Dynamics of emotional contagion in dense pedestrian crowds

Journal Pre-proof Dynamics of emotional contagion in dense pedestrian crowds

Tengfei Xu, Dongdong Shi, Juan Chen, Tao Li, Peng Lin, Jian Ma

PII:

S0375-9601(19)30979-X

DOI:

https://doi.org/10.1016/j.physleta.2019.126080

Reference:

PLA 126080

To appear in:

Physics Letters A

Received date:

10 April 2019

Revised date:

11 October 2019

Accepted date:

21 October 2019

Please cite this article as: T. Xu et al., Dynamics of emotional contagion in dense pedestrian crowds, Phys. Lett. A (2019), 126080, doi: https://doi.org/10.1016/j.physleta.2019.126080.

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Highlights • • • •

An emotional contagion model is put forward to simulate pedestrian emotion spread process. Pedestrian motion driven by emotional contagion is formulized to investigate pedestrian dynamics. Combined effects of pedestrian density, emotional influence dose factor and pedestrian perception range are discussed. Dynamics of emotional phase transition in moving crowds are investigated.

Dynamics of emotional contagion in dense pedestrian crowds Xu Tengfei1, Shi Dongdong1, Chen Juan2, Li Tao2, Lin Peng2, Ma Jian1,* 1, School of Transportation and Logistics, National Engineering Laboratory of Integrated Transportation Big Data Application Technology, Southwest Jiaotong University, Chengdu, 610031, China 2, Faculty of Geosciences and Environmental Engineering, Southwest Jiaotong University, Chengdu 610031, China Abstract: In places with high-density pedestrian movements, irrational emotions can quickly spread out under emergency, which may eventually lead to asphyxiation and crushing. It was noticed that a pedestrian’s emotion in crowd would change as a result of the influence from other pedestrians. Thus, to explore the dynamics of emotion contagion process in flowing dense pedestrians, two types of pedestrian emotions, i.e., negative and positive have been identified. Taking into account the emotional transit of a pedestrian, a crowd movement model is established in the present paper. We simulate pedestrian movement in a region with periodic boundary condition to study the dynamics of emotional contagion in flowing dense crowds. Influences of the initial negative pedestrian proportion, pedestrian crowd density, emotion influence radius, and dose factor on the transition of overall crowd emotion state have been investigated. We expect this study could provide theoretical suggestions for crowd management. Keywords: negative emotion; emotional influence radius; dose factor; pedestrian movement

1. Introduction With the development of modern society, more and more people are now living in cities. Massive gathering places such as cinemas, supermarkets, stadiums are usually over-crowded in case of major events. Sensing the crowded situation, pedestrians can be easily affected by bad emotions [1], which usually cause some special collective motions such as irrational rushing. For example, on September 30, 2008, in a temple in Jodhpur, western India, there was a rumor saying that bombs had been placed in the temple, which caused crowd riots and resulted in massive trampling. 224 people died in that disaster. On July 24, 2010, 21 people died in the “Love Parade” electronic music festival in Duisburg, Germany. For the reason of the delayed opening door, the pedestrian density on the slope connecting the underground passageways and the event entrance got too high [2]. Some pedestrians tried to get out of the over-crowded scenario, which makes more and more pedestrians get nervous. It finally resulted in crowd turbulent motion [3]. Such accidents often *

Corresponding author. E-mail address: [email protected] 1

caused catastrophic consequences, so it is of great significance to study the characteristics of crowd movement in crowded places under emergencies. To explore pedestrian and evacuation dynamics, modeling and simulation approach [4, 5] is commonly used. Typical models can be categorized into continuous model [6-8] and discrete model [9, 10]. As a representative continuous model, social force model quantifies the factor of social psychology. The model can well reproduce self-organized phenomena, e.g., “bottleneck blockage”, “faster is slow” and “intermittent flow” during escape panic. Considering the influence of relative speed, Mohcine et al. [11] formulized a new spatially continuous force-based model for simulating pedestrian dynamics. Wang et al.[12] improved social force model and proposed a model in which expected speed is revised. A self-expected speed was determined by remaining time, distance between pedestrians and the speed difference with respect to pedestrians around. The revised model can simulate large-scale crowd behavior under emergency situation in a more realistic way. Ma et al.[13] noticed there were two different groups of people in the Love parade disaster, so they introduced two kinds of people into social force model and proposed a heterogeneous contact model for massive crowd. The model can well capture the phenomenon of crowd turbulence. To study escape dynamics in fire, Yang et al.[14] proposed a basic model of personnel escape by introducing the concept of “total risk map”, it is more convenient to determine the route of escape in the fire. Song et al.[15] applied the idea of interaction force to the rules of cellular automata model, enriching the movement pattern of pedestrians. The results indicate that the interaction force in evacuation process has a great influence on the evacuation efficiency. Recently, it is noticed that emergencies often lead to mental nervousness among pedestrians in actual situations. Under some conditions, negative emotion can easily spread out in the crowds, which eventually may cause large-scale crowd riots and catastrophic consequences. Therefore, a deep understanding on the dynamics of negative emotion spread and its influence on pedestrian collective motion would benefit crowd management a lot. Although emotional or psychological state of people is quite difficult to be incorporated in a physical pedestrian movement model, some scholars have still made some progress regarding this issue. Liu et al.[16] proposed a model of emotional infection in a three-dimensional interactive simulation program. It was concluded that the key of crowd management is to control the infection of negative emotion and prevent all crowds from falling into negative emotional state. It is noticed that the emotion contagion process is similar to the SIR [17] model in the traditional epidemiological field. Thus, Fu et al.[18] introduced SIR process into a cellular automata model and developed the “CA-SIRS model.” With this model, simulations of 2

emotion contagion in the crowd without and with individual movement were performed. The results indicate that the individual movement increases the rate of emotional infection and the proportion of infected pedestrians in the final stable state. Reducing the infection duration and the probability of reinfection can significantly reduce the number of infected people. What’s more, Cao et al.[19] combined the P-SIS model with social force model and applied the new model to study crowd evacuation process, the results show that the model can simulate crowd evacuation realistically. In similar way, Liu et al.[20] introduced psychology feature into a crowd stampedes simulation model. In the model, emotional infection process in the stampedes were taking into consideration. Their results are helpful to manage and deploy massive gathering crowds. Analogously, Mao et al.[21] presented an emotion based simulation framework for complex evacuation scenarios, and the simulation successful reproduce the behavior and emotion of the agents during the evacuation in real world. Xu et al.[22] modeled the generation and contagion of panic emotion under multi-hazard circumstances. Summarizing the current studies, it can be found that quite a few studies put effort on studying characteristics of crowd movement and evacuation efficiency under emergency conditions. It is recently noticed that the influence of emotion contagion also contribute to the dynamics of crowd movement. In the present study, we establish an emotion driven pedestrian movement model. In the model, those who have negative emotions would like to wander around, while those who have positive emotions would like to stand still. Each pedestrian is affected by the neighboring pedestrians, as a consequence his emotion status change with increasing time. The model will be detailed in section 2. Section 3 will be the results and discussions. Section 4 comes to the conclusions.

2. Emotional contagion model The model we established in this section mainly consists of two aspects, crowd movements and emotion contagion. Section 2.1 will build a crowd movement model based on the characteristics of people’s motion in “Love Parade 2010” disaster, and in section 2.2 the initial configuration and pedestrian emotion infection rules are introduced. 2.1 Crowd movement As reported in Ref.[13], in the “Love Parade 2010” disaster, there was found two different kinds of pedestrians, i.e., active and inactive ones. For the active pedestrians, they wanted to get out of the crowded 3

situation, so they had a higher expected speed and they can move a relatively long distance away from their initial position. As can be found from the scenario recordings, a lane-formation phenomenon [23] was reported. That was due to the reason that the active pedestrians can perceive the surrounding other active pedestrians in a certain range and follow them. For the inactive pedestrians, they wanted to stay in their original place, but because of the repulsive force generated by the contacts of high-density pedestrians, their motion features were swing around their initial location and had a very small speed. Based on the movement features in “Love Parade” disaster, Ma et al.[13] proposed a modified model as follows,

JJG JJK JJK Fi = Fs + Fr

(1)

JK dvi JJK JJK = Fs + Fr dt

(2)

JJK v0 − vi JK Fs = vi

(3)

JJK v0 =

(4)

τ Nj JJK ¦vj j =1

K ¦v j Nj

j =1

In Equation (1),

JJK Fr

JJK Fi

r 3 JK JJK ­°k (1 − ij ) 2 rij , 2r > rij > 0 Fr = ® 2r ° 0, others ¯ represents the total force exacted on pedestrian i,

(5)

JJK Fs

is the self-driven force, and

is the repulsive force from other pedestrians. Equation (3) is the self-driven force equation, where v0

is the magnitude of the desired speed, and pedestrians in different emotional state have different expected speed,

JK vi

is pedestrian i’s velocity. The direction of the desired speed can be calculated by Equation (4).

For negative pedestrian i, he may follow other negative pedestrians within his perception range R. That is to say, his desired direction is related to the average direction of those N

j

negative pedestrians. This

approach is quite equivalent to the approach used by Shiwakoti and Bandini et al. in Ref.[24, 25]. Equation (5) reflects the repulsive force. In this equation, the coefficient k is a constant reflecting the willingness of the pedestrians to avoid physical contacts with others. The parameter r means pedestrian i’s radius, and rij is the distance between pedestrians i and j. In the proposed model, the radius of all pedestrians is a fixed value 4

of 0.2m. When the central distance between two pedestrians is greater than 2r, there is no contact and no repulsive force, otherwise repulsive force presents. It should be noted that to realize massive pedestrian movement feature, we use periodic boundary [26] conditions here in the present study, thus only pedestrian interactions exist and no walls are included in the model. 2.2 Emotional contagion According to Ma et al.[13], two types of pedestrians can be identified in the “Love Parade 2010” disaster, i.e., active and inactive pedestrians. In fact, it can be found from similar crowd disasters that some of active pedestrians may turn into inactive ones and swing in the initial position after have traveled a relatively long distance, while some of the inactive ones may turn into active ones and try to push their way out. What’s more, it was recently noticed that many researchers believe the origin of a pedestrian’s behavior is his/her emotional or mental state [27, 28]. That is to say, a pedestrian’s emotional state determines their behavior. As a consequence, we define two emotional states, i.e., negative and positive state in the present paper. For active people, they have strong willingness to push their way out because they didn’t feel comfortable under the situation, as a consequence, their emotional state is assumed to be negative emotional state. Similarly, inactive people are assumed to have positive emotional state. As a result, pedestrians in negative emotional state try their best to escape the crowded situation, so they have a higher expected velocity, which is set to be 1.2m/s in the model, and they prefer following other negative pedestrians around. On the contrary, the pedestrians in positive emotional state just want stay near their initial place, so they have a desired velocity of 0m/s. To quantify the emotional contagion process, it is further assumed that the strength of the emotion can be quantified by emotional value. Each person has both negative and positive emotional values and the sum of them is a fixed value. When a person’s negative emotional value or positive emotional value reaches a certain threshold, they will show the corresponding emotional state and perform the motion features accordingly. The two types of pedestrians could transform their emotional state in the process of movements by perceiving other pedestrian’s emotion values and adjusting his/her emotional values. Thus, it is not the expected speed that determines the emotional state, but the emotional state that determines the desired speed and pedestrian movement behavior in the proposed model. For the convenience of illustration, in the following context, we use negative and positive pedestrians to indicate those who have negative emotional 5

state or positive emotional state. Initially, each pedestrian’s emotional values, i.e., negative and positive ones are randomly generated according to the pedestrian’s emotional state. The negative emotional value en and positive emotional value e p of a pedestrian satisfy the following relationship,

0 ≤ en ≤ 1

(6)

0 ≤ ep ≤ 1

(7)

en + e p = 1

(8)

From equation (8), it can be found that if the negative emotional value en increases, the positive emotion value e p will meanwhile decrease. To model the emotional state transition process, we assume that if a person’s positive emotional value satisfies e p ≥ 0.5 + δ , then this person show a positive emotional state. Similarly, if a person’s negative emotional value satisfies en ≥ 0.5 + δ , then the person shows a negative emotional state. In the present paper, we assume δ = 0.05. In case of emergency, a moving person will perceive the neighboring pedestrians’ positive and negative emotion within a certain range and update his or her own emotional value. Meanwhile, the pedestrian’s own emotional value could be perceived by others. Taking pedestrian i as an example, N T is the total number of neighboring pedestrians within a perception range R. En and E p are the sum of those neighbors’ negative and positive emotional value, respectively. Then, we have,

En =

NT

¦

en ( j)

(9)

¦ e ( j)

(10)

j =1, j ≠i

Ep =

NT

p

j =1, j ≠i

P (en ) =

P(ep ) =

En × 100% En + E P

Ep En + EP

×100%

(11)

(12)

Obviously, P(en ) and P ( e p ) represent the proportions of these two kinds emotional value perceived by 6

pedestrian i, respectively. The neighboring pedestrians’ negative and positive emotional influence on the central pedestrian can be determined by, Nn

¦ e (i ) n

an =

×n

(13)

×p

(14)

i =1

Nn Np

¦ e (i ) p

ap =

i =1

Np

Here, N n and N P represent the number of negative and positive pedestrians within radius R. n and p actually represent the relative importance of the corresponding emotion state, so we name them dose factors hereinafter. They satisfy the following equation,

n + p =1

(15)

For each pedestrian, his emotional value will be updated according to the following rules at each time step of the simulation process, (1) If pedestrian i is in negative emotional state, when P (en ) ≥ 0.5, it means that pedestrian i is mainly affected by the neighboring pedestrians who are mainly in negative emotions. Then, emotional value of pedestrian i is updated according to the following Rule 1. Rule 1:

en = en + an

(16)

e p = e p − an

(17)

Otherwise, pedestrian i is affected by the neighboring positive emotions, then the emotion value of pedestrian i is updated according to Rule 2 as follows. Rule 2: en = en − a p

(18)

ep = ep + a p

(19)

(2) If pedestrian i is in positive emotional state and P ( e p ) ≥ 0.5, then pedestrian i is affect by positive emotion, and the emotion value is updated according to Rule 2, otherwise pedestrian i is affected by negative emotion, meaning the emotion will be updated according to Rule 1. 7

For the convenient of illustration, pedestrian emotional state transformation is indicated by changing the corresponding pedestrian’s color in the model. To make the simulation process much more clear, we further provide the flow chart of the simulation in Fig.1, and the relevant parameters are shown in Table 1.

x, y, v, direction, en , e p

v0 = 0m / s

v0 = 1.2m / s

P(en ) =

En ×100% En + EP

JK d vi JJK JJK = Fs + Fr dt

JK d vi JJK JJK = Fs + Fr dt

P(ep ) =

en = en − a p ep = ep + a p

e p = e p − an

×100%

if P(e p ) > 0.5?

if P (en ) > 0.5?

en = en + an

Ep En + EP

en = en − a p

en = en + an

ep = ep + a p

e p = e p − an

em , en

em , en

if e p > 0.5 + eps ?

if en > 0.5 + eps ?

x, y, v, direction, en , e p

Fig.1 The flow chart of simulation process

8

Table.1 Related variables and meanings.

Variables

Meanings

x

x-coordinate

y

y-coordinate

v

The value of velocity

direction

The direction of velocity

type

Emotional state(1-negative;0-positive)

em;en

em-negative emotional value;en-positive emotional value

eps

A small positive number

3 Results and discussions To capture the dynamics of emotional contagion in flowing crowds, a simulated scene with a size of 6×6m2 was set up. Periodic boundary condition was applied so the number of pedestrians during the simulation keeps a constant value. At the beginning of the simulation, the pedestrians’ initial speeds were randomly selected between 0 and 0.5m/s and they followed a uniform distribution. Meanwhile, the proportion of pedestrians in negative emotional state, denoted by z, was initially set as 33%, 50%, and 66%, respectively in different simulations. The desired speed of the initial pedestrians in negative emotional state was set to be 1.2m/s, reflecting that they are eager to get out of the crowded status, and the desired direction was the averaged motion direction of all the neighboring negative pedestrians. While the initial expected speed of positive emotional state pedestrian was set as 0m/s, indicating that this group only wants to stay in their original position. During the simulation process, if the emotional state of pedestrian have a change, then the desired speed will also change according to the emotional state of the pedestrian. Totally, the emotional state decides the desired speed. In addition, the pedestrians’ emotional values at the beginning of the simulation are randomly generated according to the method mentioned in 2.2. Other parameters were set as follows, r=0.2m, IJ=0.5s, and k=20. 3.1 Effect of initial negative pedestrian proportion We firstly set the total pedestrians N=300, the range of emotional infection R=4r, and the dose factor of 9

negative emotion n=0.1. The simulation process for z=33% and 66% are shown in Fig.2. When z=33%, we can see that at the beginning of the simulation, there were some negative pedestrians randomly distributed in the positive pedestrians. With the increase of time, the number of positive pedestrian increases, and meanwhile the number of negative pedestrian decreases. Finally, the overall crowd turns into positive emotion. When z=66%, we can see that at the beginning of the simulation, the percentage of negative pedestrians is larger than that of positive pedestrians. With the increase of time, the negative pedestrians gather together and form a lane. The gathering negative pedestrians can move collectively in the crowd and their number keeps a constant for a long simulation time. Since the purpose of the study is mainly to explore the contagion patterns of negative emotions in emergencies, the following analysis mainly focus on the negative emotion. For negative pedestrian proportions of 33%, 50%, and 66%, we plotted the number of negative pedestrians in one simulation run in Fig.3.

Fig.2 Snapshots of the simulation process for z=33% (a-c) and z=66% (d-f). Note: Gray circles and white circles stand for negative and positive pedestrians, respectively. Red line in the circle stands for every pedestrian’s direction of velocity.

10

Total number of negative pedestrians

33%

200

50%

66%

150

100

50

0 0

50

100

150

200

Time step

Fig.3 The total number of negative pedestrians with increase of time.

From Fig.3, we can see that when z=33% and 50%, the pedestrians in negative emotional state quickly turn to positive emotional state, and when z=66%, some pedestrians in negative emotional state slowly turn to positive emotional state. After 200 time steps, there are still many negative pedestrians. These differences mean that the number of pedestrians in negative emotional state at initialization does have an effect on the emotional state of the overall crowd. In some researches [6, 11], pedestrian desired speed was assumed to be around 1.5 m/s. To verify whether desired speed affect the crowd dynamics, we further set desired speed as 1.5m/s and performed some more simulations. It was found that the value of desired speed doesn’t change the trend of crowd evolving features shown in Fig.3. So, in the next simulation, the desired speed was chosen to be 1.2m/s. In order to verify that the change in the number of negative people in Fig.3 is due to emotional infection, we take z=66% as an example to extract the coordinate, velocity and negative emotional value of two pedestrian in one simulation process, as shown in Fig.4 and Fig.5.

11

6

en

5

0.2000

0.4000

4

Y-coordinate

0.000

0.6000

3

Pedestrian j

2

1.000

t=0

Pedestrian i

1

0

0.8000

0

1

t=0

2

3

4

5

6

X-coordinate

Fig.4 Trajectories of two pedestrians in simulation process. Note: the color means the negative emotional value. (a)

1.0

Velocity(m/s)

1.0

Pedestrian i Pedestrian j

0.8 0.6 0.4 0.2

Negative emotional value

1.2

(b)

Pedestrian i Pedestrian j

0.8 0.6 0.4 0.2 0.0

0.0 0

50

100

150

200

250

300

0

Time step

50

100

150

200

250

300

Time step

Fig.5 The evolution patterns of velocity and negative emotional values in one simulation process.

From Fig.4, we can see the trajectory evolution patterns for these two pedestrians are different. At time t=0, pedestrian i and j both have positive emotional state, as indicated by the corresponding color. For pedestrian i, his trajectory is compact at the beginning of the simulation, which dues to the reason that he had a positive state and thus the corresponding desired velocity is 0m/s, he just wanted to stand still, but the repulsive force still resulted in short movements. Soon afterwards, he can move quickly for a relatively long distance because he transformed into negative emotional state, as indicated by the color changing in Fig.4 and the negative emotion value evolution process in Fig.5(b). As a result of the emotional state changing, he had a lager desired speed, as shown in Fig.5(a). However, for pedestrian j, he was almost always in a positive emotional state, so his trajectory is always compact. Fig.5 shows the evolution pattern of velocity and negative emotional values of the pedestrian. It can find that pedestrian j’s velocity increased first and then decreased. This is because pedestrian j was infected and thus transited into negative emotional state at first 12

but soon after he/she transformed into positive state again, which is correctly correspond to the negative emotional value of pedestrian j in Fig.5(b).

3.2 Effect of dose factor n on negative emotion propagation In the former simulation case, the dose factor n takes value of 0.1. Here in this section, we focus on the effect of dose factor n on the change of a person’s emotional state. The simulation for each parameter combination lasted 600 time steps and at the end of the simulation, the total number of negative pedestrians was recorded. The evolution process of the crowd negative emotion values in one typical simulation run and averaged value of total negative pedestrians of 50 different runs were plotted in Fig.6. From Fig.6(a) we can see that when z=33%, the trends of crowd negative emotional value vary a lot in the simulation process with different dose factor n, although the negative emotional values at the beginning are similar. With the increase of time, the negative emotional values of n=0.9 increase by the maximum value, i.e., 300, which means all the pedestrians in the crowd are negative pedestrians. However, the negative emotional values for the case with n=0.7 and 0.8 gradually decrease. For n=0.7, the negative emotional value becomes 0 at the end of the simulation, while for n=0.8, the value stabilizes at a constant number at the end. That is to say, all pedestrians transit into positive at the end when n=0.7, and when n=0.8, there emerges a two-type pedestrian co-exists state, the overall crowd is neither completely negative nor completely positive, but with both negative and positive ones in a simulation. This phenomenon will be referred to as “intermediate state”. The occurrence probabilities of the intermediate state under different parameters were calculated and shown in Fig.7(a). As can be found in Fig.7(a), a larger dose factor means a larger probability of pedestrian co-exists for z=33%. Due to this reason, in Fig.6(d), the averaged negative pedestrian proportion for n ≥ 0.7 increases. For other dose factor n, all pedestrians in the system transit into positive emotional state. When z=50%, it can be found that when the dose factor n<0.3, all the negative pedestrians turned into positive emotional state. For n ≥ 0.6, all the positive pedestrians turn into negative pedestrians. For the dose factor n between 0.3 and 0.6, taking 0.4 and 0.5 as examples, it can be found that negative and positive pedestrians co-exist in the crowd, as shown in Fig.6(b). The probability of pedestrians co-exist can also be found in Fig.7(a). The number of negative pedestrians in the crowd increases with increasing dose factor n. When z=66%, the whole crowd is infected into negative emotional state except the situation where the dose factor n=0.1. When n=0.1, as shown in Fig.6(c), negative and positive pedestrians co-exist in the crowd. 13

From the above analysis, it can be found that the effect of dose factor n on the overall crowd emotional state has a relation with initial negative pedestrian proportion z. When the value of z is small, the dose factor barely influent overall crowd emotional state, except for very large dose factor. When the value of z is large, even if the dose factor n is small, the negative emotion can still propagates and finally infect the all crowd. 350

350

(a) z=33%

(b) z=50% 300

Crowd negative emotional values

Crowd negative emotional values

300

n=0.7 n=0.8 n=0.9

250 200 150 100 50 0

n=0.4 n=0.5

250 200 150 100 50 0

0

50

100

150

200

0

50

Time step

1.6

(c) z=66%

Averaged negative pedestrian proportion

Crowd negative emotional values

350

300

n=0.1 n=0.2

250

200

150

100

50 0

50

100

150

200

Time step

100

150

200

33%

(d)

50%

66%

1.2 0.8 0.4 0.0 -0.4 0.0

0.2

0.4

0.6

0.8

1.0

Dose factor n

Time step

Fig.6 Effect of dose factor on negative emotional values. In (a), (b) and (c), crowd negative emotional value evaluation for one simulation process with z=33%, 50% and 66% are presented. In (d), the statistical averaged result for 50 simulation runs is shown.

Intermediate state probability

50%

66%

0.16

0.12

0.08

0.04

0.00

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0 Averaged negative pedestrian proportion

33%

(a)

Dose factor n

(b)

0.8 0.6 0.4 0.2 0.0 22%

24%

26%

28%

30%

Initial negative pedestrian proportion z

Fig.7 (a) Intermediate state occurrence probability histogram and (b) the influence of different initial 14

negative pedestrian proportion.

From Fig.7(a) we can clearly see that for different value of z and n, the intermediate state occurrence probability is different. When z=33%, the probability is the highest for n=0.9. When z=66%, the probability is the highest for n=0.1. Comparing these results it can be found that in a crowd where they are only a small proportion of negative pedestrians, they should be quite influential to infect others, otherwise, the crowd could remain in positive state. However, if a crowd is composed of a large amount of negative pedestrians, the overall crowd state can easily turn into negative. For a crowd with equal number of negative and positive pedestrians, i.e., z=50%, the crowd can hardly evolve into completely negative or positive state for moderate dose factor n of 0.4, 0.5, as a result, the intermediate probability at these dose factor is highest in Fig.7. After carefully examining the simulated scenes with highest probability of intermediate state, it can be deduced that the negative emotional values of the crowd can be reduced by psychological intervention and other measures to avoid the crowd state transition, which may contribute to avoiding disasters such as stampede. To further provide theoretical guidance for crowd management, it is important to know the smallest proportion with which the negative emotion would die out when the dose factor n is large. Simulations with n=0.9 were further performed with different initial negative pedestrian proportion. As shown in Fig.7(b), with the increase of initial negative pedestrian proportion, the averaged negative pedestrian proportion increase gradually. When the initial proportion is lower than 24%, the averaged proportion of negative pedestrian in the crowd will almost be 0, so the initial number of negative pedestrians should be controlled less than 24% when emergency occurs, which could serve as a reference for crowd management. 3.3 Effect of the emotion influence range R In the previous sections, it was assumed that the emotion influence range R equals to 4r, but pedestrians are affected by height, size and environment in actual situations, his perception range could be larger or smaller. Therefore more simulations were performed with several different R values. For each influence range, we performed 50 simulation runs and calculated the averaged negative pedestrian proportion. Results can be found in Fig.8.

15

50%

66%

1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4

2r

4r

6r

8r

10r

Averaged negative pedestrian proportion

Averaged negative pedestrian proportion

33%

1.2 (a) n=0.2

33%

1.2 (b) n=0.4

0.8 0.6 0.4 0.2 0.0 -0.2 -0.4

2r

4r

50%

66%

0.6 0.4 0.2 0.0 -0.2 4r

6r

8r

10r

Averaged negative pedestrian proportion

Averaged negative pedestrian proportion

33%

0.8

2r

6r

8r

10r

Influence radius R

1.0

-0.4

66%

1.0

Influence radius R 1.2 (c) n=0.6

50%

Influence radius R

33%

1.2 (d) n=0.8

50%

66%

1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4

2r

4r

6r

8r

10r

Influence radius R

Fig.8 Relationship between averaged negative pedestrian percentage and influence range R when (a) n=0.2, (b) n=0.4, (c) n=0.6 and (d) n=0.8.

It can be seen from Fig.8 that when the proportion of negative pedestrian is small, e.g., z=33%, for n ≤ 0.5 in Fig.8(a) and 8(b), the proportion of pedestrians in negative emotional state is basically 0 at the end of the simulation, and the increase of the influence radius R does not have much influence on the proportion of negative pedestrians. For n ≥ 0.6 in Fig.8(c) and 8(d), the effect of influence radius R emerges. It can be found that there is a gradually decreases tendency in the relationship between negative pedestrian proportion and the influence range R. It can be found when R=2r, the negative pedestrian proportion value is the largest. The negative pedestrian proportion is also affected by dose factor n, in general, a larger n makes a higher proportion value. But when R ≥ 8r, the overall crowd will be finally in positive state no matter what value the dose factor n is. When the number of negative pedestrian is equal to positive pedestrians, i.e., z=50%, for n ≤ 0.5 in Fig.8(a) and 8(b), the proportion of pedestrians in negative emotional state increases with the increase of influence radius R in the steady state of the simulation. When n ≥ 0.6 as shown in Fig.8(c) and 8(d), the proportion of 16

pedestrians in negative emotional state is almost 1, meaning the negative emotion can easily propagate in the crowd. When z=66%, if n ≤ 0.4, the proportion of pedestrians in negative state increases with the increase of the infectious radius R no matter how much the value of n is. When R ≥ 6r, the crowd is ultimately in negative emotional state. 3.4 Effect of pedestrians density In this section, the influence of pedestrian density on the dynamics of emotion propagation will be investigated. The total number of pedestrian in the scenario was set to be N=60, 100, 150, 200 and 250 respectively. Each case was run 50 times and the relationship between averaged negative pedestrian proportion and dose factor n are shown in Fig.9. It can be seen from Fig.9(a) that when the density of the simulation scenario is moderate, i.e., total number of pedestrians is 100, with the increase of the dose factor n, the proportion of pedestrians in negative emotional state increases, no matter what value the initial negative pedestrian proportion is. For large dose factor, the crowd were infected and turned into negative emotional state. Further comparing Fig.9(a) with 9(b) to 9(d), it can be found that the trend of these curves looks similar but with difference. It can be found the dose factor n at which the overall crowd transits into negative state for different total number of pedestrians is different. Summarizing the transition points we got Fig.10.

(a) N=100

33%

50%

1.4

66%

1.2 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 0.0

0.2

0.4

0.6

0.8

1.0

Averaged negative pedestrian proportion

Averaged negative pedestrian proportion

1.4

Dose factor n

33%

(b) N=150

66%

1.0 0.8 0.6 0.4 0.2 0.0 -0.2 0.0

0.2

0.4

0.6

Dose factor n

17

50%

1.2

0.8

1.0

33%

(c) N=200

50%

1.4

66%

1.2 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 0.0

0.2

0.4

0.6

0.8

1.0

Averaged negative pedestrian proportion

Average negative pedestrian proportion

1.4

33%

(d) N=250

50%

66%

1.2 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 0.0

0.2

Dose factor n

0.4

0.6

0.8

1.0

Dose factor n

Fig.9 Relationship between averaged negative pedestrian percentage and dose factor n with pedestrian number of (a) N=100, (b) N=150, (c) N=200, and (d) N=250.

From Fig.10 it can be found that when z=33%, the entire crowd can only transit into negative emotional state only when the negative emotional dose factor n reach 0.9, regardless of how much the total number of pedestrians is. When z=50%, with the increase of the total number of the pedestrians, the value of state transition dose factor n firstly decrease and then keeps almost constant when the total number of pedestrians comes across 150. When z=66%, the value of the state transition dose factor n gradually decrease with the increase of the number of pedestrians. It can be conclude that when the total number of pedestrians is extremely large, e.g., N=300, the negative emotional dose factor n only needs to take a small value and the whole crowd could be turned into negative emotional state. 1.0

33%

50%

200

250

66%

Critical dose factor n

0.8

0.6

0.4

0.2

0.0

50

100

150

300

Total number of pedestrians

Fig.10 Relationship between emotion state transition dose factor n and the total number of pedestrians in the crowd. 18

In the next simulation, different emotion influence ranges were employed when the total number of pedestrians was 60, 100, 150, 200 and 250, respectively. The critical influence radius at which all pedestrians were infected into negative emotional state was plotted in Fig.11. When the initial proportion of negative pedestrian is small, e.g., z=33% and 50%, if n<0.6, the overall crowd would transit into positive pedestrians or there exists an “intermediate state”. As a result, there is no critical influential radius in Fig.11(a). When z=66%, if n<0.6, with the increase of density, the critical influence radius Rc gradually decreases, as shown in Fig.12. That is to say, in high dense crowd, when the initial negative proportion is rather higher, pedestrians would be more easily infected to transit into negative state even though their perception radius is small. Therefore, it is critical for high dense crowd to relieve the negative influence dose factor n. In addition, if n>0.6, no matter what the initial negative proportion and crowd density are, the crowd will eventually turn into a negative state even if the influence range is just 2r. It can also be concluded that the dose factor and critical influence radius jointly contribute the crowd state in different density, because the critical influence radius are different when the dose factor n differs from each other. 12

33%

(a) n=0.5

50%

12

66%

8 6 4 2 0

50%

66%

10 Critical influence radius Rc/r

Critical influence radius Rc/r

10

33%

(b) n≥0.6

50

100

150

200

250

8 6 4 2 0

300

Total number of pedestrians

50

100

150

200

250

300

Toal number of pedestrians

Fig.11 Relationship between critical influence range Rc and the total number of pedestrians in the crowd when the dose factor (a) n=0.5 and (b) n൒0.6.

19

12

50 100 150 200 250 300

Critical influence radius Rc/r

10 8 6 4 2 0.0

0.2

0.4

0.6

0.8

1.0

Dose facor n

Fig.12 Relationship between critical influence radius Rc and the dose factor n when z=66%.

4 Conclusions With the development of modern society, more and more people are now living in cities. Massive gathering places such as mass transit stations, cinemas, supermarkets, stadiums and so on are usually over-crowded in case of major events. Sensing the crowded situations, pedestrians under emergency can be easily affected by irrational emotions which will spread out and lead to asphyxiation and crushing. It is thus assumed that a pedestrian’s emotion in crowd would evolve as a result of other pedestrians’ influence. To explore the dynamics of emotion contagion process in flowing dense pedestrians, two types of pedestrian emotions, i.e., negative and positive have been identified. Negative pedestrian would like to push their way out of the crowded situation, while positive ones would like to stay near their original locations. During the movement process, both negative and positive pedestrians would be influenced by those who are nearby. In this way, the emotional transit of a pedestrian in flowing crowd is modeled in the present paper. Pedestrian movement in a region with periodical boundary condition was simulated to investigate the dynamics of emotional contagion. Influential factors including the initial negative pedestrian proportion, pedestrian crowd density, emotion influence radius, and dose factor have been studied. Their influence on the transition of overall crowd emotion state have been explored. The overall crowd state could be pure negative, mixed state and pure positive. The transition from mixed state to pure negative was carefully examined. Critical phase transition parameters, i.e., the critical dose factor n and critical influence radius Rc in different density crowd, have been identified to prevent the crowd from transiting into a negative state. These results indicate that try 20

to stabilize the emotional state by reducing dose factor n could be a sound theoretical suggestions for the management of flowing dense crowds under emergency situation. It should be noted that it is not easy to incorporate emotions in a physical model. Actually, thanks to the recent findings reported by peer scholars including Gallup et al.[29], Fu et al.[18] and so on, it is possible to further investigate the special empirical observations for the “Love Parade 2010”. The model proposed in the current paper will be validated by performing virtual reality massive gathering experiments in the future. Acknowledgements The authors deeply acknowledge the support from the National Natural Science Foundation of China (Nos. 71871189 and 71473207), the National Key Research and Development Program (2017YFC0804906). The authors express their gratitude to the editor and anonymous reviewers for the valuable comments and suggestions which helped us improving the manuscript.

References [1].

Fan R., Xu K., Zhao J. An agent-based model for emotion contagion and competition in online social media. Physica A: Statistical Mechanics and its Applications, 495:245-259 (2018).

[2].

Helbing D., Mukerji P. Crowd disasters as systemic failures: analysis of the Love Parade disaster. EPJ Data

[3].

Ma J., Song W. G., Lo S. M., Fang Z. M. New insights into turbulent pedestrian movement pattern in

Science, 1(1):7 (2012). crowd-quakes. Journal of Statistical Mechanics: Theory and Experiment, (02):P02028 (2013). [4].

Chen C., Tong Y., Shi C., Qin W. An extended model for describing pedestrian evacuation under the threat of artificial attack. Physics Letters A, 382(35):2445-2454 (2018).

[5].

Dias C., Lovreglio R. Calibrating cellular automaton models for pedestrians walking through corners. Physics

[6].

Helbing D., Molnar P. Social force model for pedestrian dynamics. Phys Rev E Stat Phys Plasmas Fluids Relat

Letters A, 382(19):1255-1261 (2018). Interdiscip Topics, 51(5):4282-4286 (1995). [7].

Hoogendoorn S. P., van Wageningen-Kessels F. L. M., Daamen W., Duives D. C. Continuum modelling of pedestrian flows: From microscopic principles to self-organised macroscopic phenomena. Physica A: Statistical Mechanics and its Applications, 416:684-694 (2014).

[8].

Vicsek T., Czirok A., Ben-Jacob E., Cohen I. I., Shochet O. Novel type of phase transition in a system of self-driven particles. Phys Rev Lett, 75(6):1226-1229 (1995).

[9]. [10].

J. S., A. B. Theory of Self-Reproducing Automata. Urbana: University of Illinois Press,

(1966).

Cao S., Song W., Lv W. Modeling pedestrian evacuation with guiders based on a multi-grid model. Physics Letters A, 380(4):540-547 (2016).

[11].

Chraibi M., Seyfried A., Schadschneider A. Generalized centrifugal-force model for pedestrian dynamics. Phys Rev E Stat Nonlin Soft Matter Phys, 82(4):046111 (2010).

[12].

wang l., cai y., xu q. Modifications to Social Force Model. Journal of Nanjing University of Science and Technology( Natural Science), 25(1):144-149 (2011). 21

[13].

Ma J., Song W., Lo S., Simulation of Crowd-Quakes with Heterogeneous Contact Model, in Traffic and Granular Flow '13. 2015. p. 103-110.

[14].

Yang L., Fang W., Huang R. A model of pedestrians escape in fire based on cellular automata. Chinese Science Bulletin, 47(12):896-901 (2002).

[15].

Song W. A cellular automata evacuation model considering friction and repulsion. Science in China Series E,

[16].

Liu z., Jin w., Huang p. An Emotion Contagion Simulation Model for Crowds Events. Journal of Computer

48(4):403-413 (2005). Research and Development, 50(12):2578-2589 (2013). [17].

Dodds P. S., Watts D. J. A generalized model of social and biological contagion. J Theor Biol, 232(4):587-604 (2005).

[18].

Fu L., Song W., Lv W., Lo S. Simulation of emotional contagion using modified SIR model: A cellular automaton approach. Physica A: Statistical Mechanics and its Applications, 405:380-391 (2014).

[19].

X C. M., J Z. G., S W. M. A method of emotion contagion for crowd evacuation. Physica A: Statistical Mechanics and its Applications, 483:250-258 (2017).

[20].

Liu T., Liu Z., Chai Y. Method for Crowd Trampling and Crushing Simulation Based on Psychological Model. Journal of System Simulation, 28(10):2448-2454 (2016).

[21].

Mao Y., Du X., Li Y., He W. An emotion based simulation framework for complex evacuation scenarios. Graphical Models, 102:1-9 (2019).

[22].

Xu M., Xie X., Lv P. Crowd Behavior Simulation With Emotional Contagion in Unexpected Multihazard Situations. IEEE Transactions on Systems, Man, and Cybernetics: Systems,

[23].

(2019).

Ma J., Song W.-g., Zhang J., Lo S.-m., Liao G.-x. k-Nearest-Neighbor interaction induced self-organized pedestrian counter flow. Physica A: Statistical Mechanics and its Applications, 389(10):2101-2117 (2010).

[24].

S B., F R., G V., An agent model of pedestrian group dynamics : Experiments on Group Cohesion, in Congress of the Italian association for artificial intelligence. 2011. p. 104-116.

[25].

Shiwakoti N., Sarvi M., Rose G., Burd M. Animal dynamics based approach for modeling pedestrian crowd egress under panic conditions. Procedia - Social and Behavioral Sciences, 17:438-461 (2011).

[26].

Burstedde C., Klauck K., Schadschneider A., Zittartz J. Simulation of pedestrian dynamics using a two-dimensional cellular automaton. Physica A: Statistical Mechanics and its Applications, 295(3):507-525 (2001).

[27].

X P., D B., M S., Computer Based Video and Virtual Environments in the Study of the Role of Emotions in Moral Behavior, in International Conference on Affective Computing and Intelligent Interaction. Springer, . 2011: Berlin, Heidelberg. p. 52-61.

[28].

Robb A., Kopper R., Ambani R. Leveraging Virtual Humans to Effectively Prepare Learners for Stressful Interpersonal Experiences. IEEE transactions on visualization and computer graphics, 19(4):662-670 (2013).

[29].

Gallup A. C., Hale J. J., Sumpter D. J., Garnier S., Kacelnik A., Krebs J. R., and Couzin I. D. Visual attention and the acquisition of information in human crowds. Proc Natl Acad Sci U S A, 109(19):7245-7250 (2012).

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Declaration of interests ‫ ܈‬The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. տThe authors declare the following financial interests/personal relationships which may be considered as potential competing interests: