A particle model of crowd behavior: Exploring the relationship between alcohol, crowd dynamics and violence

Aggression and Violent Behavior 13 (2008) 413–422

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Aggression and Violent Behavior

A particle model of crowd behavior: Exploring the relationship between alcohol, crowd dynamics and violence Simon C. Moore a,⁎, Mario Flajšlik b, Paul L. Rosin c, David Marshall c a b c

Violence & Society Research Group, Applied Clinical Research & Public Health, School of Dentistry, Cardiff University, Cardiff, CF14 4XY, UK Department of Electrical Engineering, Stanford University, United States School of Computer Science, Cardiff University, UK

a r t i c l e

i n f o

Article history: Received 18 June 2008 Received in revised form 26 June 2008 Accepted 26 June 2008 Available online 8 July 2008 Keywords: Alcohol use Violent crime Particle model

a b s t r a c t Aggressive behavior is more frequent in drunk crowds compared to sober crowds. However, there exists no predicative theory on why intoxicated crowds should display greater levels of violence as crowd density increases. This paper presents such a model. It is argued that intoxication disrupts social interactions between individuals. As emergent affiliative behaviors, such as line formation, that serve to increase flow and minimize invasions to personal space and therefore goal attainment, are a product of individual level interactions it is argued that intoxication increases individuals levels of stress and therefore aggression. This model is illustrated by a particle model of crowd behavior. Models of crowd behavior, derived from particle physics, have been successfully developed to account for collective emergent features in both human and non-human organisms through modeling individual level interactions. Simulations are consistent with the hypothesis that intoxication disrupts the emergence of affiliative behavior. © 2008 Elsevier Ltd. All rights reserved.

Contents 1. Introduction . . . . . . . . . . . . . 2. Drunkenness and crowds. . . . . . . 3. Alcohol, crowd dynamics and violence 4. Data and environment . . . . . . . . 5. Particle models . . . . . . . . . . . 6. Simulations . . . . . . . . . . . . . 7. Conclusion . . . . . . . . . . . . . References . . . . . . . . . . . . . . . .

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413 414 415 416 416 419 420 421

1. Introduction As composition of emergent social behaviors involves interacting individuals the psychological processes that regulate such interactions will contribute to the nature of those social behaviors. Emergence of affiliative social phenomena is evident in crowd behavior and includes temporary coalitions between individuals that facilitate shared goals, coalitions that are also observed across

⁎ Corresponding author. Tel.: +44 (0)29 20462894; fax: +44 (0)29 20746489. E-mail address: [email protected] (S.C. Moore). 1359-1789/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.avb.2008.06.004

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a range of organisms (Helbing, Johansson, & Al-Abideen, 2007; Helbing & Molnár, 1995; Hoogendoorn & Bovy, 2003). Computational models which describe the relationship between local interactions and emergent phenomena have been developed using principles from particle physics (Vicsek, Czirók, Ben-Jacob, Cohen, & Shochet, 1995). However, such models do not account for individual differences that may affect the emergence of social phenomena. In the current paper, a theoretical model that accounts for the observation that violence is observed more frequently in areas where drunk crowds gather compared to sober crowds is developed, and we further implement a particle model to assess predictions. While particle models have been used to understand crowd behavior, the role of alcohol in such models is novel. Crowds that congregate in and around licensed premises are associated with drunk and disorderly behavior (Livingston, 2008; Zhu, Gorman, & Horel, 2004). The ensuing violence imposes considerable costs on victims, through physical and psychological harm, on the institutions charged with policing those crowds and the health services that treat the injured (Dolan & Moore, 2007; Moore, 2006). A better understanding of the conditions that contribute toward disorder in crowds may therefore offer means through which some of these harms can be reduced by using theoretically informed interventions. Although elementary models of individual behavior have been used to successfully account for and predict complex patterns of movement in sober crowds and these models have successfully informed subsequent interventions that reduced harm (Helbing et al., 2007; Helbing & Molnár, 1995), such models have not been considered for crowds where individuals are drunk. This is an important omission as understanding disorder in and around licensed premises is a priority in many local jurisdictions. The purpose of this paper is threefold: first, to review the relationship between crowding, alcohol, and violence in the night time economy and second, suggest a model which might capture the relationship between crowding, drunkenness, and violence. Finally, the emphasis here is to present a model that would motivate further research and model development. While there are numerous psychological and sociological studies relating alcohol to aggression and violence, this paper's focus is narrower, allowing the opportunity to discuss particle models generally, something that might not be possible should greater attention be paid to behavioral science. Emergent behaviors, e.g., flocking in birds (Olfati-Saber, 2004) and swarming in insects (Buhl et al., 2006), appear coordinated. However, simple models that take the repeated interaction of individuals as their starting point indicate that complex emergent behaviors can be understood at an elementary level (Helbing, Farkas, & Vicsek, 2000; Helbing & Molnár, 1995; Vicsek et al., 1995) and further suggest that simple mathematical models might allow researchers to compare emergent behaviors across a diverse range of organisms (Nicolis & Prigogine, 1977). These self-propelled particle (SPP) models typically start out by characterizing each individual as a simple automaton which responds to local circumstances. They are termed self-propelled, as each particle is propelled in a particular direction from forces originating in the particle towards a goal (e.g., the pub), propulsion that is further modified by forces in the local environment, e.g., the force applied by walking into another person or object. Simple rules regulate these propulsive and reactive forces. This elegant and parsimonious approach to understanding collective action in organisms has also been applied to crowds of humans, most notably in panicking crowds (Helbing et al., 2000; Helbing & Molnár, 1995; Treuille, Cooper, & Popović, 2006). The approach used with humans is similar to those used with insects and other organisms, except that the particles are constrained in two dimensions rather than three sometimes seen in models of flocks. Such simulations have played an important role in designing areas so as to minimize pedestrian congestion, which is of considerable practical value if those crowds are escaping a life threatening events, such as a fire or are panicked for other reasons. As an example, Helbing et al. (2007) modeled crowding at the Hajj, an annual pilgrimage of Muslims to Mecca in Saudi Arabia. Such crowding in 2006 resulted in 362 deaths. Helbing developed a SPP model of crowd movement and applied it in a simulation of the Hajj environment. Simulations were run to test changes to the environment and their predicted effect on crowd density, identified as a key variable in eliciting harm in that context. No lives were lost at the subsequent pilgrimage possibly because Helbing's model had informed a series of crowd control measures and environmental changes which modified congestion. Another example of the practical application of these models was presented by Batty, Desyllas, and Duxbury (2003), who developed a simple model of the Notting Hill Carnival, a carnival in London, UK, which attracts over 1 million visitors to an area no greater than 3 km2. Using this model practitioners and researchers were able to assess how changing characteristics of the carnival's route might ease crowd congestion and therefore reduce probability of harm. In addition to their practical value, SPP models also reveal some of the mechanisms that govern production of emergent social phenomena such as the orderly cooperative behavior that emerges in complex interacting systems (Hoogendoorn & Bovy, 2003). As density increases in swarming locust larvae, individuals appear to line up behind one another (Buhl et al., 2006). This transition from chaotic to orderly behavior is typical in flocking birds, swarming insects, and people (Sumpter, 2006), and typically emerge as density increases. Phase transitions to orderly behavior are also highly advantageous. For example, with human crowds, in narrow alleyways, where individuals are moving in opposing directions, laminar flow emerges. People line up behind other pedestrians who are going in the same direction, a strategy that serves to reduce unintentional conflict, increase flow rate, and unintentional invasions to personal space under conditions when space is limited (Hoogendoorn & Bovy, 2003). Thus, individual level interactions produce emergent behaviors that, in turn, benefit pedestrians through reducing the costs associated with goal attainment. 2. Drunkenness and crowds There are at least two reasons why drunkenness in crowds is important. First, there has been a rapid growth in city centers offering night time leisure activities, typically involving alcohol, to large numbers of young people (Hobbs, 2003). Many thousands of individuals often congregate to drink and socialize in cities around the world, which lead to numerous problems, including violence, in these locations (Giesbrecht & Pederson, 1992; Homel, Hauritz, McIlwain, Wortley, & Carvolth, 1997). Similar to how

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models of panicking crowds can provide insights into how harm might be reduced, modeling crowds in the night time economy (NTE) may also provide important insights which may help reduce violence. Furthermore, SPP models could offer a simple means to assess how various changes to an environment might change crowd dynamics (e.g., Helbing et al., 2007) and therefore possibly the congestion that contributes to aggression and violence. Second, the effect of alcohol on behavior becomes an important factor in SPP models of the licensed premises environment because there are known relationships between drunkenness and cognitive function (Lane, Cherek, Pietras, & Tcheremissine, 2004). Symptoms of heightened levels of intoxication are individuals' loss of balance and therefore staggering gait (Perham, Moore, Shepherd, & Cusens, 2007). This effect of alcohol in causing movement to become more erratic suggests that the emergence of adaptive orderly behavior evident in sober crowds, and which aids flow in crowded areas, may be affected by drunkenness, the hypothesis explored in this paper. Critically, there has been little attempt to conduct causal empirical analyses into the relationship between crowd behavior and disorder, either in the context of behaviorally informed particle models or more generally (Jager, Popping, & van de Sande, 2001). This is an important omission as it is plausible that crowding in and of itself may not be causally related to violence. Drunkenness has been associated with violence (Bushman & Cooper, 1990) and evidence suggests that there is an underlying personality trait, impulsivity, which is associated with both an increase in the desire to drink and levels of aggression (Manuck et al., 1998; Petry, 2001; Vuchinich & Simpson, 1990). These individualistic perspectives suggest that as the number of individuals in a crowd increases a greater proportion of potentially violent individuals will be present if it is assumed that violent traits do not govern crowd membership. The observed relationship between crowd size and number of violent offenses may then be best described according to these individualistic accounts, and that it is not crowding in and of itself that promotes violence. Furthermore, interventions that are designed to affect changes in crowd behavior to reduce disorder would therefore be inappropriate. Evidence, however, suggests a more complex mechanism. Zhu's study of licensed premises density, a proxy to crowd density, suggests an association between density and violence (Zhu et al., 2004). A result confirmed by Livingston (2008) who found a non-linear relationship between premises density and assault frequency, a result inconsistent with an individualistic account of crowd violence. Despite lack of direct empirical evidence and models of crowd behavior relating crowd dynamics to violence policy, changes have been implemented across, for example, the UK. These are changes to licensing laws governing when licensed premises can operate, indeed, the stated motivation of policy makers was that relaxing operating hours would reduce congestion and therefore violence. 3. Alcohol, crowd dynamics and violence It is argued here that one mechanism through which variations in crowd dynamics can elevate levels of aggression is through high entropy (a state characterized by randomness) pedestrian movement eliciting stress in and increasing competition and goal blocking between crowd members which in turn increases aggression. Social stresses, in particular those resulting from an undesired or uncontrollable event frequently trigger anger (Cohen, 1980; Roseman, 2004; Roseman & Evdokas, 2004) and under certain conditions crowd dynamics can represent undesired or uncontrollable events. For example, competition between crowd members is heightened when crowd density is high and resources scarce (Jain, 1987). Moreover, size and design of licensed premises contribute to their atmosphere. Small enclosed venues can promote inefficient service and overcrowding resulting in impatience, combined with increased physical contact between patrons and therefore, aggressive behavior (Macintyre & Homel, 1997). Another stressor candidate concerns the distance deemed acceptable between social group members. Strangers tend to keep 4 to 12 ft apart; more intimate groups keep 18 in. to 4 ft apart (Hall, 1966). As the number of individuals in a space increases, then it follows that available distance between individuals must fall, increasing incidence that personal space is invaded and therefore elevating levels of stress. The relationship between invasions to personal space and stress has received empirical attention. Kanaga and Flynn (1981) found self report stress levels were increased in interviewees when their personal space was invaded by interviewers compared to control conditions. More recently, Evans and Wener (2007) used an array of objective (including salivary cortisol) and subjective measures to examine how crowding affected stress in commuting train passengers. Their analyses suggested that it was not so much the density of crowds that elevated levels of stress but rather the number of times their personal space was invaded. In a controlled study of male university students, Sundstrom (1975) examined how aspects of crowding (crowd density, intrusions to personal space and goal blocking) affected self-reported stress and affiliative behaviors. While the dependant measures in Sundstrom's study mostly concerned interpersonal interactions (for example affiliative behaviors included the frequency of head nodding and talking between participants), the study provides general support of the hypothesis that crowding increases stress and reduces affiliative behavior. In the NTE, where many thousands of individuals compete for resources, stranger proximity and the frequency of unintentional physical interactions will increase as crowd density increases. Crowding and the associated invasions of personal space elicit stress (Evans & Wener, 2007; Heller, Groff, & Solomon, 1977; Middlemist, Knowles, & Matter, 1976) and stress is strongly implicated in the etiology of anger, aggression, and violence (Barnett, Fagan, & Booker, 1991; Kruk, Halasz, Meelis, & Haller, 2004; MartimportuguesGoyenechea & Gomez-Jacinto, 2005; Nelson & Trainor, 2007). This relationship between stress, aggression and violence is both intuitive and empirically demonstrated in humans (Barnett et al., 1991; Kruk et al., 2004; Martimportugues-Goyenechea & GomezJacinto, 2005; Nelson & Trainor, 2007) but has also been identified in rats. Kruk et al. (2004) found that an increase in indicators (the adrenortical stress response) associated with stress increased indicators associated with violence (hypothalamic aggression) and vice versa (aggression increased stress). The implication of Kruk's results is that they explain why some people are often quick to lash out but slow to cool down in stressful situations and also how aggression can elicit an ongoing stress response which is

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difficult to stop (Nelson & Trainor, 2007). Generalizing Kruk's (Kruk et al., 2004) findings to the NTE suggests that features of crowd dynamics that cause invasions in personal space may partly explain the prevalence of violence in such environments. Phase transitions to orderly behavior in crowds are adaptive and serve to increase flow and reduce congestion (Dussutour, Fourcassié, Helbing, & Deneubourg, 2004; Hoogendoorn & Bovy, 2003). A staple of law enforcers' tools to detect drunkenness in, for example, car drivers has been the field sobriety test. This test requires those who are suspected of driving under the influence of alcohol to complete a series of tasks that will help determine whether their level of drunkenness to too high for safe use of a vehicle. One of these tests requires participants to walk in a straight line, something which those who are heavily intoxicated are unable to do. Empirical evidence (Perham et al., 2007) supporting the notion that intoxication is associated with a staggering gait comes from a novel breathalyzer survey of NTE drinkers. Researchers assessed whether drinkers were displaying a staggering gait and recorded their breath alcohol level. Analyses clearly indicated that a staggering gait emerged when estimated percent blood alcohol level reached 0.18% (the driving limit in the US and UK is 0.08%) and was evident in over 20% of their sample. 4. Data and environment So that simulations were realistic models were developed using survey data collected in a typical UK city center environment, an environment which was characterized by having numerous licensed premises. The survey, led by Moore (Moore, Perham, Shepherd, & Cusens, 2007), provided data on drunkenness and behavior in NTE crowds. Twenty four surveys were completed between December 2004 and December 2005 from 11pm to 3am on one Friday and one Saturday evening, each month and involved breathalyzing 1038 respondents. Median blood alcohol concentration (BAC) in men was 0.13% (min = 0%, max = 0.33%) and in women was 0.09% (min = 0%, max = 0.27%). About 7% of the sample provided breathalyzer scores of zero. To contextualize these data, a score of 0.08% is the drink driving limit in the UK and US, at risk drinking occurs from 0.15% upwards (NIAAA, 2004). Surveyors also recorded whether drinkers were walking normally or staggering. Of interest here is the BAC at which transition from walking normally to staggering occurs. Moore and colleagues (Perham et al., 2007) found that the BAC transition point from walking normally to staggering occurred at 0.186% BAC (0.184% BAC for men and 0.189% BAC for women) and that approximately 25% of all drinkers exhibited a staggering gait. Other aspects of the survey which are relevant concern gender and age: systematic differences in human crowd motion have been observed between men and women and between different age groups (Henderson, 1971; Henderson & Lyons, 1972; Ishaque & Noland, 2008). Moore describes a surprising result in the street surveys which suggest that for every one woman in this environment there were two men. These results were evident both in the survey data and subsequent head counts initiated to verify that result (Perham et al., 2007). However, most drinkers were in their twenties suggesting variations in age were not critical for models of NTE crowds. This is important as gender predicts aspects of pedestrian trajectory (discussed further below). Another important feature of NTE crowds concerns social groups. As with many other animals (Couzin, Krause, Franks, & Levin, 2005; Dussutour et al., 2004), humans' social behavior is usually conducted within groups. In particle models, groups may be important as they represent an additional structure within a crowd. Usually groups try to stay together, their behaviors are correlated and they persist irrespective of crowd density. Demers et al. (2002) provide estimates suggesting social group sizes in people socializing around licensed premises mostly comprise of four to nine individuals (37.9%) with fewer groups having being made up of two to three (23.8%) and more than ten (25.8%) individuals and only 12.5% of groups contain just one person. Assuming a Poisson distribution, f ðk; λÞ ¼

e−λ λk k!

ð1Þ

for these data and fitting (using OLS) to category medians, a probability density function for group size was derived (λ≈5) and subsequently informed our development of the particle model (see below). In sum, the available research into NTE crowds suggested that 60% of all particles in a SPP model should be male, that 25% stagger due to drunkenness and that median social group size is about four. 5. Particle models Most models of human crowd behavior assume that individuals are simple information processors reacting to events in the immediate environment. These assumptions are appropriate where density is such that individuals have a very restricted choice set; distal events are irrelevant when individual behavior is wholly determined by adjacent pedestrians. Pedestrian movement in less confined areas, however, is more varied and results from more complex interactions. Exogenous factors which effect pedestrian walking speed, for example, includes age, gender, physical size, and environmental characteristics including weather conditions and pedestrian density (Hoogendoorn & Bovy, 2003; Ishaque & Noland, 2008). Moreover, at road crossings where traffic flow is controlled by lights, pedestrians walk more slowly when lights indicated that it is safe for them to cross compared to other times (Ishaque & Noland, 2008). Pedestrians both react to proximal events and plan their trajectory through an understanding of the environment. These two aspects of pedestrian movement are components of Hoogendoorn's model, which includes both a physical and control model. The physical model accounts for proximal forces exerted on individuals by crowds and adjacent individuals similar to the force models developed by Helbing and others (Helbing et al., 2000, 2007; Helbing & Molnár, 1995). Hoogendoorn's control model (Hoogendoorn & Bovy, 2003) develops a game theoretic perspective such that pedestrians are realized as utility

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maximizing agents where control is exerted on velocity (velocity is described here as a vector containing information on the individuals speed and direction) so that subjective expected utility is optimized. In this control model, individuals 1. 2. 3. 4. 5.

continuously reconsider their walking choices using current observations and are therefore feedback-oriented controllers, they are anisotropic, they react only to stimuli in front of them, they anticipate the behavior of other pedestrians by predicting their trajectories, predictions are limited to pedestrians in their immediate locality, they avoid straying from planned trajectories, unless necessary.

Hoogendoorn's more psychologically plausible model acted as the starting point for the model presented in the current paper. The current particle model was adapted from Helbing's generalized force model (Helbing & Molnár, 1995), Hoogendoorn's control model (Hoogendoorn & Bovy, 2003) and programmed in MatLab (“MatLab”, 2007). Particles were realized as a noncollapsible circle on a two-dimensional plane, with radius r, mass m. Particle velocity was governed by forces exerted on the particle and towards the particle's goal or forces exerted by objects and other particles in its immediate environment resulting in a ⇀ ⇀ net force vector, f . Thus, for each time step the change in velocity, v , was given by →

→

Δv ¼

f  Δt ; m

ð2Þ

where t is time, and velocity at a subsequent time steps by →

→

v tþ1 ¼ v t þ

→

f  Δt : m

ð3Þ

It is unrealistic to expect all particles within an environment to influence all other particles particularly as the distance between particles increases. Each particle's proximal ‘attention’ was therefore limited spatially such that particles influenced one another only when they were within a pre-determined area (within 20r, although, as discussed below, near and far particles in this range were treated differently) and the force exerted when they were within this range was proportional to the distance to that particle, given by, →

f ¼

1 d−2r

ð4Þ

where d is the distance between the center of both particles. This did mean, however, that d − r b 0 was possible. For this reason particles were treated→as elastic and allowed to bounce off objects and each other. This was achieved by finding the component of ⇀ v in the direction of d , perpendicular to the object or other particle, and reversing that component. A dampening factor was also ⇀ ⇀ placed on v to reduce v by 80% to account for the imperfect elasticity of the human body. Pedestrians are able to predict each others movement and can alter trajectories based on estimated future states. Properties of ⇀ ellipses were used to implement this feature and applied to the all particles within a 5r radius. First, from v the first particle's coordinates were estimated at t + 5. Predicted and current locations were used as foci, ft and ft + 5, in an ellipse with an eccentricity such that the boundary passed through the second, to be avoided, particle's center. The sum of the line segments, dt and dt + 5, from ft and ft + 5 to the second particle center are, due to the properties of an ellipse, constant. The force exerted by the second particle was in proportion to the sum of dt and dt + 5, delip (Eq. (5)). In effect, forces were regulated in such a manner to maintain a ‘personal space’ (dp) which was partly defined by a future expected state. More formally, delip was defined as delip ¼

jdt j þ jdtþ5 j ; 2

ð5Þ

⇀

and f was proportional to
 ⇀ dp f ≈− −1 d t : delip

→

ð6Þ

The predictive ability of a particle was extended further to consider cases where other particles' future states were estimated, in particular whether other particles (within the 20r to 5r band) would cross paths. To determine if convergence between two particles would ⇀ ⇀ occur the component of the second particle's velocity, v 2, parallel to the first particle's velocity, v 1, was calculated. The time at convergence, tc, could then be determined, where convergence was the point where the particles would pass one another if their trajectories were parallel and opposing. Using tc and the second particle's velocity the actual location at tc of the second particle could be deduced. If the second particle at tc was within a corridor of width 2r+r/2, from the first particle to it's goal then the estimated position of the second particle at tc exerted a force on the first particle as if a particle were in that position. In social environments it is plausible that objects occlude the line of sight to a goal. Stationary particles were treated as objects and groups of stationary particles were treated as a single object if the gaps between particles were no larger than 2r + r/2. If an object lay within a corridor of width 2r + r/2 between particle and goal then it was avoided. Referring to Fig. 1, the force exerted towards the goal was rotated to find a clear path past the object but keeping the deviation from the optimal trajectory, θ, at a minimum. In this example, the force was rotated such that the particle moved past point A rather than point C, which would have involved a larger → value of θ. Avoidance calculations were computed at each time step meaning that f was continually updated so that additional features of the object, e.g. corner B, influenced the particles trajectory once they entered line of sight.

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Fig. 1. Avoiding objects.

Socializing involves maintaining spatial proximity to friends (Demers et al., 2002). Whereas previous particle models have been mostly concerned with individual behavior under extreme conditions, such as panicking crowds in a confined space (Helbing et al., 2000; Helbing, Johansson, & Al-Abideen, 2007), the environment considered in this paper affords more space to individuals. Therefore social cohesion, which bonds individuals to their respective social group, should play an important role. It is important as social grouping implies that particles will show differential preferences depending on whether adjacent particles are part of their social group or not. Models developed to mimic flocking in non-human organisms were adapted (Reynolds, 1987). Reynolds (1987) suggested three rules which govern flock behavior: 1. flock centering: attempt to stay close to nearby flock mates, 2. obstacle avoidance: avoid collisions with nearby flock mates, 3. velocity matching: attempt to match velocity with nearby flock mates. Rule two, had already been implemented. The implementation of rules one and three was developed in two stages. First, it was assumed that one advantage of social groups is that members can delegate responsibility. We simplified social interactions by designating one particle as the lead particle and applying goal-specific forces only on that one particle (algorithms governing avoidance were unaffected). Lead particles were defined as those which were closest to the group's goal and for this reason designations (and the application of goal-specific forces) changed as the group moved through the simulation. Second, group cohesion was defined by imposing forces on each particle towards the group's spatial center. Spatial center was calculated as the average x–y coordinates of group particles. A motivation of the current model was to incorporate into the particle model some of the numerous individual differences that exist between individuals. In particular, women are more easily deflected from their optimal trajectory by other people than men (Henderson, 1971; Henderson & Lyons, 1972). A straightforward realization of this gender difference was to simply increase the average mass of male particles (by 10%) relative to female particles. Drunkenness, the focus of the current simulation, was defined as having a staggering gait (Perham et al., 2007). In the current particle model, drunkenness was defined as a loss of balance leading to the drinkers' inability to keep their center of mass stable. To prevent falling drinkers try to keep their feet under their center of mass. Thus, to model a staggering gait a small random force, frnd, was applied at each time step to particles designated drunk. The actual force applied to the particle was smoothed and calculated from, where ft is the total drunkenness force at time t, ftþ1 ¼ 0:5  frnd þ 0:5  ft :

ð7Þ

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Fig. 2. Groups of particles were sent both directions down a narrow space (A), when all particles were sober (B) laminar flow was evident whereas in simulations where all particles were drunk and staggering (C) behavior was less organized.

This meant that the force applied at the current time step depended on all the previous random forces generated but that dependence falls away exponentially with time (e.g. a force from six time steps back only accounts for 1.5% of the current force). 6. Simulations The hypothesis under exploration concerned the role of alcohol in destabilizing individual trajectories and therefore the transition point and emergence of adaptive crowd behaviors. If emergence of phenomena such as laminar flow, a phenomenon which reduces some of the problems associated with crowding, was affected by intoxication then this would increase the prevalence of aggression. To assess the relationship between intoxication and emergent behavior in particle models of crowd movement a series of simulations was run. In these simulations the proportion of particles drunk and staggering was the independent variable. In addition, particles were randomly assigned gender, weighted so that 60% would be male and 40% female, and whether they were staggering or not, again weighted so that the desired proportion (25%) of staggering particles would be achieved. The simulation environment was a narrow alleyway, where three groups of particles were created at each end (two groups are shown in Fig. 2A for simplicity). Each group was attracted to one of three positions at the other end of the alleyway. Referring to Fig. 2, initial simulations appeared to suggest that laminar flow was less evident in simulations where all particles were drunk (Fig. 2C). Whereas laminar flow was evident, as predicted in simulations where all particles were sober (Fig. 2B). Qualitatively, these initial simulations indicated that emergent laminar flow was a property of this SPP model, that the model's performance was consistent with similar models (Hoogendoorn & Bovy, 2003) and that intoxication effected the emergence of these adaptive behaviors. Subsequent simulations were developed to empirically assess these observations. Laminar flow serves to optimize flow rate as density increases. It therefore follows that if drunkenness affects the emergence of laminar flow then velocity should be reduced compared to sober conditions and therefore suggests a method to quantitatively test

Table 1 Mean velocity (and standard deviation) for simulations with varying proportions of staggering particles and for congested and non-congested conditions Percent drunk

No congestion

Congestion

Percent baseline

0% 20% 40% 60% 80% 100%

7.26 (0.12) 6.94 (0.25) 6.50 (0.36) 6.26 (0.35) 5.87 (0.29) 5.50 (0.21)

5.94 5.42 4.97 4.63 4.05 3.69

82% 78% 76% 74% 69% 67%

Percent baseline expresses congested velocities as a percentage of non-congested velocities.

(0.28) (0.38) (0.49) (0.45) (0.46) (0.44)

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the effect of drunkenness on crowd dynamics. Forty simulations were run for each of two (congested and not congested) by six (proportion of staggering particles from 0% to 100% in 20% intervals) conditions (see Table 1). In the non-congested condition three groups of particles all traveled in the same direction in the simulation environment unopposed. Each particle's velocity towards its goal was usually at some angle to the optimal trajectory, a line describing the shortest distance between the particle and goal, due to obstructions. The value of the projection of velocity onto the optimal trajectory gave the speed, s, at which the particle is moving towards its goal. The square-mean-root (SMR) of all particles at time t was calculated 0nparticles 1 pffiffiffiffiffiffiffiffiffi 2 si ðt ÞC B ∑ B C savg ðt Þ ¼ B i¼1 C : @ nparticles A

ð8Þ

SMR was preferred to root-mean-square (RMS) as the former places greater weight on smaller values where greater differences were anticipated. Because, in Eq. (8), s b 0 is feasible the SMR was modified to account for negative values, 0nparticles 1 pffiffiffiffiffiffiffiffiffiffiffiffi 2 B ∑ sgnðsi ðt ÞÞ jsi ðt ÞjC B C savg ðt Þ ¼ B i¼1 C @ A nparticles

ð9Þ

Table 1 presents the means and standard deviations of velocity for the forty simulations run in each condition. In both congestion and non-congestion conditions particle velocity reduced as the proportion of drunk particles increased. A two (congested, not congested) by six (percent drunk) analysis of variance test yielded significant main effects of both conditions on velocity (percent drunk: F(5, 468) = 350.34, p b 0.0001; congestion: F(1, 468) = 2429.70, p b 0.0001) and a significant interaction effect (F(5, 468) = 5.77, p b 0.0001). Critically, the relative speed in the congested condition, as a percentage of baseline, decreased as the proportion of drunk particles increased. Our hypothesis was that that there should be an interaction between drunkenness and congestion and therefore on velocity: drunkenness affects the emergence of laminar flow and therefore particle velocity and this effect is in addition to the effect of drunkenness, seen in the non-congested simulations, on velocity. To confirm this interaction empirically, the proportion of drunken particles was regressed onto velocity for both the congested and non-congested conditions. If there is no interaction with congestion on velocity then the linear relationship between the proportion of drunken particles and velocity would be similar for both conditions. In both regression models the proportion of drunken particles significantly predicted velocity (congested: R2 = 0.77, F(1, 238) = 793.01, p b 0.0001; non-congested: R2 = 0.82, F(1, 238) = 1118.94, p b 0.0001). The coefficients on drunkenness for the congested condition (β = −2.24, t = −28.16, p b 0.0001) was greater than in the non-congested condition (β = −1.75, t = −33.45, p b 0.0001). Using a Wald test to test the equality of regression coefficients for both models yielded a significant effect (χ2 = 35.97, p b 0.0001) meaning we could reject the null that there was no additional effect of congestion on velocity and consistent with the hypothesis that emergent laminar flow was disrupted by drunkenness. 7. Conclusion The purpose of the analysis presented here was to examine the effect of alcohol on emergent phenomena in an SPP model of crowd dynamics. Alcohol is known to effect behavior but the impact of individual variation on emergent phenomena had not been previously considered in SPP models. It was further hypothesized that perturbations in individual interactions elicited by intoxication might not only affect the emergence of adaptive temporary coalitions and affiliative behaviors which facilitate shared goals, i.e., laminar flow, they might also explain why violence is associated with crowds in and around licensed premises. Empirical evidence had suggested an additional effect of crowding on rates of violence which were inconsistent with purely individualistic accounts. We, therefore, reasoned that perturbations in emergent phenomena might contribute towards individuals' levels of aggression by interfering with goal directed behavior. This paper, therefore, sought to unify disparate research concerning the relationship between alcohol, crowding, and violence. As previous SPP models had been successfully used to reduce harm in sober crowds we further reasoned that SPP models of crowds in the NTE might be similarly used to inform policies relating to the management of areas around licensed premises. Simulations were consistent with the hypothesis that crowding effects crowd dynamics in a manner which is associated with eliciting anger. The SPP model developed here has value to practitioners involved with managing crowded areas, in particular areas where intoxication may affect crowd dynamics. It would be possible to model a specific NTE with particles entering and exiting points akin to people entering and exiting bars and seeking other resources such as food outlets. Crowd flows under different conditions could then be assessed and the implication of changes to that environment, such as changing a licensed premise's opening hours, assessed. NTEs are diverse and developing local interventions specific to each NTE requires considerable planning, expense and should be evidence based. Predictive models derived from SPP models of NTE crowd dynamics may offer a methodology through which interventions in the NTE can be assessed before committing resources. While the model presented here offers opportunities to better understand important features of crowd dynamics in the NTE the modeling process should be regarded as an iterative process with the model presented above a first stage. There are two strands of work which would refine methods. First, the design of the model was informed by other researchers' work in this area but with

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sober crowds, non-human organisms and theoretical investigation. These are all very different from the crowds that gather in and around licensed premises. For these reasons further model development would require behavioral data from drunken crowds in the NTE to both compare model predictions at the individual level with actual observed interactions between individuals in a crowd and the models predicted emergent features under varying conditions (e.g., density). In particular, data from studies that examined how variations in personality affected crowd dynamics should be included. Second, further attention should be given to measuring emergent phenomena. It is likely that pattern recognition algorithms could be used with both images of actual crowd behavior and model outputs to properly quantify the nature and durability of such coalitions. 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