in fects ofmaking different communication topology on collective deBogacz et al.on (2006) andofdrift diffusion model (DDM) cision process for the TAFC task. Results obtained Poulakakis et al. (2010) have been further enhanced in pared based physics decision making process in in cision making process for the TAFC task. Results obtained Bogacz et al.on (2006) andofdrift diffusion model (DDM) in Poulakakis (2010) been further in Poulakakis et et al.al.(2016) tohave characterize the enhanced information This work isal. supported by National Natural Science Foundation of Bogacz et (2006) and drift diffusion model (DDM) in Poulakakis et al. (2010) have been further enhanced in This work is supported by National Natural Science Foundation of Poulakakis et al. (2016)oftoeach characterize theis information centrality and ordering node which very useful China (Grant No. 61374055), Natural science foundation of Jiangsu This work is supported by National Natural Science Foundation of Poulakakis et al. (2016) to characterize the information centrality and ordering of each node which is very useful China (Grant No. 61374055), Natural science foundation of Jiangsu Province (Grant No. BK20161520, BK20131381), China Postdocin selecting leaders based on topology for TAFC problem. centrality and ordering ofon each node which is very useful China (Grant No.No. 61374055), Natural science foundation ofPostdocJiangsu Province (Grant BK20161520, BK20131381), in selecting leaders based topology for TAFC problem. toral Science Foundation funded project (Grant No.China 2013M541663), Coupled drift-diffusion model under network interactions Province (Grant No. BK20161520, BK20131381), China Postdocin selecting leaders based on topology for TAFC problem. toral Science Foundation funded project (Grant No. 2013M541663), Jiangsu Planned Projects for Postdoctoral Research Funds (Grant Coupled drift-diffusion modeland under network interactions are also adopted in Srivastava Leonard (2014) to study toral Science Foundation project (Grant No. Coupled drift-diffusion modeland under network interactions Jiangsu Planned Projectsfunded for Postdoctoral Research Funds (Grant No. 1202015C), Scientific Research Foundation for2013M541663), the Returned are also adopted in Srivastava Leonard (2014) to study the trade-off between speed and accuracy of the evidence Jiangsu Planned Projects for Postdoctoral Research Funds (Grant No. 1202015C), Scientific Research Foundation for the Returned are also adopted in Srivastava andaccuracy Leonard of (2014) to study Overseas Chinese Scholars, State Education Ministry (Grant No. the trade-off between speedreduced and the evidence aggregation process where DDM has been used to No. 1202015C), Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education MinistryUniversity (Grant No. BJ213022), Scientific Research Foundation of Nanjing of the trade-offprocess between speedreduced and accuracy of the evidence aggregation where DDM has been used to Overseas Chinese Scholars, State Education Ministry (Grant No. approximate the original DDM in performance analysis. BJ213022), Scientific Research Foundation of Nanjing University of Posts and Telecommunications (Grant No. NY214075, XJKY14004). aggregation process where DDM reduced DDM has beenanalysis. used to approximate the original in performance BJ213022), Scientific Research Foundation of Nanjing University of Posts and Telecommunications (Grant No. NY214075, XJKY14004). approximate the original DDM in performance analysis. Posts and Telecommunications (Grant No. NY214075, XJKY14004).
Copyright © 2017, 2017 IFAC 12261 2405-8963 © IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Copyright © 2017 IFAC 12261 Peer review under responsibility of International Federation of Automatic Control. Copyright © 2017 IFAC 12261 10.1016/j.ifacol.2017.08.1985
Proceedings of the 20th IFAC World Congress Toulouse, France, July 9-14, 2017 Ke-cai Cao et al. / IFAC PapersOnLine 50-1 (2017) 11764–11769
11765
Comprise between internal opinion and interactions from neighbor agents has been included in Franci et al. (2015) to modeling the two alternatives problem where rich nonlinear dynamics such as bifurcations due to these interactions have been studied and implemented to validate empirically observed behavior. Based on decision making model of integer order for TAFC, Langevin equation with additive noise has been proposed in West et al. (2014a) to describe the collective decision process of all-to-all network when number of individuals goes to infinity. For the two alternative choices problem, some other aspects such as influence of the number of informed agents (Leaders) and the location of these leaders have also been considered in Couzin et al. (2005) and Franci et al. (2015), respectively.
1.2 Overview of this paper
For efficient evacuation of crowds of pedestrians, decision making process has been incorporated in the study of evacuation problem that makes the obtained model is more close to the reality of human beings. Social force model proposed by Dirk and Peter (1995) has been enhanced using information about the crowds at the exit and that of the following crowds in Zainuddin and Shuaib (2010) where the decision-making process has been embedded in the evacuation problem of pedestrians; Influence of strength of interactions, distribution of initial seeds (leaders) and even memory of human beings on the evacuation decision making has been included in social contagion model in Hasan and V.Ukkusuri (2011) where conditions for cascade propagation of information have been studied through simulations; Choosing between two asymmetric exits for pedestrians in one corridor are also conducted in Heliovaara et al. (2012) using experimental study and it seems reasonable to allocate people with similar physical abilities to use the same egress routes for purpose of collective and efficient evacuation; Influences of different interactions on evacuation pattern and evacuation time has been considered in Lo et al. (2006) using non-cooperative game theory; Games theory was also utilized to prove that mutual cooperation will contribute to the efficient evacuation in Bouzat and Kuperman (2014) and Mean field games has been used to modeling the evacuating crowds in Burger et al. (2013) where macroscopic optimal control problem has been solved under nonlinear mobilities of pedestrians. Previous study on evacuation of crowds of pedestrians has shown that there are some more “human” effects should be considered in the modeling of decision making process besides the popular effects of interacting distance. Remark 1. Remark on previous research:
This paper is organized as follows: In Section 2 the definition of fractional calculus and some useful lemmas are firstly presented. Problem formulation is also shown in this section; Section 3 is devoted to show the model of fractional order under symmetric and asymmetric interactions where local stability analysis of obtained fractional model is also given; Finally we present a discussion of the relationship between our model of fractional order and previous model of integer order for decision making process and future work based on the work of this paper.
No matter research on isolated decision making or that on networked decision making, the evidence aggregation process for the Two-Alternative Choice problem has been considered in the framework of calculus of integer order. But more and more published evidence has shown that it is much more appropriate to model the dynamics of decision making process using calculus of fractional order than calculus of integer order such as memory effects in Couzin et al. (2002) and Guzman-Vargas and Hernandez-Peerez (2006), inverse power-law distribution in time and spatial scale of movement of human being and even long-range interactions in Bialek et al. (2011) for group of animals.
The Caputo fractional order differentiation for f (t) with order α is defined by ˆ t 1 f (n) (τ ) C α dτ t0 Dt f (t) = Γ(n − α) t0 (t − τ )α−n+1
Based on the above statements, dynamics of the decision making process for evacuating of crowds of pedestrians has been considered in this paper using the calculus of fractional order. Dynamic model of fractional order for choosing between two exits has been given under symmetric interactions and asymmetric interactions. As generalization of the decision making model of integer order that has been proposed in West et al. (2014b), stability analysis of obtained dynamic model has been studied using linearization around the desired equilibrium. Relationship with previous model and discussion of future work on the obtained model are also given for completeness.
2. PROBLEM FORMULATION 2.1 Definition of Fractional Calculus and Useful Lemmas Definition 2. Two-alternative forced choice (2AFC) (Wikipedia) Two-alternative forced choice (2AFC) is a method for measuring the subjective experience of a person or animal through their pattern of choices and response times. For fractional calculus shown in Podlubny (1999), there are mainly two widely used fractional operators: Caputo and Riemann–Liouville (R–L) fractional operators where the traditional definitions of the integral and derivative of a function are generalized from integer orders to real or complex orders. The following Caputo definition for fractional derivative is adopted in this book because the Laplace transform of the Caputo derivative allows utilization of initial values of classical integer-order derivatives with known physical interpretations. Definition 3. Caputo’s Fractional Derivative(Podlubny (1999))
where n is an integer satisfying n − 1 < α < n. For fractional-order linear time-invariant (FOLTI) system in state-space model C q 0 Dt x(t) = Ax(t) + bu(t) , (1) y(t) = Cx(t)
the stability of the commensurate system (1) is guaranteed by the following Lemma. 12262
Proceedings of the 20th IFAC World Congress 11766 Ke-cai Cao et al. / IFAC PapersOnLine 50-1 (2017) 11764–11769 Toulouse, France, July 9-14, 2017
to choose the other exit is denoted by gmn (t) that depends on his/her neighbor’s choice gmn (t) = (πm (t) − πn (t))2 where m = n = 1, 2 and Ns (t) πs (t) = N where N is the number of neighbor of ith pedestrian and Ns (t) is the number of the ith pedestrian’s neighbor that choosing the sth exit at time t for s = 1, 2. Similarly, Denote the following probability Ωs (t) Σs (t) = Ω where Ω is the total number of the whole network and Ωs (t) is the number of the pedestrians in the total network that choosing the sth exit at time t for s = 1, 2. Fig. 1. Evacuation of crowds of pedestrians in a room with two exits Lemma 4. (Petras (2011)) System (1) is stable if the following condition is satisfied π |arg(eig(A))| > q , 2 where 0 < q < 2 and eig(A) represents the eigenvalue of matrix A. 2.2 Problem considered in this paper We analyze the evacuation problem in a parallelogram room with two symmetrical exits as shown in Figure 1. The choice of each pedestrian is the results of interactions with his/her neighbors and balancing between these two exits. For the pedestrian in yellow color as shown in Figure 1, the choices between left exit and right exit has been made based on the distribution of red ones and blue ones. In order to describe the collective opinion, each pedestrian has been assumed be homogeneous in the dynamics of decision making process in this paper. Both symmetric interactions and asymmetric interactions have been considered.
3. FRACTIONAL DECISION MAKING MODEL Assumption 5. The communication topology among each pedestrian is assumed to be all-to-all in the decision making problem in this paper. Due to the presence of memory, statistic characteristics of time and space and long range interactions in the behavior of large crowds, the dynamic model for decision making process for each pedestrian in Two-Alternative Choice Tasks is modeled by the following coupled master equation of fractional order C α t0 Dt pi1 (t) = −g12 (t)pi1 (t) + g21 (t)pi2 (t) (2) C α t0 Dt pi2 (t) = g12 (t)pi1 (t) − g21 (t)pi2 (t)
where α ∈ (0, 2), pij (t) ∈ [0, 1] is the probability of the ith pedestrian being in state of choosing the jth exit for j = 1, 2. The transition probability from choosing one exit
3.1 Symmetric Interaction Similar to the relationship between action and reaction of forces, g12 (t) = g21 (t) has been assumed in this section. Denote the following ξ(t) as one global variable 1 (3) ξ(t) = Σ1 (t) − Σ2 (t) = [Ω1 (t) − Ω2 (t)]. Ω which means the differences between the two alternative choices of the whole network. As in an network of all to all communications, πs (t) = Σs (t) is satisfied. Then Σs (t) will be one reasonable approximation of the average opinion of the whole network if the number of pedestrians goes to infinity. Thus based on the fractional DMM that prescribed in (2), the opinion of the whole network can be described as pi1 (t) pi2 (t) pi1 (t) α C = −g + g D (t) (t) 12 21 t0 t N N N p pi2 (t) p (t) (t) i2 i1 C α D = g12 (t) − g21 (t) t0 t N N N (4)
Then dynamic equation of the global variable ξ(t) in (3) can be obtained through subtracting the second line from the first line of system in (2) pi1 (t) pi2 (t) C α + 2g D ξ(t) = −2g (t) (t) 12 21 t0 t N N or
= −2g12 (t)ξ(t). (5) Theorem 6. Under the assumption of all to all network, the decision making model (5) of fractional order α ∈ (0, 2) that proposed for the evacuation problem with two exits is asymptotically stable under any symmetric interaction rules. C α t0 Dt ξ(t)
Proof. The equilibrium zero is asymptotically stable that can be found in references concerning about Fractional Calculus in Podlubny (1999); Petras (2011) and is omitted here. Remark 7. As the probability of choosing either of these two exits is equal to each other, there is no difference between random particles and each pedestrian. The number
12263
Proceedings of the 20th IFAC World Congress Toulouse, France, July 9-14, 2017 Ke-cai Cao et al. / IFAC PapersOnLine 50-1 (2017) 11764–11769
of pedestrians that evacuated from each exit will be the same as time goes on.
11767
0.04 0.03
3.2 Asymmetric Interaction Similar to derivations in the previous section, the dynamics of the opinion of the whole network under asymmetric interactions can also be obtained based on the fractional microscopic model (4). The dynamic description of the whole network’s opinion can be obtained as pi1 (t) pi2 (t) C α + g D ξ(t) = −g (t) (t) 12 21 t0 t N N (1 − pi2 (t)) (1 − pi1 (t)) − g12 (t) + g21 (t) N N pi1 (t) pi2 (t) + g21 (t) = −g12 (t) N N pi2 (t) pi1 (t) + g21 (t) 1 − − g12 (t) 1 − N N pi2 (t) pi1 (t) = − (g12 (t) + g21 (t)) − N N + (g21 (t) − g12 (t)) . Then the generalized and fractional version of decision making model that proposed in West et al. (2014b) can be obtained as follows C α t0 Dt ξ(t) = − (g12 (t) + g21 (t)) ξ(t) + (g21 (t) − g12 (t)) . (6) The following asymmetric interactions rules that proposed in West et al. (2014b) gij (t) = g0 exp [K {πj (x, t) − πi (x, t)}] (7) are still used to model the transition probability between different choices of each pedestrian where i = j = 1, 2. Under the assumption of all-to-all network and infinite number of pedestrians, the transition probability gij (t) in (7) can also be written as g12 = g0 exp(−Kξ(t)) (8) g21 = g0 exp(kξ(t)) where preference in choosing one of the two exits has been greatly amplified or suppressed. Thus system (6) can also be written as C α t0 Dt ξ(t) = −g0 (exp(kξ(t)) + exp(−kξ(t))) ξ(t) + g0 (exp(kξ(t)) − exp(−kξ(t))) or
Potential Function V(ξ)
0.02 0.01 0 equilibrium \sqrt(2(k-1))/k
-0.01
equilibrium -\sqrt(2(k-1))/k equilibrium 0
-0.02
k=1.1 k=1.85
-0.03
k=3 -0.04 -0.05 -1.5
-1
-0.5 0 0.5 difference variable ξ(t)
1
1.5
Fig. 2. Potential Function V (ξ) as a function of ξ for g0 = 0.01 and different K The three equilibrium ξeq of system (9) can be computed as 2(K − 1) ξeq = 0; and ξeq = ± K after setting the right side of (9) to zero and expanding the hyperbolic function using Taylor series as shown in Figure 2. Stability of these equilibrium are shown in the following theorem. Theorem 8. Under the assumption of all to all network, the decision making model (9) of fractional order α ∈ (0, 2) that proposed for the evacuation problem with two exits is locally and asymptotically stable around each equilibrium even under asymmetric interaction rules (8). Proof.
(1) Stability analysis of ξeq = 0; When the global difference variable of ξ(t) is close to zero, the asymmetric interaction is also similar to symmetric interactions as g12 = g0 exp(−Kξ(t)) → g0 . g21 = g0 exp(kξ(t)) → g0 Thus system (6) can be reduced to C α t0 Dt ξ(t)
= −2g0 ξ(t). (10) It is easy to see that system (10) is asymptotically stable based on Lemma 4 for α √ ∈ (0, 2).
2(k−1) (2) Stability analysis of ξeq = ± ; k C α After substituting the following linear approximat0 Dt ξ(t) = −2g0 cosh(kξ(t))ξ(t) + 2g0 sinh(kξ(t)) (9) tion of hyperbolic function For stability analysis of (9), the following potential funccosh Kξ = cosh Kξeq + sinh Kξeq (ξ − ξeq ) tion sinh Kξ = sinh Kξeq + cosh Kξeq (ξ − ξeq ) 2g0 K + 1 into system (9), system (9) can be written as the V (ξ) = cosh(kξ) − ξ sinh(Kξ) , K K following form around its equilibrium C α is constructed firstly. Contrary to the Figure 3.8 for V (ξ) t0 Dt ξ(t) = −2g0 (cosh Kξeq + sinh Kξeq (ξ − ξeq ))ξ(t) that shown in West et al. (2014b), the Figure for potential + 2g0 (sinh Kξeq + cosh Kξeq (ξ − ξeq )) function and its equilibrium is redrawn in Figure 2 to = −2g0 (cosh Kξeq + sinh Kξeq (ξ − ξeq ))ξ(t) provide an overview for stability analysis where not only + 2g0 cosh Kξeq ξ ξ ∈ [ −1 1 ] but also some values of ξ that is out of the interval [ −1 1 ] are given in the Figure 2. = −2g0 sinh Kξeq (ξ − ξeq )ξ(t). 12264
Proceedings of the 20th IFAC World Congress 11768 Ke-cai Cao et al. / IFAC PapersOnLine 50-1 (2017) 11764–11769 Toulouse, France, July 9-14, 2017
As sinh Kξeq = ξeq cosh Kξeq is satisfied for each equilibrium ξeq , linearization of system (9) can be derived as C α t0 Dt η(t) = −2g0 sinh Kξeq η(t)(η(t) + ξeq )
Due to the characteristics that mentioned above for decision making of each pedestrian, there are too much limitations to consider the aggregation and diffusion process of evidences only in the framework of calculus of integer order. Advantages of adopting of calculus of fractional = −2g0 ξeq sinh Kξeq η(t) − 2g0 sinh Kξeq η 2 (t), order in modeling of the drift-diffusion process lie in where η(t) = ξ(t) − ξeq . that much more human effects can be easily described or √ 2(k−1) modeled such as memory, statistic characteristics of time Thus around equilibrium ξeq = ± , systems k and space in decision making and long-range interactions (9) can be approximated using in individual interactions. As generalization of dynamic C α (11) model of integer order, the dynamic model of fractional t0 Dt η(t) = −2g0 ξeq sinh Kξeq η(t). As sinh(·) is an odd function, the term ξeq sinh Kξeq > order will provide us much more freedom in characterizing 0 in system (11) for ξeq = 0. Based on the stability and understanding the complexities of crowd’s decision of linear time invariant system of fractional order making process. that shown in Lemma 4,√it is easy to see that these 4.2 Future Topics 2(k−1) two equilibrium ξeq = ± are also locally and k asymptotically stable. Related topics for future work based on the work of this Remark 9. Based on the results obtained in Theorem 8, paper are listed as follows. final value of the global difference variable for the whole (1) Homogeneity in decision making process has been crowd’s decision is dependent on the initial distribution of assumed for each pedestrian in this paper. Due to crowd’s choice. heterogeneity in the aggregation and dissemination of • As shown in the above proof, the asymmetric interevidence in decision making process, dynamic model action can be reduced to symmetric interaction if of variable order or distributed order that proposed in the initial distribution of the whole crowd’s choice is Mainardi et al. (2008); Jiao et al. (2012) will be useful closed to zero and it will be kept as time goes on. In in solving the challenges caused by heterogeneity; other words, if half of the crowds choose to evacuate (2) As all-to-all framework and infinite number of pedesfrom exit 1 and the other half of the crowds choose to trians have been assumed in this paper, it is more evacuate from exit 2, this situation will be kept under reasonable to consider the decision making of crowds symmetric interactions; with finite number of pedestrians and some other • It is often the case that the number of crowds choosing communication topology. The derivation and analysis different exits is not equal to each other. This can under this kind of assumption will be much preferred be validated using the results presented in Theorem in the real evacuation of crowds in reality. 8 under symmetric interaction. If one of the exits (3) Stability of multiple equilibrium of the fractional is much preferred for some of the crowds, this kind decision making model have been analyzed locally of preference can be amplified while choices of the without external forces. How to drive the whole other exit will be suppressed using the interaction crowd’s opinion to move from one equilibrium to rules proposed in (8). This kind of situation can another equilibrium is an interesting problem that is be approved by the observation in real evacuation worthy of putting much more efforts. Research related process that a lot of people try to exit from one exit to this topic will contribute to the efficient and safe while there are not so many people in the other exits; evacuation of crowds of pedestrians. • In evacuation of crowds of pedestrians, it is much preferred that the difference between choosing these two 5. CONCLUSION exits converges to zero non matter the interactions is symmetric or asymmetric. How to make the system Drift-diffusion modeling of evidence accumulating has (9) converge to ξeq = 0 is an interesting problem been further extended from calculus of integer order to caland will be helpful for management of crowds so culus of fractional order. Under the framework of calculus that fluent and efficient evacuation can be realized of fractional order, not only much more freedom is given in in reality. modeling of the decision making of each evacuee but also much more appropriate model can be obtained after some 4. DISCUSSION characteristics of the crowds are explicitly included using the calculus of fractional order. Stability of the decision 4.1 Relationship with previous model making model of fractional order has been studied under symmetric and asymmetric interactions respectively. It Six kinds of model have been proposed and compared in was found that the final evacuation opinion of the whole Bogacz et al. (2006) for modeling of the dynamic model crowds depends on the initial distribution of all pedesof decision making process. All these models except one trian’s choice under asymmetric interaction rules while can be described using drift-diffusion model of integer it is not the case under symmetric interactions. Stability order. The characterizing and analysis of the diffusion analysis in the entire state space without linearization is in decision making process have also been conducted in much preferred and switching condition between different some references such as Ratcliff and McKoon (2008); equilibrium needs further consideration which will play an Poulakakis et al. (2010, 2012); Srivastava and Leonard important role in the efficient and safe evacuation of large crowds of pedestrians. (2014); Poulakakis et al. (2016). 12265
Proceedings of the 20th IFAC World Congress Toulouse, France, July 9-14, 2017 Ke-cai Cao et al. / IFAC PapersOnLine 50-1 (2017) 11764–11769
REFERENCES Nicola Bellomo and Christian Dogbe. On the modeling of traffic and crowds a survey of models, speculations, and perspectives. SIAM Review, 53(3):409–463, 2011. Nicola Bellomo, Benedetto Piccoli, and Andrea Tosin. Modeling crowd dynamics from a complex system viewpoint. Mathematical Models and Methods in Applied Sciences, 22:1–29, 2012. William Bialek, Andrea Cavagna, Irene Giardina, Thierry Mora, Edmondo Silvestri, Massimiliano Viale, and Aleksandra M Walczak. Statistical mechanics for natural flocks of birds. Proceedings of the National Academy of Sciences of the United States of America, 109(13):4786– 91, 2011. Rafal Bogacz, Eric Brown, Jeff Moehlis, Philip Holmes, and Jonathan D. Cohen. The physics of optimal decision making: A formal analysis of models of performance in two-alternative forced-choice tasks. Psychological Review, 113(4):700–765, 2006. S Bouzat and M N Kuperman. Game theory in models of pedestrian room evacuation. Physical Review E, 89(3): 32806–32806, 2014. Martin Burger, Marco Di Francesco, Peter A. Markowich, and Marie therese Wolfram. Mean field games with nonlinear mobilities in pedestrian dynamics. Physics, 19(5):1311–1333, 2013. Kecai Cao, Yangquan Chen, Dan Stuart, and Dong Yue. Cyber-physical modeling and control of crowd of pedestrians: a review and new framework. IEEE/CAA Journal of Automatica Sinica, 2(3):334–344, 2015. Kecai Cao, YangQuan Chen, and Daniel Stuart. A fractional micromacro model for crowds of pedestrians based on fractional mean field games. IEEE/CAA Journal of Automatica sinica, 3(3):261–270, 2016. Iain D. Couzin, Jens Krause, Richard James, Graeme D. Ruxton, and Nigel R. Franks. Collective memory and spatial sorting in animal groups. Journal of Theoretical Biology, 218:1–11, 2002. Iain D. Couzin, Jens Krause, Nigel R. Franks, and Simon A. Levin. Effective leadership and decision-making in animal groups on the move. Nature, 433:513–516, 2005. Helbing Dirk and Molnar Peter. Social force model for pedestrian dynamics. Physical Review E, 51(5):4282– 4286, 1995. Alessio Franci, Vaibhav Srivastava, Naomi Ehrich, and Leonard. A realization theory for bio-inspired collective decision-making. Mathematics, 2015. L. Guzman-Vargas and R. Hernandez-Peerez. Small-world topology and memory effects on decision time in opinion dynamics. Physica A: Statistical Mechanics and its Applications, 372(2):326–332, 2006. Samiul Hasan and Satish V.Ukkusuri. A threshold model of social contagion process for evacuation decision making. Transportation Research Part B: Methodological, 45 (10):1590–1605, 2011. Dirk Helbing, Illes Farkas, and Tamas Vicsek. Simulating dynamical features of escape panic. Nature, 407(28): 487–490, 2000. Simo Heliovaara, Juha-Matti Kuusinen, Tuomo Rinne, Timo Korhonen, and Harri Ehtamo. Pedestrian behavior and exit selection in evacuation of a corridor an experimental study. Safety Science, 50(2):221–227,
11769
2012. Vladimir G Ivancevic and Tijana T Ivancevic. Control and synchronization in complex systems and human crowds. In New Trends in Control Theory, volume Volume 19 of Series on Stability, Vibration and Control of Systems, Series A, pages 445–574. World Scientific, 2012. Zhuang Jiao, YangQuan Chen, and Igor Podlubny. Distributed-Order Dynamic Systems Stability, Simulation, Applications and Perspectives. Springer, 2012. Pushkin Kachroo. Optimal and Feedback Control for Hyperbolic Conservation Laws. PhD thesis, Virginia Polytechnic Institute and State University, 2007. Pushkin Kachroo, Sadeq J. Al-nasur, Sabiha Amin Wadoo, and Apoorva Shende. Pedestrian Dynamics Feedback Control of Crowd Evacuation. Springer-Verlag Berlin Heidelberg, 2008. S.M. Lo, H.C. Huang, P. Wang, and K.K. Yuen. A game theory based exit selection model for evacuation. Fire Safety Journal, 41(5):364–369, 2006. Francesco Mainardi, Antonio Mura, Gianni Pagnini, and Rudolf Gorenflo. Time-fractional diffusion of distributed order. Journal of Vibration and Control, 14(9-10):1267– 1290, 2008. Ivo Petras. Fractional-Order Nonlinear Systems Modeling, Analysis and Simulation. Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg, 2011. Igor Podlubny. Fractional differential equations. Academic Press, 1999. I. Poulakakis, L. Scardovi, and N. E. Leonard. Coupled stochastic differential equations and collective decision making in the two-alternative forced-choice task. 2010 American Control Conference, pages 69–74, 2010. Ioannis Poulakakis, Luca Scardovi, and Naomi Ehrich Leonard. Node classification in networks of stochastic evidence accumulators. 2012. doi: arXiv:1210.4235v1. Ioannis Poulakakis, George F. Young adn Luca Scardovi, and Naomi Ehrich Leonard. Information centrality and ordering of nodes for accuracy in noisy decision-making networks. IEEE Transactions on Automatic Control, 61 (4):1040–1045, 2016. Roger Ratcliff and Gail McKoon. The diffusion decision model: Theory and data for two-choice decision tasks. Neural Computation, 20:873–922, 2008. Vaibhav Srivastava and Naomi Ehrich Leonard. Collective decision-making in ideal networks: The speed-accuracy tradeoff. IEEE Transactions on Control of Network Systems, 1(1):121–132, 2014. Malgorzata Turalska, Mirko Lukovic, Bruce J. West, and Paolo Grigolini. Complexity and synchronization. Physical Review E, 80(2), 2009. B.J. West, M. Turalska, and P. Grigolini. Fractional calculus ties the microscopic and macroscopic scales of complex network dynamics. 2014a. Bruce J. West, Malgorzata Turalska, and Paolo Grigolini. Networks of Echoes Imitation, Innovation and Invisible Leaders, volume Computatio. Springer International Publishing Switzerland, 2014b. Z Zainuddin and M Shuaib. Modification of the decisionmaking capability in the social force model for the evacuation process. Transport Theory and Statistical Physics, 39(1):47–70, 2010.
12266
ScienceDirect
IFAC PapersOnLine 50-1 (2017) 11764–11769 Fractional Decision Making Model for Fractional Decision Making Model for Fractional Decision Making Model for Crowds of Pedestrians in Two-Alternative Crowds of Pedestrians in Two-Alternative Crowds of Pedestrians in Two-Alternative Choice Evacuation Choice Evacuation Choice∗∗ Evacuation ∗∗∗ ∗ ∗∗∗∗
Ke-cai Cao ∗ CaiBin Zeng ∗∗ YangQuan Chen ∗∗∗ Dong Yue ∗∗∗∗ Ke-cai Cao ∗∗ CaiBin Zeng ∗∗ YangQuan Chen ∗∗∗ Dong Yue ∗∗∗∗ Ke-cai∗ Cao CaiBin Zeng ∗∗ YangQuan Chen ∗∗∗ Dong Yue ∗∗∗∗ College of Automation, Nanjing University of Posts and ∗ ∗ College of Automation, Nanjing of Posts and Telecommunications, Nanjing, P.R.University China, 210023, (e-mail: ∗ College of Automation, Nanjing University of Posts and Telecommunications, Nanjing, P.R. China, 210023, (e-mail: [email protected]). Telecommunications, Nanjing, P.R. China, 210023, (e-mail: ∗∗ [email protected]). School of Sciences, South China University of Technology, ∗∗ [email protected]). ∗∗ School of Sciences, South(e-mail: China University of Technology, Guangzhou, 510640, China [email protected]) ∗∗ School of510640, Sciences, South(e-mail: China University of Technology, ∗∗∗Guangzhou, China [email protected]) Mechatronics Embedded Systems and Automation Lab, School of ∗∗∗Guangzhou, 510640, China (e-mail: [email protected]) ∗∗∗ Mechatronics Embedded Systems and Automation Lab, School of ∗∗∗Engineering, University of California, Merced, CA, USA 95343 MechatronicsUniversity EmbeddedofSystems and Merced, Automation Lab, Engineering, California, CA, USASchool 95343 of (e-mail: [email protected]) Engineering, University of California, Merced, CA, USA 95343 ∗∗∗∗ (e-mail: [email protected]) College of Automation, Nanjing University of Posts and ∗∗∗∗ (e-mail: [email protected]) ∗∗∗∗ College of Automation, Nanjing University of Posts and Telecommunications, Nanjing, P.R. China, 210023, (e-mail: ∗∗∗∗ College of Automation, Nanjing University of Posts and Telecommunications, Nanjing, P.R. China, 210023, (e-mail: [email protected]) Telecommunications, Nanjing, P.R. China, 210023, (e-mail: [email protected]) [email protected]) Abstract: Modeling of the two-alternative decision making process for evacuation of crowds Abstract: Modeling of the two-alternative decision making process for under evacuation of crowds of pedestrians from bounded room has been considered in this paper the framework Abstract: Modelingbounded of the two-alternative decision making process for under evacuation of crowds of pedestrians room has been considered in this paper the order framework of Calculus of from Fractional Order. Dynamic decision making models of fractional have of pedestrians from bounded room has been considered in this paper under the order framework of Calculus of Fractional Order. Dynamic decision making models of fractional have been presented under symmetric interactions and asymmetric interactions, respectively. Stability of Calculus of under Fractional Order.interactions Dynamic and decision makinginteractions, models of fractional order have been presented symmetric asymmetric respectively. Stability analysis under asymmetric interactions has been studied using linear approximation. Final value been presented under symmetric interactions and asymmetric interactions, respectively. Stability analysis under asymmetric interactions has been studied is using linear approximation. Final value of the collective opinion under asymmetric interactions dependent on the initial distribution analysis under asymmetric interactions has been studied is using linear approximation. Final value of the collective opinion dependent on the initial distribution of crowd’s opinions whileunder it is asymmetric not the caseinteractions for symmetric interactions. Future work based on of the collective opinion under asymmetric interactions is dependent on the initial distribution of crowd’s opinions while it is not the case for symmetric interactions. Future work based on the results of this paper are also discussed at the end of this paper. of whileare it also is not the caseatfor Future work based on thecrowd’s results opinions of this paper discussed thesymmetric end of thisinteractions. paper. the results of (International this paper are also discussed at the end ofHosting this paper. © 2017, IFAC Federation of Automatic Control) by Elsevier Ltd. All rights reserved. Keywords: Decision Making Model of Fractional order, Evacuation of Crowds, Keywords: Decision Making Model of Fractional order, Evacuation of Crowds, Two-Alternative Choice, Interactions. Keywords: Decision Making Model of Fractional order, Evacuation of Crowds, Two-Alternative Choice, Interactions. Two-Alternative Choice, Interactions. 1. INTRODUCTION has been proved to be a better choice for describing the has been proved to be amaking better model choice (DMM) for describing the 1. INTRODUCTION TAFC Tasks. Decision of integer 1. INTRODUCTION has been proved to be amaking better model choice (DMM) for describing the TAFC Tasks. Decision of integer for Two-Alternative Choice Tasks has also been The phenomenon of decision making has attracted a lot of order TAFC Tasks. Decision making model (DMM) of integer Two-Alternative Choice Tasks the hascooperative also been The phenomenon decision makingresearch has attracted a lot of order shown for in Turalska et al. (2009) to describe attention in recentofyears. Extensive has been done for Two-Alternative Choice Tasks the hascooperative also been The phenomenon ofyears. decision makingresearch has attracted a lot of order in Turalska et al. (2009) to describe attention in recentdynamic Extensive hasanimals been done interaction of individuals in network where the value of for this complex process from social to shown shown in Turalska et al. (2009) to describe thethe cooperative attention in recentdynamic years. Extensive research hasanimals been done interaction of individuals in network where of for this complex process from social to of interaction has played an importantvalue role in human beings that scattered from modeling in Dirk and strength of individuals in played network where the value of for this beings complex dynamic process from socialinanimals to interaction strength of interaction has an important role in human that scattered from modeling Dirk and collective behavior. Compared with research Peter (1995); Bellomo and Dogbe (2011); Bellomo et al. generating strength of interaction has played an important role in human beings that scattered from modeling in Dirk and generating collective behavior. Compared with research Peter Bellomo and et Dogbe (2011); Bellomo et al. on isolated decision making process, a lot of efforts of (2012),(1995); analysis in Helbing al. (2000); Cao et al. (2015, collective behavior.process, Compared with research Peter (1995); Bellomo and et Dogbe (2011); Bellomo et al. generating isolatedand decision making lotmodeling of efforts of (2012),and analysis in Helbing al. (2000); Cao et al. (2015, biologists engineers has been putaon and 2016) even control in Kachroo (2007); Kachroo et al. on on isolatedand decision making process, aon lotmodeling of effortsand of (2012), analysis in Helbing et al. (2000); Cao et al. (2015, biologists engineers has been put 2016) and even control in Kachroo (2007); Kachroo et al. analysis of the collective decision making process so that (2008); Ivancevic and Ivancevic (2012). biologists and engineers has been put on modeling and 2016) and even control in Kachroo (2007); Kachroo et al. analysis of the collective decision making process so that (2008); Ivancevic and Ivancevic (2012). we can understand and control the collective behavior of analysis of the collective decision making process so that (2008); Ivancevic and Ivancevic (2012). we can understand and control the collective behavior of animals, man-made agents and even that of human beings. 1.1 Dynamic Modeling for Decision Making we can understand and control the that collective behavior of animals, man-made agents and even of human beings. 1.1 Dynamic Modeling for Decision Making animals, man-made agentsMaking and even Model that of human beings. Networked Decision for TAFC: 1.1 Dynamic Modeling for Decision Making Isolated Decision Making Model for Two-Alternative Networked Making for onTAFC: Based on the Decision drift-diffusion modelModel for TAFC micro Isolated Decision Making Although Model fora Two-Alternative Decision Making Model for TAFC: Forced-Choice (TAFC): lot of modeling Networked Based on the drift-diffusion model for TAFC scale, coupled stochastic differential equations on has micro been Isolated Decision Making Model for Two-Alternative on the drift-diffusion model for TAFC on micro Forced-Choice (TAFC): Although a lot of of the modeling methods have been proposed for modeling Two- Based scale, coupled stochasticetdifferential hasthe been employed by Poulakakis al. (2010) equations in analyzing efForced-Choice (TAFC): Although a lot of modeling coupled stochasticetdifferential equations hasthe been methods have been problem, proposedthese for modeling of been the TwoAlternative Choices models have com- scale, employed by Poulakakis al. (2010) in analyzing effects of different communication topology on collective demethods have been proposed for modeling of the Twoemployed by Poulakakis et al. (2010) in analyzing the deefAlternative problem, these models have process been compared basedChoices on physics of decision making in fects ofmaking different communication topology on collective cision process for the TAFC task. Results obtained Alternative Choices problem, these models have process been compared based physics decision making in fects ofmaking different communication topology on collective deBogacz et al.on (2006) andofdrift diffusion model (DDM) cision process for the TAFC task. Results obtained Poulakakis et al. (2010) have been further enhanced in pared based physics decision making process in in cision making process for the TAFC task. Results obtained Bogacz et al.on (2006) andofdrift diffusion model (DDM) in Poulakakis (2010) been further in Poulakakis et et al.al.(2016) tohave characterize the enhanced information This work isal. supported by National Natural Science Foundation of Bogacz et (2006) and drift diffusion model (DDM) in Poulakakis et al. (2010) have been further enhanced in This work is supported by National Natural Science Foundation of Poulakakis et al. (2016)oftoeach characterize theis information centrality and ordering node which very useful China (Grant No. 61374055), Natural science foundation of Jiangsu This work is supported by National Natural Science Foundation of Poulakakis et al. (2016) to characterize the information centrality and ordering of each node which is very useful China (Grant No. 61374055), Natural science foundation of Jiangsu Province (Grant No. BK20161520, BK20131381), China Postdocin selecting leaders based on topology for TAFC problem. centrality and ordering ofon each node which is very useful China (Grant No.No. 61374055), Natural science foundation ofPostdocJiangsu Province (Grant BK20161520, BK20131381), in selecting leaders based topology for TAFC problem. toral Science Foundation funded project (Grant No.China 2013M541663), Coupled drift-diffusion model under network interactions Province (Grant No. BK20161520, BK20131381), China Postdocin selecting leaders based on topology for TAFC problem. toral Science Foundation funded project (Grant No. 2013M541663), Jiangsu Planned Projects for Postdoctoral Research Funds (Grant Coupled drift-diffusion modeland under network interactions are also adopted in Srivastava Leonard (2014) to study toral Science Foundation project (Grant No. Coupled drift-diffusion modeland under network interactions Jiangsu Planned Projectsfunded for Postdoctoral Research Funds (Grant No. 1202015C), Scientific Research Foundation for2013M541663), the Returned are also adopted in Srivastava Leonard (2014) to study the trade-off between speed and accuracy of the evidence Jiangsu Planned Projects for Postdoctoral Research Funds (Grant No. 1202015C), Scientific Research Foundation for the Returned are also adopted in Srivastava andaccuracy Leonard of (2014) to study Overseas Chinese Scholars, State Education Ministry (Grant No. the trade-off between speedreduced and the evidence aggregation process where DDM has been used to No. 1202015C), Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education MinistryUniversity (Grant No. BJ213022), Scientific Research Foundation of Nanjing of the trade-offprocess between speedreduced and accuracy of the evidence aggregation where DDM has been used to Overseas Chinese Scholars, State Education Ministry (Grant No. approximate the original DDM in performance analysis. BJ213022), Scientific Research Foundation of Nanjing University of Posts and Telecommunications (Grant No. NY214075, XJKY14004). aggregation process where DDM reduced DDM has beenanalysis. used to approximate the original in performance BJ213022), Scientific Research Foundation of Nanjing University of Posts and Telecommunications (Grant No. NY214075, XJKY14004). approximate the original DDM in performance analysis. Posts and Telecommunications (Grant No. NY214075, XJKY14004).
Copyright © 2017, 2017 IFAC 12261 2405-8963 © IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Copyright © 2017 IFAC 12261 Peer review under responsibility of International Federation of Automatic Control. Copyright © 2017 IFAC 12261 10.1016/j.ifacol.2017.08.1985
Proceedings of the 20th IFAC World Congress Toulouse, France, July 9-14, 2017 Ke-cai Cao et al. / IFAC PapersOnLine 50-1 (2017) 11764–11769
11765
Comprise between internal opinion and interactions from neighbor agents has been included in Franci et al. (2015) to modeling the two alternatives problem where rich nonlinear dynamics such as bifurcations due to these interactions have been studied and implemented to validate empirically observed behavior. Based on decision making model of integer order for TAFC, Langevin equation with additive noise has been proposed in West et al. (2014a) to describe the collective decision process of all-to-all network when number of individuals goes to infinity. For the two alternative choices problem, some other aspects such as influence of the number of informed agents (Leaders) and the location of these leaders have also been considered in Couzin et al. (2005) and Franci et al. (2015), respectively.
1.2 Overview of this paper
For efficient evacuation of crowds of pedestrians, decision making process has been incorporated in the study of evacuation problem that makes the obtained model is more close to the reality of human beings. Social force model proposed by Dirk and Peter (1995) has been enhanced using information about the crowds at the exit and that of the following crowds in Zainuddin and Shuaib (2010) where the decision-making process has been embedded in the evacuation problem of pedestrians; Influence of strength of interactions, distribution of initial seeds (leaders) and even memory of human beings on the evacuation decision making has been included in social contagion model in Hasan and V.Ukkusuri (2011) where conditions for cascade propagation of information have been studied through simulations; Choosing between two asymmetric exits for pedestrians in one corridor are also conducted in Heliovaara et al. (2012) using experimental study and it seems reasonable to allocate people with similar physical abilities to use the same egress routes for purpose of collective and efficient evacuation; Influences of different interactions on evacuation pattern and evacuation time has been considered in Lo et al. (2006) using non-cooperative game theory; Games theory was also utilized to prove that mutual cooperation will contribute to the efficient evacuation in Bouzat and Kuperman (2014) and Mean field games has been used to modeling the evacuating crowds in Burger et al. (2013) where macroscopic optimal control problem has been solved under nonlinear mobilities of pedestrians. Previous study on evacuation of crowds of pedestrians has shown that there are some more “human” effects should be considered in the modeling of decision making process besides the popular effects of interacting distance. Remark 1. Remark on previous research:
This paper is organized as follows: In Section 2 the definition of fractional calculus and some useful lemmas are firstly presented. Problem formulation is also shown in this section; Section 3 is devoted to show the model of fractional order under symmetric and asymmetric interactions where local stability analysis of obtained fractional model is also given; Finally we present a discussion of the relationship between our model of fractional order and previous model of integer order for decision making process and future work based on the work of this paper.
No matter research on isolated decision making or that on networked decision making, the evidence aggregation process for the Two-Alternative Choice problem has been considered in the framework of calculus of integer order. But more and more published evidence has shown that it is much more appropriate to model the dynamics of decision making process using calculus of fractional order than calculus of integer order such as memory effects in Couzin et al. (2002) and Guzman-Vargas and Hernandez-Peerez (2006), inverse power-law distribution in time and spatial scale of movement of human being and even long-range interactions in Bialek et al. (2011) for group of animals.
The Caputo fractional order differentiation for f (t) with order α is defined by ˆ t 1 f (n) (τ ) C α dτ t0 Dt f (t) = Γ(n − α) t0 (t − τ )α−n+1
Based on the above statements, dynamics of the decision making process for evacuating of crowds of pedestrians has been considered in this paper using the calculus of fractional order. Dynamic model of fractional order for choosing between two exits has been given under symmetric interactions and asymmetric interactions. As generalization of the decision making model of integer order that has been proposed in West et al. (2014b), stability analysis of obtained dynamic model has been studied using linearization around the desired equilibrium. Relationship with previous model and discussion of future work on the obtained model are also given for completeness.
2. PROBLEM FORMULATION 2.1 Definition of Fractional Calculus and Useful Lemmas Definition 2. Two-alternative forced choice (2AFC) (Wikipedia) Two-alternative forced choice (2AFC) is a method for measuring the subjective experience of a person or animal through their pattern of choices and response times. For fractional calculus shown in Podlubny (1999), there are mainly two widely used fractional operators: Caputo and Riemann–Liouville (R–L) fractional operators where the traditional definitions of the integral and derivative of a function are generalized from integer orders to real or complex orders. The following Caputo definition for fractional derivative is adopted in this book because the Laplace transform of the Caputo derivative allows utilization of initial values of classical integer-order derivatives with known physical interpretations. Definition 3. Caputo’s Fractional Derivative(Podlubny (1999))
where n is an integer satisfying n − 1 < α < n. For fractional-order linear time-invariant (FOLTI) system in state-space model C q 0 Dt x(t) = Ax(t) + bu(t) , (1) y(t) = Cx(t)
the stability of the commensurate system (1) is guaranteed by the following Lemma. 12262
Proceedings of the 20th IFAC World Congress 11766 Ke-cai Cao et al. / IFAC PapersOnLine 50-1 (2017) 11764–11769 Toulouse, France, July 9-14, 2017
to choose the other exit is denoted by gmn (t) that depends on his/her neighbor’s choice gmn (t) = (πm (t) − πn (t))2 where m = n = 1, 2 and Ns (t) πs (t) = N where N is the number of neighbor of ith pedestrian and Ns (t) is the number of the ith pedestrian’s neighbor that choosing the sth exit at time t for s = 1, 2. Similarly, Denote the following probability Ωs (t) Σs (t) = Ω where Ω is the total number of the whole network and Ωs (t) is the number of the pedestrians in the total network that choosing the sth exit at time t for s = 1, 2. Fig. 1. Evacuation of crowds of pedestrians in a room with two exits Lemma 4. (Petras (2011)) System (1) is stable if the following condition is satisfied π |arg(eig(A))| > q , 2 where 0 < q < 2 and eig(A) represents the eigenvalue of matrix A. 2.2 Problem considered in this paper We analyze the evacuation problem in a parallelogram room with two symmetrical exits as shown in Figure 1. The choice of each pedestrian is the results of interactions with his/her neighbors and balancing between these two exits. For the pedestrian in yellow color as shown in Figure 1, the choices between left exit and right exit has been made based on the distribution of red ones and blue ones. In order to describe the collective opinion, each pedestrian has been assumed be homogeneous in the dynamics of decision making process in this paper. Both symmetric interactions and asymmetric interactions have been considered.
3. FRACTIONAL DECISION MAKING MODEL Assumption 5. The communication topology among each pedestrian is assumed to be all-to-all in the decision making problem in this paper. Due to the presence of memory, statistic characteristics of time and space and long range interactions in the behavior of large crowds, the dynamic model for decision making process for each pedestrian in Two-Alternative Choice Tasks is modeled by the following coupled master equation of fractional order C α t0 Dt pi1 (t) = −g12 (t)pi1 (t) + g21 (t)pi2 (t) (2) C α t0 Dt pi2 (t) = g12 (t)pi1 (t) − g21 (t)pi2 (t)
where α ∈ (0, 2), pij (t) ∈ [0, 1] is the probability of the ith pedestrian being in state of choosing the jth exit for j = 1, 2. The transition probability from choosing one exit
3.1 Symmetric Interaction Similar to the relationship between action and reaction of forces, g12 (t) = g21 (t) has been assumed in this section. Denote the following ξ(t) as one global variable 1 (3) ξ(t) = Σ1 (t) − Σ2 (t) = [Ω1 (t) − Ω2 (t)]. Ω which means the differences between the two alternative choices of the whole network. As in an network of all to all communications, πs (t) = Σs (t) is satisfied. Then Σs (t) will be one reasonable approximation of the average opinion of the whole network if the number of pedestrians goes to infinity. Thus based on the fractional DMM that prescribed in (2), the opinion of the whole network can be described as pi1 (t) pi2 (t) pi1 (t) α C = −g + g D (t) (t) 12 21 t0 t N N N p pi2 (t) p (t) (t) i2 i1 C α D = g12 (t) − g21 (t) t0 t N N N (4)
Then dynamic equation of the global variable ξ(t) in (3) can be obtained through subtracting the second line from the first line of system in (2) pi1 (t) pi2 (t) C α + 2g D ξ(t) = −2g (t) (t) 12 21 t0 t N N or
= −2g12 (t)ξ(t). (5) Theorem 6. Under the assumption of all to all network, the decision making model (5) of fractional order α ∈ (0, 2) that proposed for the evacuation problem with two exits is asymptotically stable under any symmetric interaction rules. C α t0 Dt ξ(t)
Proof. The equilibrium zero is asymptotically stable that can be found in references concerning about Fractional Calculus in Podlubny (1999); Petras (2011) and is omitted here. Remark 7. As the probability of choosing either of these two exits is equal to each other, there is no difference between random particles and each pedestrian. The number
12263
Proceedings of the 20th IFAC World Congress Toulouse, France, July 9-14, 2017 Ke-cai Cao et al. / IFAC PapersOnLine 50-1 (2017) 11764–11769
of pedestrians that evacuated from each exit will be the same as time goes on.
11767
0.04 0.03
3.2 Asymmetric Interaction Similar to derivations in the previous section, the dynamics of the opinion of the whole network under asymmetric interactions can also be obtained based on the fractional microscopic model (4). The dynamic description of the whole network’s opinion can be obtained as pi1 (t) pi2 (t) C α + g D ξ(t) = −g (t) (t) 12 21 t0 t N N (1 − pi2 (t)) (1 − pi1 (t)) − g12 (t) + g21 (t) N N pi1 (t) pi2 (t) + g21 (t) = −g12 (t) N N pi2 (t) pi1 (t) + g21 (t) 1 − − g12 (t) 1 − N N pi2 (t) pi1 (t) = − (g12 (t) + g21 (t)) − N N + (g21 (t) − g12 (t)) . Then the generalized and fractional version of decision making model that proposed in West et al. (2014b) can be obtained as follows C α t0 Dt ξ(t) = − (g12 (t) + g21 (t)) ξ(t) + (g21 (t) − g12 (t)) . (6) The following asymmetric interactions rules that proposed in West et al. (2014b) gij (t) = g0 exp [K {πj (x, t) − πi (x, t)}] (7) are still used to model the transition probability between different choices of each pedestrian where i = j = 1, 2. Under the assumption of all-to-all network and infinite number of pedestrians, the transition probability gij (t) in (7) can also be written as g12 = g0 exp(−Kξ(t)) (8) g21 = g0 exp(kξ(t)) where preference in choosing one of the two exits has been greatly amplified or suppressed. Thus system (6) can also be written as C α t0 Dt ξ(t) = −g0 (exp(kξ(t)) + exp(−kξ(t))) ξ(t) + g0 (exp(kξ(t)) − exp(−kξ(t))) or
Potential Function V(ξ)
0.02 0.01 0 equilibrium \sqrt(2(k-1))/k
-0.01
equilibrium -\sqrt(2(k-1))/k equilibrium 0
-0.02
k=1.1 k=1.85
-0.03
k=3 -0.04 -0.05 -1.5
-1
-0.5 0 0.5 difference variable ξ(t)
1
1.5
Fig. 2. Potential Function V (ξ) as a function of ξ for g0 = 0.01 and different K The three equilibrium ξeq of system (9) can be computed as 2(K − 1) ξeq = 0; and ξeq = ± K after setting the right side of (9) to zero and expanding the hyperbolic function using Taylor series as shown in Figure 2. Stability of these equilibrium are shown in the following theorem. Theorem 8. Under the assumption of all to all network, the decision making model (9) of fractional order α ∈ (0, 2) that proposed for the evacuation problem with two exits is locally and asymptotically stable around each equilibrium even under asymmetric interaction rules (8). Proof.
(1) Stability analysis of ξeq = 0; When the global difference variable of ξ(t) is close to zero, the asymmetric interaction is also similar to symmetric interactions as g12 = g0 exp(−Kξ(t)) → g0 . g21 = g0 exp(kξ(t)) → g0 Thus system (6) can be reduced to C α t0 Dt ξ(t)
= −2g0 ξ(t). (10) It is easy to see that system (10) is asymptotically stable based on Lemma 4 for α √ ∈ (0, 2).
2(k−1) (2) Stability analysis of ξeq = ± ; k C α After substituting the following linear approximat0 Dt ξ(t) = −2g0 cosh(kξ(t))ξ(t) + 2g0 sinh(kξ(t)) (9) tion of hyperbolic function For stability analysis of (9), the following potential funccosh Kξ = cosh Kξeq + sinh Kξeq (ξ − ξeq ) tion sinh Kξ = sinh Kξeq + cosh Kξeq (ξ − ξeq ) 2g0 K + 1 into system (9), system (9) can be written as the V (ξ) = cosh(kξ) − ξ sinh(Kξ) , K K following form around its equilibrium C α is constructed firstly. Contrary to the Figure 3.8 for V (ξ) t0 Dt ξ(t) = −2g0 (cosh Kξeq + sinh Kξeq (ξ − ξeq ))ξ(t) that shown in West et al. (2014b), the Figure for potential + 2g0 (sinh Kξeq + cosh Kξeq (ξ − ξeq )) function and its equilibrium is redrawn in Figure 2 to = −2g0 (cosh Kξeq + sinh Kξeq (ξ − ξeq ))ξ(t) provide an overview for stability analysis where not only + 2g0 cosh Kξeq ξ ξ ∈ [ −1 1 ] but also some values of ξ that is out of the interval [ −1 1 ] are given in the Figure 2. = −2g0 sinh Kξeq (ξ − ξeq )ξ(t). 12264
Proceedings of the 20th IFAC World Congress 11768 Ke-cai Cao et al. / IFAC PapersOnLine 50-1 (2017) 11764–11769 Toulouse, France, July 9-14, 2017
As sinh Kξeq = ξeq cosh Kξeq is satisfied for each equilibrium ξeq , linearization of system (9) can be derived as C α t0 Dt η(t) = −2g0 sinh Kξeq η(t)(η(t) + ξeq )
Due to the characteristics that mentioned above for decision making of each pedestrian, there are too much limitations to consider the aggregation and diffusion process of evidences only in the framework of calculus of integer order. Advantages of adopting of calculus of fractional = −2g0 ξeq sinh Kξeq η(t) − 2g0 sinh Kξeq η 2 (t), order in modeling of the drift-diffusion process lie in where η(t) = ξ(t) − ξeq . that much more human effects can be easily described or √ 2(k−1) modeled such as memory, statistic characteristics of time Thus around equilibrium ξeq = ± , systems k and space in decision making and long-range interactions (9) can be approximated using in individual interactions. As generalization of dynamic C α (11) model of integer order, the dynamic model of fractional t0 Dt η(t) = −2g0 ξeq sinh Kξeq η(t). As sinh(·) is an odd function, the term ξeq sinh Kξeq > order will provide us much more freedom in characterizing 0 in system (11) for ξeq = 0. Based on the stability and understanding the complexities of crowd’s decision of linear time invariant system of fractional order making process. that shown in Lemma 4,√it is easy to see that these 4.2 Future Topics 2(k−1) two equilibrium ξeq = ± are also locally and k asymptotically stable. Related topics for future work based on the work of this Remark 9. Based on the results obtained in Theorem 8, paper are listed as follows. final value of the global difference variable for the whole (1) Homogeneity in decision making process has been crowd’s decision is dependent on the initial distribution of assumed for each pedestrian in this paper. Due to crowd’s choice. heterogeneity in the aggregation and dissemination of • As shown in the above proof, the asymmetric interevidence in decision making process, dynamic model action can be reduced to symmetric interaction if of variable order or distributed order that proposed in the initial distribution of the whole crowd’s choice is Mainardi et al. (2008); Jiao et al. (2012) will be useful closed to zero and it will be kept as time goes on. In in solving the challenges caused by heterogeneity; other words, if half of the crowds choose to evacuate (2) As all-to-all framework and infinite number of pedesfrom exit 1 and the other half of the crowds choose to trians have been assumed in this paper, it is more evacuate from exit 2, this situation will be kept under reasonable to consider the decision making of crowds symmetric interactions; with finite number of pedestrians and some other • It is often the case that the number of crowds choosing communication topology. The derivation and analysis different exits is not equal to each other. This can under this kind of assumption will be much preferred be validated using the results presented in Theorem in the real evacuation of crowds in reality. 8 under symmetric interaction. If one of the exits (3) Stability of multiple equilibrium of the fractional is much preferred for some of the crowds, this kind decision making model have been analyzed locally of preference can be amplified while choices of the without external forces. How to drive the whole other exit will be suppressed using the interaction crowd’s opinion to move from one equilibrium to rules proposed in (8). This kind of situation can another equilibrium is an interesting problem that is be approved by the observation in real evacuation worthy of putting much more efforts. Research related process that a lot of people try to exit from one exit to this topic will contribute to the efficient and safe while there are not so many people in the other exits; evacuation of crowds of pedestrians. • In evacuation of crowds of pedestrians, it is much preferred that the difference between choosing these two 5. CONCLUSION exits converges to zero non matter the interactions is symmetric or asymmetric. How to make the system Drift-diffusion modeling of evidence accumulating has (9) converge to ξeq = 0 is an interesting problem been further extended from calculus of integer order to caland will be helpful for management of crowds so culus of fractional order. Under the framework of calculus that fluent and efficient evacuation can be realized of fractional order, not only much more freedom is given in in reality. modeling of the decision making of each evacuee but also much more appropriate model can be obtained after some 4. DISCUSSION characteristics of the crowds are explicitly included using the calculus of fractional order. Stability of the decision 4.1 Relationship with previous model making model of fractional order has been studied under symmetric and asymmetric interactions respectively. It Six kinds of model have been proposed and compared in was found that the final evacuation opinion of the whole Bogacz et al. (2006) for modeling of the dynamic model crowds depends on the initial distribution of all pedesof decision making process. All these models except one trian’s choice under asymmetric interaction rules while can be described using drift-diffusion model of integer it is not the case under symmetric interactions. Stability order. The characterizing and analysis of the diffusion analysis in the entire state space without linearization is in decision making process have also been conducted in much preferred and switching condition between different some references such as Ratcliff and McKoon (2008); equilibrium needs further consideration which will play an Poulakakis et al. (2010, 2012); Srivastava and Leonard important role in the efficient and safe evacuation of large crowds of pedestrians. (2014); Poulakakis et al. (2016). 12265
Proceedings of the 20th IFAC World Congress Toulouse, France, July 9-14, 2017 Ke-cai Cao et al. / IFAC PapersOnLine 50-1 (2017) 11764–11769
REFERENCES Nicola Bellomo and Christian Dogbe. On the modeling of traffic and crowds a survey of models, speculations, and perspectives. SIAM Review, 53(3):409–463, 2011. Nicola Bellomo, Benedetto Piccoli, and Andrea Tosin. Modeling crowd dynamics from a complex system viewpoint. Mathematical Models and Methods in Applied Sciences, 22:1–29, 2012. William Bialek, Andrea Cavagna, Irene Giardina, Thierry Mora, Edmondo Silvestri, Massimiliano Viale, and Aleksandra M Walczak. Statistical mechanics for natural flocks of birds. Proceedings of the National Academy of Sciences of the United States of America, 109(13):4786– 91, 2011. Rafal Bogacz, Eric Brown, Jeff Moehlis, Philip Holmes, and Jonathan D. Cohen. The physics of optimal decision making: A formal analysis of models of performance in two-alternative forced-choice tasks. Psychological Review, 113(4):700–765, 2006. S Bouzat and M N Kuperman. Game theory in models of pedestrian room evacuation. Physical Review E, 89(3): 32806–32806, 2014. Martin Burger, Marco Di Francesco, Peter A. Markowich, and Marie therese Wolfram. Mean field games with nonlinear mobilities in pedestrian dynamics. Physics, 19(5):1311–1333, 2013. Kecai Cao, Yangquan Chen, Dan Stuart, and Dong Yue. Cyber-physical modeling and control of crowd of pedestrians: a review and new framework. IEEE/CAA Journal of Automatica Sinica, 2(3):334–344, 2015. Kecai Cao, YangQuan Chen, and Daniel Stuart. A fractional micromacro model for crowds of pedestrians based on fractional mean field games. IEEE/CAA Journal of Automatica sinica, 3(3):261–270, 2016. Iain D. Couzin, Jens Krause, Richard James, Graeme D. Ruxton, and Nigel R. Franks. Collective memory and spatial sorting in animal groups. Journal of Theoretical Biology, 218:1–11, 2002. Iain D. Couzin, Jens Krause, Nigel R. Franks, and Simon A. Levin. Effective leadership and decision-making in animal groups on the move. Nature, 433:513–516, 2005. Helbing Dirk and Molnar Peter. Social force model for pedestrian dynamics. Physical Review E, 51(5):4282– 4286, 1995. Alessio Franci, Vaibhav Srivastava, Naomi Ehrich, and Leonard. A realization theory for bio-inspired collective decision-making. Mathematics, 2015. L. Guzman-Vargas and R. Hernandez-Peerez. Small-world topology and memory effects on decision time in opinion dynamics. Physica A: Statistical Mechanics and its Applications, 372(2):326–332, 2006. Samiul Hasan and Satish V.Ukkusuri. A threshold model of social contagion process for evacuation decision making. Transportation Research Part B: Methodological, 45 (10):1590–1605, 2011. Dirk Helbing, Illes Farkas, and Tamas Vicsek. Simulating dynamical features of escape panic. Nature, 407(28): 487–490, 2000. Simo Heliovaara, Juha-Matti Kuusinen, Tuomo Rinne, Timo Korhonen, and Harri Ehtamo. Pedestrian behavior and exit selection in evacuation of a corridor an experimental study. Safety Science, 50(2):221–227,
11769
2012. Vladimir G Ivancevic and Tijana T Ivancevic. Control and synchronization in complex systems and human crowds. In New Trends in Control Theory, volume Volume 19 of Series on Stability, Vibration and Control of Systems, Series A, pages 445–574. World Scientific, 2012. Zhuang Jiao, YangQuan Chen, and Igor Podlubny. Distributed-Order Dynamic Systems Stability, Simulation, Applications and Perspectives. Springer, 2012. Pushkin Kachroo. Optimal and Feedback Control for Hyperbolic Conservation Laws. PhD thesis, Virginia Polytechnic Institute and State University, 2007. Pushkin Kachroo, Sadeq J. Al-nasur, Sabiha Amin Wadoo, and Apoorva Shende. Pedestrian Dynamics Feedback Control of Crowd Evacuation. Springer-Verlag Berlin Heidelberg, 2008. S.M. Lo, H.C. Huang, P. Wang, and K.K. Yuen. A game theory based exit selection model for evacuation. Fire Safety Journal, 41(5):364–369, 2006. Francesco Mainardi, Antonio Mura, Gianni Pagnini, and Rudolf Gorenflo. Time-fractional diffusion of distributed order. Journal of Vibration and Control, 14(9-10):1267– 1290, 2008. Ivo Petras. Fractional-Order Nonlinear Systems Modeling, Analysis and Simulation. Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg, 2011. Igor Podlubny. Fractional differential equations. Academic Press, 1999. I. Poulakakis, L. Scardovi, and N. E. Leonard. Coupled stochastic differential equations and collective decision making in the two-alternative forced-choice task. 2010 American Control Conference, pages 69–74, 2010. Ioannis Poulakakis, Luca Scardovi, and Naomi Ehrich Leonard. Node classification in networks of stochastic evidence accumulators. 2012. doi: arXiv:1210.4235v1. Ioannis Poulakakis, George F. Young adn Luca Scardovi, and Naomi Ehrich Leonard. Information centrality and ordering of nodes for accuracy in noisy decision-making networks. IEEE Transactions on Automatic Control, 61 (4):1040–1045, 2016. Roger Ratcliff and Gail McKoon. The diffusion decision model: Theory and data for two-choice decision tasks. Neural Computation, 20:873–922, 2008. Vaibhav Srivastava and Naomi Ehrich Leonard. Collective decision-making in ideal networks: The speed-accuracy tradeoff. IEEE Transactions on Control of Network Systems, 1(1):121–132, 2014. Malgorzata Turalska, Mirko Lukovic, Bruce J. West, and Paolo Grigolini. Complexity and synchronization. Physical Review E, 80(2), 2009. B.J. West, M. Turalska, and P. Grigolini. Fractional calculus ties the microscopic and macroscopic scales of complex network dynamics. 2014a. Bruce J. West, Malgorzata Turalska, and Paolo Grigolini. Networks of Echoes Imitation, Innovation and Invisible Leaders, volume Computatio. Springer International Publishing Switzerland, 2014b. Z Zainuddin and M Shuaib. Modification of the decisionmaking capability in the social force model for the evacuation process. Transport Theory and Statistical Physics, 39(1):47–70, 2010.
12266