Pedestrian, Crowd, and Evacuation Dynamics
PPedestrian, Crowd, and Evacuation Dynamics
Dirk Helbing , and Anders Johansson Dresden University of Technology, Andreas-Schubert-Str. 23, 01062 Dresden,Germany Collegium Budapest – Institute for Advanced Study, Szenth´aroms´ag utca 2, 1014Budapest, Hungary
This contribution describes efforts to model the behavior of individual pedes-trians and their interactions in crowds, which generate certain kinds of self-organized patterns of motion. Moreover, this article focusses on the dynamics ofcrowds in panic or evacuation situations, methods to optimize building designsfor egress, and factors potentially causing the breakdown of orderly motion.
Collective Intelligence:
Emergent functional behavior of a large number ofpeople that results from interactions of individuals rather than from individualreasoning or global optimization.
Crowd:
Agglomeration of many people in the same area at the same time.The density of the crowd is assumed to be high enough to cause continuousinteractions with or reactions to other individuals.
Crowd Turbulence:
Unanticipated and unintended irregular motion of indi-viduals into different directions due to strong and rapidly changing forces incrowds of extreme density.
Emergence:
Spontaneous establishment of a qualitatively new behavior throughnon-linear interactions of many objects or subjects.
Evolutionary Optimization:
Gradual optimization based on the effect of fre-quently repeated random mutations and selection processes based on some suc-cess function (“fitness”).
Faster-is-Slower Effect:
This term reflects the observation that certain pro-cesses (in evacuation situations, production, traffic dynamics, or logistics) takemore time if performed at high speed. In other words, waiting can often help tocoordinate the activities of several competing units and to speed up the averageprogress.
Freezing-by-Heating Effect:
Noise-induced blockage effect caused by the break-down of direction-segregated walking patterns (typically two or more “lanes”characterized by a uniform direction of motion). “Noise” means frequent vari-ations of the walking direction due to nervousness or impatience in the crowd,e.g. also frequent overtaking maneuvers in dense, slowly moving crowds. a r X i v : . [ phy s i c s . s o c - ph ] S e p Panic:
Breakdown of ordered, cooperative behavior of individuals due to anx-ious reactions to a certain event. Often, panic is characterized by attemptedescape of many individuals from a real or perceived threat in situations of immi-nent danger, which may end up in trampling or crushing of people in a crowd.Definitions of the term “panic” are controversial and depend on the disciplineor community of people using it.
Self-Organization:
Spontaneous organization (i.e. formation of ordered pat-terns) not induced by initial or boundary conditions, by regulations or con-straints. Self-organization is a result of non-linear interactions between manyobjects or subjects, and it often causes different kinds of spatio-temporal pat-terns of motion.
Social Force:
Vector describing acceleration or deceleration effects that arecaused by social interactions rather than by physical interactions or fields.
The modeling of pedestrian motion is of great theoretical and practical interest.Recent experimental efforts have revealed quantitative details of pedestrian inter-actions, which have been successfully cast into mathematical equations. Further-more, corresponding computer simulations of large numbers of pedestrians havebeen compared with the empirically observed dynamics of crowds. Such studieshave led to a deeper understanding of how collective behavior on a macroscopicscale emerges from individual human interactions. Interestingly enough, thenon-linear interactions of pedestrians lead to various complex, spatio-temporalpattern-formation phenomena. This includes the emergence of lanes of uniformwalking direction, oscillations of the pedestrian flow at bottlenecks, and the for-mation of stripes in two intersecting flows. Such self-organized patterns of motiondemonstrate that an efficient, “intelligent” collective dynamics can be based onsimple, local interactions. Under extreme conditions, however, coordination maybreak down, giving rise to critical crowd conditions. Examples are “freezing-by-heating” and “faster-is-slower” effects, but also the transition to “turbulent”crowd dynamics. These observations have important implications for the opti-mization of pedestrian facilities, in particular for evacuation situations.
The emergence of new, functional or complex collective behaviors in social sys-tems has fascinated many scientists. One of the primary questions in this fieldis how cooperation or coordination patterns originate based on elementary indi-vidual interactions. While one could think that these are a result of intelligenthuman actions, it turns out that much simpler models assuming automatic re-sponses can reproduce the observations very well. This suggests that humans areusing their intelligence primarily for more complicated tasks, but also that simpleinteractions can lead to intelligent patterns of motion. Of course, it is reasonableto assume that these interactions are the result of a previous learning process that has optimized the automatic response in terms of minimizing collisions anddelays. This, however, seems to be sufficient to explain most observations.Note, however, that research into pedestrian and crowd behavior is highlymulti-disciplinary. It involves activities of traffic scientists, psychologists, soci-ologists, biologists, physicists, computer scientists, and others. Therefore, it isnot surprising that there are sometimes different or even controversial views onthe subject, e.g. with regard to the concept of “panic”, the explanation of collec-tive, spatio-temporal patterns of motion in pedestrian crowds, the best modelingconcept, or the optimal number of parameters of a model.In this contribution, we will start with a short history of pedestrian mod-eling and, then, introduce the wide-spread “social force model” of pedestrianinteractions to illustrate further issues such as, for example, model calibrationby video tracking data. Next, we will turn to the subject of crowd dynamics,since one typically finds the formation of large-scale spatio-temporal patterns ofmotion, when many pedestrians interact with each other. These patterns will bediscussed in some detail before we will turn to evacuation situations and casesof extreme densities, where one can sometimes observe the breakdown of co-ordination. Finally, we will address possibilities to design improved pedestrianfacilities, using special evolutionary algorithms.
Pedestrians have been empirically studied for more than four decades [1,2,3].The evaluation methods initially applied were based on direct observation, pho-tographs, and time-lapse films. For a long time, the main goal of these studieswas to develop a level-of-service concept [4], design elements of pedestrian facil-ities [5,6,7,8], or planning guidelines [9,10]. The latter have usually the form of regression relations , which are, however, not very well suited for the prediction ofpedestrian flows in pedestrian zones and buildings with an exceptional architec-ture, or in challenging evacuation situations. Therefore, a number of simulationmodels have been proposed, e.g. queueing models [11], transition matrix mod-els [12], and stochastic models [13], which are partly related to each other. Inaddition, there are models for the route choice behavior of pedestrians [14,15].None of these concepts adequately takes into account the self-organizationeffects occuring in pedestrian crowds. These are the subject of recent experi-mental studies [8,16,17,18,19,20]. Most pedestrian models, however, were formu-lated before. A first modeling approach that appears to be suited to reproducespatio-temporal patterns of motion was proposed by Henderson [21], who con-jectured that pedestrian crowds behave similar to gases or fluids (see also [22]).This could be partially confirmed, but a realistic gas-kinetic or fluid-dynamictheory for pedestrians must contain corrections due to their particular interac-tions (i.e. avoidance and deceleration maneuvers) which, of course, do not obeymomentum and energy conservation. Although such a theory can be actually formulated [23,24], for practical applications a direct simulation of individual pedestrian motion is favourable, since this is more flexible. As a consequence,pedestrian research mainly focusses on agent-based models of pedestrian crowds,which also allow one to consider local coordination problems. The “social forcemodel” [25,26] is maybe the most well-known of these models, but we also liketo mention cellular automata of pedestrian dynamics [27,28,29,30,31,32,33] and
AI-based models [34,35].
In the following, we shall shortly introduce the social force concept, which repro-duces most empirical observations in a simple and natural way. Human behavioroften seems to be “chaotic”, irregular, and unpredictable. So, why and underwhat conditions can we model it by means of forces? First of all, we need tobe confronted with a phenomenon of motion in some (quasi-)continuous space,which may be also an abstract behavioral space such as an opinion scale [36].Moreover, it is favourable to have a system where the fluctuations due to un-known influences are not large compared to the systematic, deterministic partof motion. This is usually the case in pedestrian traffic, where people are con-fronted with standard situations and react “automatically” rather than takingcomplicated decisions, e.g. if they have to evade others.This “automatic” behavior can be interpreted as the result of a learningprocess based on trial and error [37], which can be simulated with evolution-ary algorithms [38]. For example, pedestrians have a preferred side of walking,since an asymmetrical avoidance behavior turns out to be profitable [25,37].The related formation of a behavioral convention can be described by means of evolutionary game theory [25,39].Another requirement is the vectorial additivity of the separate force termsreflecting different environmental influences. This is probably an approximation,but there is some experimental evidence for it. Based on quantitative measure-ments for animals and test persons subject to separately or simultaneously ap-plied stimuli of different nature and strength, one could show that the behaviorin conflict situations can be described by a superposition of forces [40,41]. Thisfits well into a concept by Lewin [42], according to which behavioral changes areguided by so-called social fields or social forces , which has later on been put intomathematical terms [25,43]. In some cases, social forces, which determine theamount and direction of systematic behavioral changes, can be expressed as gra-dients of dynamically varying potentials, which reflect the social or behavioralfields resulting from the interactions of individuals. Such a social force conceptwas applied to opinion formation and migration [43], and it was particularlysuccessful in the description of collective pedestrian behavior [8,25,26,37].For reliable simulations of pedestrian crowds, we do not need to know whethera certain pedestrian, say, turns to the right at the next intersection. It is suffi-cient to have a good estimate what percentage of pedestrians turns to the right.This can be either empirically measured or estimated by means of route choicemodels [14]. In some sense, the uncertainty about the individual behaviors is averaged out at the macroscopic level of description. Nevertheless, we will usethe more flexible microscopic simulation approach based on the social force con-cept. According to this, the temporal change of the location r α ( t ) of pedestrian α obeys the equation d r α ( t ) dt = v α ( t ) . (1)Moreover, if f α ( t ) denotes the sum of social forces influencing pedestrian α andif ξ α ( t ) are individual fluctuations reflecting unsystematic behavioral variations,the velocity changes are given by the acceleration equation d v α dt = f α ( t ) + ξ α ( t ) . (2)A particular advantage of this approach is that we can take into account the flex-ible usage of space by pedestrians, requiring a continuous treatment of motion.It turns out that this point is essential to reproduce the empirical observations ina natural and robust way, i.e. without having to adjust the model to each singlesituation and measurement site. Furthermore, it is interesting to note that, if thefluctuation term is neglected, the social force model can be interpreted as a par-ticular differential game , i.e. its dynamics can be derived from the minimizationof a special utility function [44]. The social force model for pedestrians assumes that each individual α is tryingto move in a desired direction e α with a desired speed v α , and that it adaptsthe actual velocity v α to the desired one, v α = v α e α , within a certain relaxationtime τ α . The systematic part f α ( t ) of the acceleration force of pedestrian α isthen given by f α ( t ) = 1 τ α ( v α e α − v α ) + (cid:88) β ( (cid:54) = α ) f αβ ( t ) + (cid:88) i f αi ( t ) , (3)where the terms f αβ ( t ) and f αi ( t ) denote the repulsive forces describing at-tempts to keep a certain safety distance to other pedestrians β and obstacles i . In very crowded situations, additional physical contact forces come into play(see Sec. 7.3). Further forces may be added to reflect attraction effects betweenmembers of a group or other influences. For details see Ref. [37].First, we will assume a simplified interaction force of the form f αβ ( t ) = f (cid:0) d αβ ( t ) (cid:1) , (4)where d αβ = r α − r β is the distance vector pointing from pedestrian β to α . Angular-dependent shielding effects may be furthermore taken into accountby a prefactor describing the anisotropic reaction to situations in front of as compared to behind a pedestrian [26,45], see Sec. 5.4. However, we will startwith a circular specification of the distance-dependent interaction force, f ( d αβ ) = A α e − d αβ /B α d αβ (cid:107) d αβ (cid:107) , (5)where d αβ = (cid:107) d αβ (cid:107) is the distance. The parameter A α reflects the interactionstrength , and B α corresponds to the interaction range . While the dependence on α explicitly allows for a dependence of these parameters on the single individual,we will assume a homogeneous population, i.e. A α = A and B α = B in thefollowing. Otherwise, it would be hard to collect enough data for parametercalibration. Elliptical specification:
Note that it is possible to express Eq. (5) as gra-dient of an exponentially decaying potential V αβ . This circumstance can be usedto formulate a generalized, elliptical interaction force via the potential V αβ ( b αβ ) = AB e − b αβ /B , (6)where the variable b αβ denotes the semi-minor axis b αβ of the elliptical equipo-tential lines. This has been specified according to2 b αβ = (cid:113) ( (cid:107) d αβ (cid:107) + (cid:107) d αβ − ( v β − v α ) ∆t (cid:107) ) − (cid:107) ( v β − v α ) ∆t (cid:107) , (7)so that both pedestrians α and β are treated symmetrically. The repulsive forceis related to the above potential via f αβ ( d αβ ) = − ∇ d αβ V αβ ( b αβ ) = − dV αβ ( b αβ ) db αβ ∇ d αβ b αβ ( d αβ ) , (8)where ∇ d αβ represents the gradient with respect to d αβ . Considering the chainrule, (cid:107) z (cid:107) = √ z , and ∇ z (cid:107) z (cid:107) = z / √ z = z / (cid:107) z (cid:107) , this leads to the explicitformula f αβ ( d αβ ) = A e − b αβ /B · (cid:107) d αβ (cid:107) + (cid:107) d αβ − y αβ (cid:107) b αβ · (cid:18) d αβ (cid:107) d αβ (cid:107) + d αβ − y αβ (cid:107) d αβ − y αβ (cid:107) (cid:19) (9)with y αβ = ( v β − v α ) ∆t . We used ∆t = 0 . ∆t = 0, we regain theexpression of Eq. (5).The elliptical specification has two major advantages compared to the circu-lar one: First, the interactions depend not only on the distance, but also on therelative velocity. Second, the repulsive force is not strictly directed from pedes-trian β to pedestrian α , but has a lateral component. As a consequence, thisleads to less confrontative, smoother (“sliding”) evading maneuvers. Note thatfurther velocity-dependent specifications of pedestrian interaction forces havebeen proposed [7,26], but we will restrict to the above specifications, as theseare sufficient to demonstrate the method of evolutionary model calibration. Forsuggested improvements regarding the specification of social forces see, for ex-ample, Refs. [46,47]. In reality, of course, pedestrian interactions are not isotropic, but dependent onthe angle ϕ αβ of the encounter, which is given by the formulacos( ϕ αβ ) = v α (cid:107) v α (cid:107) · − d αβ (cid:107) d αβ (cid:107) . (10)Generally, pedestrians show little response to pedestrians behind them. This canbe reflected by an angular-dependent prefactor w ( ϕ αβ ) of the interaction force[45]. Empirical results are represented in Fig. 2 (right). Reasonable results areobtained for the following specification of the prefactor: w (cid:0) ϕ αβ ( t ) (cid:1) = (cid:18) λ α + (1 − λ α ) 1 + cos( ϕ αβ )2 (cid:19) , (11)where λ α with 0 ≤ λ α ≤ λ ≈ . For parameter calibration, several video recordings of pedestrian crowds in dif-ferent natural environments have been used. The dimensions of the recordedareas were known, and the floor tiling or environment provided something like a“coordinate system”. The heads were automatically determined by seaching forround moving structures, and the accuracy of tracking was improved by compar-ing actual with linearly extrapolated positions (so it would not happen so easilythat the algorithm interchanged or “lost” closeby pedestrians). The trajectoriesof the heads were then projected on two-dimensional space in a way correctingfor distortion by the camera perspective. A representative plot of the resultingtrajectories is shown in Fig. 1. Note that trajectory data have been obtainedwith infra-red sensors [48] or video cameras [49,50] for several years now, butalgorithms that can simultaneously handle more than one thousand pedestrianshave become available only recently [51].For model calibration, it is recommended to use a hybrid method fusing em-pirical trajectory data and microscopic simulation data of pedestrian movementin space. In corresponding algorithms, a virtual pedestrian is assigned to eachtracked pedestrian in the simulation domain. One then starts a simulation for atime period T (e.g. 1.5 seconds), in which one pedestrian α is moved accordingto a simulation of the social force model, while the others are moved exactlyaccording to the trajectories extracted from the videos. This procedure is per-formed for all pedestrians α and for several different starting times t , using afixed parameter set for the social force model.Each simulation run is performed according to the following scheme: Fig. 1.
Video tracking used to extract the trajectories of pedestrians from videorecordings close to two escalators (after [45]). Left: Illustration of the trackingof pedestrian heads. Right: Resulting trajectories after being transformed ontothe two-dimensional plane.1. Define a starting point and calculate the state (position r α , velocity v α , andacceleration a α = d v α /dt ) for each pedestrian α .2. Assign a desired speed v α to each pedestrian, e.g. the maximum speed dur-ing the pedestrian tracking time. This is sufficiently accurate, if the overallpedestrian density is not too high and the desired speed is constant in time.3. Assign a desired goal point for each pedestrian, e.g. the end point of thetrajectory.4. Given the tracked motion of the surrounding pedestrians β , simulate thetrajectory of pedestrian α over a time period T based on the social forcemodel, starting at the actual location r α ( t ).After each simulation run, one determines the relative distance error (cid:107) r simulated α ( t + T ) − r tracked α ( t + T ) (cid:107)(cid:107) r tracked α ( t + T ) − r tracked α ( t ) (cid:107) . (12)After averaging the relative distance errors over the pedestrians α and startingtimes t , 1 minus the result can be taken as measure of the goodness of fit (the“fitness”) of the parameter set used in the pedestrian simulation. Hence, the bestpossible value of the “fitness” is 1, but any deviation from the real pedestriantrajectories implies lower values.One result of such a parameter optimization is that, for each video, there is abroad range of parameter combinations of A and B which perform almost equallywell [45]. This allows one to apply additional goal functions in the parameteroptimization, e.g. to determine among the best performing parameter values suchparameter combinations, which perform well for several video recordings, usinga fitness function which equally weights the fitness reached in each single video.This is how the parameter values listed in Table 1 were determined. It turns outthat, in order to reach a good model performance, the pedestrian interactionforce must be specified velocity dependent, as in the elliptical model. Model A [m/s ] B [m] λ FitnessExtrapolation 0 – – 0.34Circular 0.42 ± ± ± ± ± ± Table 1.
Interaction strength A and interaction range B resulting from ourevolutionary parameter calibration for the circular and elliptical specificationof the interaction forces between pedestrians (see main text), with an assumedangular dependence according to Eq. (11). A comparison with the extrapolationscenario, which assumes constant speeds, allows one to judge the improvement inthe goodness of fit (“fitness”) by the specified interaction force. The calibrationwas based on three different video recordings, one for low crowd density, onefor medium, and one for high density (see Ref. [45] for details). The parametervalues are specified as mean value ± standard deviation. The best fitness valueobtained with the elliptical specification for the video with the lowest crowddensity was as high as 0.9.Note that our evolutionary fitting method can be also used to determineinteraction laws without prespecified interaction functions. For example, onecan obtain the distance dependence of pedestrian interactions without a pre-specified function. For this, one adjusts the values of the force at given distances d k = kd (with k ∈ { , , , ... } ) in an evolutionary way. To get some smoothness,linear interpolation is applied. The resulting fit curve is presented in Fig. 2 (left).It turns out that the empirical dependence of the force with distance can be wellfitted by an exponential decay. When the density is low, pedestrians can move freely, and the observed crowddynamics can be partially compared with the behavior of gases. At medium andhigh densities, however, the motion of pedestrian crowds shows some strikinganalogies with the motion of fluids:1. Footprints of pedestrians in snow look similar to streamlines of fluids [15].2. At borderlines between opposite directions of walking one can observe “vis-cous fingering” [52,53].3. The emergence of pedestrian streams through standing crowds [7,37,54] ap-pears analogous to the formation of river beds [55,56].At high densities, however, the observations have rather analogies with drivengranular flows. This will be elaborated in more detail in Secs. 7.3 and 7.4. In sum-mary, one could say that fluid-dynamic analogies work reasonably well in normalsituations, while granular aspects dominate at extreme densities. Nevertheless, F o r c e −0.2 0 0.2 0.4 0.6 0.8 1 1.2−1−0.8−0.6−0.4−0.200.20.40.60.81 x (m) y ( m ) Fig. 2.
Results of an evoluationary fitting of pedestrian interactions. Left: Em-pirically determined distance dependence of the interaction force between pedes-trians. An exponential decay fits the empirical data quite well. The dashed fitcurve corresponds to Eq. (5) with the parameters A = 0 .
53 and B = 1 .
0. Right:Angular dependence of the influence of other pedestrians. The direction alongthe positive x axis corresponds to the walking direction of pedestrians, y to theperpendicular direction. (After [45].)the analogy is limited, since the self-driven motion and the violation of momen-tum conservation imply special properties of pedestrian flows. For example, oneusually does not observe eddies. Despite its simplifications, the social force model of pedestrian dynamics de-scribes a lot of observed phenomena quite realistically. Especially, it allows oneto explain various self-organized spatio-temporal patterns that are not externallyplanned, prescribed, or organized, e.g. by traffic signs, laws, or behavioral con-ventions [7,8,37]. Instead, the spatio-temporal patterns discussed below emergedue to the non-linear interactions of pedestrians even without assuming strategi-cal considerations, communication, or imitative behavior of pedestrians. Despitethis, we may still interpret the forming cooperation patterns as phenomena thatestablish social order on short time scales. It is actually surprising that strangerscoordinate each other within seconds, if they have grown up in a similar envi-roment. People from different countries, however, are sometimes irritated aboutlocal walking habits, which indicates that learning effects and cultural back-grounds still play a role in social interactions as simple as random pedestrianencounters. Rather than on particular features, however, in the following we willfocus on the common, internationally reproducible observations.
Lane formation:
In pedestrian flows one can often observe that oppositelymoving pedestrians are forming lanes of uniform walking direction (see Fig. 3)[8,20,25,26]. This phenomenon even occurs when there is not a large distance to N u m b e r o f P e r s on s Passing Time (s)Flow 1Flow 2
Fig. 3.
Self-organization of pedestrian crowds. Left: Photograph of lanes formedin a shopping center. Computer simulations reproduce the self-organization ofsuch lanes very well. Top right: Evaluation of the cumulative number of pedes-trians passing a bottleneck from different sides. One can clearly see that thenarrowing is often passed by groups of people in an oscillatory way rather thanone by one. Bottom right: Multi-agent simulation of two crossing pedestrianstreams, showing the phenomenon of stripe formation. This self-organized pat-tern allows pedestrians to pass the other stream without having to stop, namelyby moving sidewards in a forwardly moving stripe. (After Ref. [8].)separate each other, e.g. on zebra crossings. However, the width of lanes increases(and their number decreases), if the interaction continues over longer distances(and if perturbations, e.g. by flows entering or leaving on the sides, are low;otherwise the phenomenon of lane formation may break down [57]).Lane formation may be viewed as segregation phenomenon [58,59]. Althoughthere is a weak preference for one side (with the corresponding behavioral con-vention depending on the country), the observations can only be well reproducedwhen repulsive pedestrian interactions are taken into account. The most relevantfactor for the lane formation phenomenon is the higher relative velocity of pedes-trians walking in opposite directions. Compared to people following each other,oppositely moving pedestrians have more frequent interactions until they havesegregated into separate lanes by stepping aside whenever another pedestrian isencountered. The most long-lived patterns of motion are the ones which changethe least. It is obvious that such patterns correspond to lanes, as they minimizethe frequency and strength of avoidance maneuvers. Interestingly enough, as computer simulations show, lane formation occurs also when there is no prefer-ence for any side.Lanes minimize frictional effects, accelerations, energy consumption, and de-lays in oppositely moving crowds. Therefore, one could say that they are apattern reflecting “collective intelligence”. In fact, it is not possible for a sin-gle pedestrian to reach such a collective pattern of motion. Lane formation is aself-organized collaborative pattern of motion originating from simple pedestrianinteractions. Particularly in cases of no side preference, the system behavior can-not be understood by adding up the behavior of the single individuals. This isa typical feature of complex, self-organizing systems and, in fact, a wide-spreadcharacteristics of social systems. It is worth noting, however, that it does notrequire a conscious behavior to reach forms of social organization like the seg-regation of oppositely moving pedestrians into lanes. This organization occursautomatically, although most people are not even aware of the existence of thisphenomenon. Oscillatory flows at bottlenecks:
At bottlenecks, bidirectional flows of mod-erate density are often characterized by oscillatory changes in the flow direction(see Fig. 3) [8,26]. For example, one can sometimes observe this at entrancesof museums during crowded art exhibitions or at entrances of staff canteensduring lunch time. While these oscillatory flows may be interpreted as an ef-fect of friendly behavior (“you go first, please”), computer simulations of thesocial force model indicate that the collective behavior may again be under-stood by simple pedestrian interactions. That is, oscillatory flows can even occurin the absence of communication, although it may be involved in reality. Theinteraction-based mechanism of oscillatory flows suggests to interpret them asanother self-organization phenomenon, which again reduces frictional effects anddelays. That is, oscillatory flows have features of “collective intelligence”.While this may be interpreted as result of a learning effect in a large num-ber of similar situations (a “repeated game”), our simulations suggest an evensimpler, “many-particle” interpretation: Once a pedestrian is able to pass thenarrowing, pedestrians with the same walking direction can easily follow. Hence,the number and “pressure” of waiting, “pushy” pedestrians on one side of thebottleneck becomes less than on the other side. This eventually increases theirchance to occupy the passage. Finally, the “pressure difference” is large enoughto stop the flow and turn the passing direction at the bottleneck. This reversesthe situation, and eventually the flow direction changes again, giving rise tooscillatory flows.At bottlenecks, further interesting observations can be made: Hoogendoornand Daamen [60] report the formation of layers in unidirectional bottleneckflows. Due to the partial overlap of neighboring layers, there is a zipper effect.Moreover, Kretz et al. [61] have observed that the specific flow through a narrowbottleneck decreases with a growing width of the bottleneck, as long as it canbe passed by one person at a time only. This is due to mutual obstructions, iftwo people are trying to enter the bottleneck simultaneously. If the opening is large enough to be entered by several people in parallel, the specific flow staysconstant with increasing width. Space is then used in a flexible way. Stripe formation in intersecting flows:
In intersection areas, the flow ofpeople often appears to be irregular or “chaotic”. In fact, it can be shown thatthere are several possible collective patterns of motion, among them rotary andoscillating flows. However, these patterns continuously compete with each other,and a temporarily dominating pattern is destroyed by another one after a shorttime. Obviously, there has not evolved any social convention that would establishand stabilize an ordered and efficient flow at intersections.Self-organized patterns of motion, however, are found in situations wherepedestrian flows cross each other only in two directions. In such situations, thephenomenon of stripe formation is observed [62]. Stripe formation allows twoflows to penetrate each other without requiring the pedestrians to stop. For anillustration see Fig. 3. Like lanes, stripes are a segregation phenomenon, but not astationary one. Instead, the stripes are density waves moving into the direction ofthe sum of the directional vectors of both intersecting flows. Naturally, the stripesextend sidewards into the direction which is perpendicular to their direction ofmotion. Therefore, the pedestrians move forward with the stripes and sidewards within the stripes. Lane formation corresponds to the particular case of stripeformation where both directions are exactly opposite. In this case, no intersectiontakes place, and the stripes do not move systematically. As in lane formation,stripe formation allows to minimize obstructing interactions and to maximizethe average pedestrian speeds, i.e. simple, repulsive pedestrian interactions againlead to an “intelligent” collective behavior.
While the previous section has focussed on the dynamics of pedestrian crowdsin normal situations, we will now turn to the description of situations in whichextreme crowd densities occur. Such situations may arise at mass events, partic-ularly in cases of urgent egress. While most evacuations run relatively smoothlyand orderly, the situation may also get out of control and end up in terriblecrowd disasters (see Tab. 2). In such situations, one often speaks of “panic”,although, from a scientific standpoint, the use of this term is rather controver-sial. Here, however, we will not be interested in the question whether “panic”actually occurs or not. We will rather focus on the issue of crowd dynamics athigh densities and under psychological stress.
Computer models have been also developed for emergency and evacuation situa-tions [32,63,64,65,66,67,68,69,70,71]. Most research into panic, however, has beenof empirical nature (see, e.g. Refs. [72,73,74]), carried out by social psychologistsand others. With some exceptions, panic is thought to occur in cases of scarce or dwin-dling resources [75,76], which are either required for survival or anxiously desired.They are usually distinguished into escape panic (“stampedes”, bank or stockmarket panic) and acquisitive panic (“crazes”, speculative manias) [77,78], butin some cases this classification is questionable [79].It is often stated that panicking people are obsessed by short-term personalinterests uncontrolled by social and cultural constraints [76,77]. This is possiblya result of the reduced attention in situations of fear [76], which also causesthat options like side exits are mostly ignored [72]. It is, however, mostly at-tributed to social contagion [73,75,76,77,78,79,80,81,82,83,84], i.e., a transitionfrom individual to mass psychology, in which individuals transfer control overtheir actions to others [78], leading to conformity [85]. This “herding behavior”is in some sense irrational, as it often leads to bad overall results like dangerousovercrowding and slower escape [72,78,79]. In this way, herding behavior canincrease the fatalities or, more generally, the damage in the crisis faced.The various socio-psychological theories for this contagion assume hypnoticeffects, rapport, mutual excitation of a primordial instinct, circular reactions, so-cial facilitation (see the summary by Brown [83]), or the emergence of normativesupport for selfish behavior [84]. Brown [83] and Coleman [78] add another expla-nation related to the prisoner’s dilemma [86,87] or common goods dilemma [88],showing that it is reasonable to make one’s subsequent actions contingent uponthose of others. However, the socially favourable behavior of walking orderly isunstable, which normally gives rise to rushing by everyone. These thoughtfulconsiderations are well compatible with many aspects discussed above and withthe classical experiments by Mintz [75], which showed that jamming in escapesituations depends on the reward structure (“payoff matrix”).Nevertheless and despite of the frequent reports in the media and manypublished investigations of crowd disasters (see Table 2), a quantitative under-standing of the observed phenomena in panic stampedes was lacking for a longtime. The following sections will close this gap.
Panic stampede is one of the most tragic collective behaviors [73,74,75,77,78,80,81,82,83,84],as it often leads to the death of people who are either crushed or trampled downby others. While this behavior may be comprehensible in life-threatening situa-tions like fires in crowded buildings [72,76], it is hard to understand in cases of arush for good seats at a pop concert [79] or without any obvious reasons. Unfor-tunately, the frequency of such disasters is increasing (see Table 2), as growingpopulation densities combined with easier transportation lead to greater massevents like pop concerts, sport events, and demonstrations. Nevertheless, system-atic empirical studies of panic [75,90] are rare [76,77,79], and there is a scarcity ofquantitative theories capable of predicting crowd dynamics at extreme densities[32,63,64,67,68,71]. The following features appear to be typical [57,91]:1. In situations of escape panic, individuals are getting nervous, i.e. they tendto develop blind actionism. Table 2.
Incomplete list of major crowd disasters since 1970 after J. F. Dickiein Ref. [89], , http://SportsIllustrated.CNN.com/soccer/world/news/2000/07/09/stadiumdisasters ap/ , and other internet sources, excluding fires, bomb attacks, andtrain or plane accidents. The number of injured people was usually a multipleof the fatalities. Date Place Venue Deaths Reason1971 Ibrox, UK Stadium 66 Collapse of barriers1974 Cairo, Egypt Stadium 48 Crowds break barriers1982 Moscow, USSR Stadium 340 Re-entering fans after last minute goal1988 Katmandu, Nepal Stadium 93 Stampede due to hailstorm1989 Hillsborough, Sheffield,UK Stadium 96 Fans trying to force their way into thestadium1990 New York City Bronx 87 Illegal happy land social club1990 Mena, Saudi Arabia PedestrianTunnel 1426 Overcrowding1994 Mena, Saudi Arabia JamaratBridge 266 Overcrowding1996 Guatemala City,Guatemala Stadium 83 Fans trying to force their way into thestadium1998 Mena, Saudi Arabia 118 Overcrowding1999 Kerala, India Hindu Shrine 51 Collapse of parts of the shrine1999 Minsk, Belarus Subway Sta-tion 53 Heavy rain at rock concert2001 Ghana, West Africa Stadium >
100 Panic triggered by tear gas2004 Mena, Saudi Arabia JamaratBridge 251 Overcrowding2005 Wai, India Religious Pro-cession 150 Overcrowding (and fire)2005 Bagdad, Iraque Religious Pro-cession >
640 Rumors regarding suicide bomber2005 Chennai, India Disaster Area 42 Rush for flood relief supplies2006 Mena, Saudi Arabia JamaratBridge 363 Overcrowding2006 Pilippines Stadium 79 Rush for game show tickets2006 Ibb, Yemen Stadium 51 Rally for Yemeni president6
2. People try to move considerably faster than normal [9].3. Individuals start pushing, and interactions among people become physicalin nature.4. Moving and, in particular, passing of a bottleneck frequently becomes inco-ordinated [75].5. At exits, jams are building up [75]. Sometimes, intermittent flows or archingand clogging are observed [9].6. The physical interactions in jammed crowds add up and can cause danger-ous pressures up to 4,500 Newtons per meter [72,89], which can bend steelbarriers or tear down brick walls.7. The strength and direction of the forces acting in large crowds can suddenlychange [51], pushing people around in an uncontrollable way. This may causepeople to fall.8. Escape is slowed down by fallen or injured people turning into “obstacles”.9. People tend to show herding behavior, i.e., to do what other people do [76,81].10. Alternative exits are often overlooked or not efficiently used in escape situ-ations [72,76].
Additional, physical interaction forces f ph αβ come into play when pedestriansget so close to each other that they have physical contact (i.e. d αβ < r αβ = r α + r β , where r α means the “radius” of pedestrian α ) [91]. In this case, whichis mainly relevant to panic situations, we assume also a “body force” k ( r αβ − d αβ ) n αβ counteracting body compression and a “sliding friction force” κ ( r αβ − d αβ ) ∆v tβα t αβ impeding relative tangential motion. Inspired by the formulas forgranular interactions [92,93], we assume f ph αβ ( t ) = kΘ ( r αβ − d αβ ) n αβ + κΘ ( r αβ − d αβ ) ∆v tβα t αβ , (13)where the function Θ ( z ) is equal to its argument z , if z ≥
0, otherwise 0. More-over, t αβ = ( − n αβ , n αβ ) means the tangential direction and ∆v tβα = ( v β − v α ) · t αβ the tangential velocity difference, while k and κ represent large constants.(Strictly speaking, friction effects already set in before pedestrians touch eachother, because of the psychological tendency not to pass other individuals witha high relative velocity, when the distance is small.)The interactions with the boundaries of walls and other obstacles are treatedanalogously to pedestrian interactions, i.e., if d αi ( t ) means the distance to obsta-cle or boundary i , n αi ( t ) denotes the direction perpendicular to it, and t αi ( t ) thedirection tangential to it, the corresponding interaction force with the boundaryreads f αi = { A α exp[( r α − d αi ) /B α ] + kΘ ( r α − d αi ) } n αi − κΘ ( r α − d αi )( v α · t αi ) t αi . (14)Finally, fire fronts are reflected by repulsive social forces similar those de-scribing walls, but they are much stronger. The physical interactions, however,are qualitatively different, as people reached by the fire front become injuredand immobile ( v α = ). Inspired by the observations discussed in Sec. 7.2, we have simulated situationsof “panic” escape in the computer, assuming the following features:1. People are getting nervous, resulting in a higher level of fluctuations.2. They are trying to escape from the source of panic, which can be reflectedby a significantly higher desired velocity v α .3. Individuals in complex situations, who do not know what is the right thingto do, orient at the actions of their neighbours, i.e. they tend to do whatother people do. We will describe this by an additional herding interaction.We will now discuss the fundamental collective effects which fluctuations, in-creased desired velocities, and herding behavior can have according to simula-tions. Note that, in contrast to other approaches, we do not assume or implythat individuals in panic or emergency situations would behave relentless andasocial, although they sometimes do. Herding and ignorance of available exits:
If people are not sure what isthe best thing to do, there is a tendency to show a “herding behavior”, i.e. toimitate the behavior of others. Fashion, hypes and trends are examples for this.The phenomenon is also known from stock markets, and particularly pronouncedwhen people are anxious. Such a situation is, for example, given if people needto escape from a smoky room. There, the evacuation dynamics is very differentfrom normal leaving (see Fig. 4).Under normal visibility, everybody easily finds an exit and uses more or lessthe shortest path. However, when the exit cannot be seen, evacuation is muchless efficient and may take a long time. Most people tend to walk relativelystraight into the direction in which they suspect an exit, but in most cases, theyend up at a wall. Then, they usually move along it in one of the two possibledirections, until they finally find an exit [18]. If they encounter others, thereis a tendency to take a decision for one direction and move collectively. Alsoin case of accoustic signals, people may be attracted into the same direction.This can lead to over-crowded exits, while other exits are ignored. The same canhappen even for normal visibility, when people are not well familiar with theirenvironment and are not aware of the directions of the emergency exits.Computer simulations suggest that neither individualistic nor herding behav-ior performs well [91]. Pure individualistic behavior means that each pedestrianfinds an exit only accidentally, while pure herding behavior implies that the com-plete crowd is eventually moving into the same and probably congested direc-tion, so that available emergency exits are not efficiently used. Optimal chancesof survival are expected for a certain mixture of individualistic and herding be-havior, where individualism allows some people to detect the exits and herdingguarantees that successful solutions are imitated by small groups of others [91]. Fig. 4.
Left: Normal leaving of a room, when the exit is well visible. Snapshotsof a video-recorded experiment with 10 people after (a) t = 0 seconds (initialcondition), (b) t = 1 sec., (c) t = 3 sec., and (d) t = 5 seconds. The face directionsare indicated by arrows. Right: Escape from a room with no visibility, e.g. due todense smoke or a power blackout. Snapshots of an experiment with test persons,whose eyes were covered by masks, after t = 0 seconds (initial condition), t = 5sec., (c) t = 10 sec., and (d) t = 15 seconds. (After Ref. [18].) “Freezing by heating”: Another effect of getting nervous has been investi-gated in Ref. [57]. Let us assume the individual fluctuation strength, i.e. thestandard deviation of the noise term ξ α , is given by η α = (1 − n α ) η + n α η max , (15)where n α with 0 ≤ n α ≤ α . Theparameter η means the normal and η max the maximum fluctuation strength.It turns out that, at sufficiently high pedestrian densities, lanes are destroyedby increasing the fluctuation strength (which is analogous to the temperature).However, instead of the expected transition from the “fluid” lane state to adisordered, “gaseous” state, a “solid” state is formed. It is characterized by ablocked, “frozen” situation so that one calls this paradoxial transition “freezingby heating” (see Fig. 5). Notably enough, the blocked state has a higher degreeof order, although the internal energy is increased [57]. Fig. 5.
Result of the noise-induced formation of a “frozen” state in a (periodic)corridor used by oppositely moving pedestrians (after Ref. [57]).The preconditions for this unusual freezing-by-heating transition are the driv-ing term v α e α /τ α and the dissipative friction − v α /τ α , while the sliding frictionforce is not required. Inhomogeneities in the channel diameter or other impuri-ties which temporarily slow down pedestrians can further this transition at therespective places. Finally note that a transition from fluid to blocked pedestriancounter flows is also observed, when a critical density is exceeded, as impatientpedestrians enter temporary gaps in the opposite lane to overtake others [31,57].However, in contrast to computer simulations, resulting deadlocks are usuallynot permanent in real crowds, as turning the bodies (shoulders) often allowspedestrians to get out of the blocked area. Intermittent flows, faster-is-slower effect, and “phantom panic”:
Ifthe overall flow towards a bottleneck is higher than the overall outflow fromit, a pedestrian queue emerges [94]. In other words, a waiting crowd is formedupstream of the bottleneck. High densities can result, if people keep headingforward, as this eventually leads to higher and higher compressions. Particularlycritical situations may occur if the arrival flow is much higher than the departureflow, especially if people are trying to get towards a strongly desired goal (“aquis-itive panic”) or away from a perceived source of danger (“escape panic”) with an increased driving force v α e α /τ . In such situations, the high density causescoordination problems, as several people compete for the same few gaps. Thistypically causes body interactions and frictional effects, which can slow downcrowd motion or evacuation (“faster is slower effect”) .A possible consequence of these coordination problems are intermittent flows.In such cases, the outflow from the bottleneck is not constant, but it is typicallyinterrupted. While one possible origin of the intermittent flows are cloggingand arching effects as known from granular flows through funnels or hoppers[92,93], stop-and-go waves have also been observed in more than 10 meter widestreets and in the 44 meters wide entrance area to the Jamarat Bridge duringthe pilgrimage in January 12, 2006 [51], see Fig. 6. Therefore, it seems to beimportant that people do not move continuously, but have minimum strides [25].That is, once a person is stopped, he or she will not move until some space opensup in front. However, increasing impatience will eventually reduce the minimumstride, so that people eventually start moving again, even if the outflow throughthe bottleneck is stopped. This will lead to a further compression of the crowd.In the worst case, such behavior can trigger a “phantom panic” , i.e. a crowddisaster without any serious reasons (e.g., in Moscow, 1982). For example, dueto the “faster-is-slower effect” panic can be triggered by small pedestrian coun-terflows [72], which cause delays to the crowd intending to leave. Consequently,stopped pedestrians in the back, who do not see the reason for the temporaryslowdown, are getting impatient and pushy. In accordance with observations[25,7], one may model this by increasing the desired velocity, for example, bythe formula v α ( t ) = [1 − n α ( t )] v α (0) + n α ( t ) v max α . (16)Herein, v max α is the maximum desired velocity and v α (0) the initial one, corre-sponding to the expected velocity of leaving. The time-dependent parameter n α ( t ) = 1 − v α ( t ) v α (0) (17)reflects the nervousness, where v α ( t ) denotes the average speed into the desireddirection of motion. Altogether, long waiting times increase the desired speed v α or driving force v α ( t ) e α /τ , which can produce high densities and inefficientmotion. This further increases the waiting times, and so on, so that this tragicfeedback can eventually trigger so high pressures that people are crushed orfalling and trampled. It is, therefore, imperative, to have sufficiently wide exitsand to prevent counterflows, when big crowds want to leave [91]. Transition to stop-and-go waves:
Recent empirical studies of pilgrim flowsin the area of Makkah, Saudi Arabia, have shown that intermittent flows occurnot only when bottlenecks are obvious. On January 12, 2006, pronounced stop-and-go waves have been even observed upstream of the 44 meter wide entranceto the Jamarat Bridge [51]. While the pilgrim flows were smooth and continuous(“laminar”) over many hours, at 11:53am stop-and-go waves suddenly appeared Time (h)
Fig. 6.
Top: Long-term photograph showing stop-and-go waves in a denselypacked street. While stopped people appear relatively sharp, people moving fromright to left have a fuzzy appearance. Note that gaps propagate from left to right.Middle: Empirically observed stop-and-go waves in front of the entrance to theJamarat Bridge on January 12, 2006 (after [51]), where pilgrims moved from leftto right. Dark areas correspond to phases of motion, light colors to stop phases.The “location” coordinate represents the distance to the beginning of the nar-rowing, i.e. to the cross section of reduced width. Bottom left: Illustration of the“shell model” (see Ref. [94]), in particular of situations where several pedestrianscompete for the same gap, which causes coordination problems. Bottom right:Simulation results of the shell model. The observed stop-and-go waves result fromthe alternation of forward pedestrian motion and backward gap propagation. and propagated over distances of more than 30 meters (see Fig. 6). The suddentransition was related to a significant drop of the flow, i.e. with the onset ofcongestion [51]. Once the stop-and-go waves set in, they persisted over morethan 20 minutes.This phenomenon can be reproduced by a recent model based on two continu-ity equations, one for forward pedestrian motion and another one for backwardgap propagation [94]. The model was derived from a “shell model” (see Fig. 6)and describes very well the observed alternation between backward gap propa-gation and forward pedestrian motion. Transition to “crowd turbulence”:
On the same day, around 12:19, thedensity reached even higher values and the video recordings showed a suddentransition from stop-and-go waves to irregular flows (see Fig. 7). These irregularflows were characterized by random, unintended displacements into all possibledirections, which pushed people around. With a certain likelihood, this causedthem to stumble. As the people behind were moved by the crowd as well andcould not stop, fallen individuals were trampled, if they did not get back on theirfeet quickly enough. Tragically, the area of trampled people grew more and morein the course of time, as the fallen pilgrims became obstacles for others [51]. Theresult was one of the biggest crowd disasters in the history of pilgrimage.How can we understand this transition to irregular crowd motion? A closerlook at video recordings of the crowd reveals that, at this time, people wereso densely packed that they were moved involuntarily by the crowd. This isreflected by random displacements into all possible directions. To distinguishthese irregular flows from laminar and stop-and-go flows and due to their visualappearance, we will refer to them as “crowd turbulence” .As in certain kinds of fluid flows, “turbulence” in crowds results from asequence of instabilities in the flow pattern. Additionally, one finds a sharplypeaked probability density function of velocity increments V τx = V x ( r , t + τ ) − V x ( r , t ) , (18)which is typical for turbulence [95], if the time shift τ is small enough [51]. Onealso observes a power-law scaling of the displacements indicating self-similarbehaviour [51]. As large eddies are not detected, however, the similarity with fluid turbulence is limited, but there is still an analogy to turbulence at cur-rency exchange markets [95]. Instead of vortex cascades like in turbulent fluids,one rather finds a hierarchical fragmentation dynamics: At extreme densities,individual motion is replaced by mass motion, but there is a stick-slip instabil-ity which leads to “rupture” when the stress in the crowd becomes too large.That is, the mass splits up into clusters of different sizes with strong velocitycorrelations inside and distance-dependent correlations between the clusters.“Crowd turbulence” has further specific features [51]. Due to the physicalcontacts among people in extremely dense crowds, we expect commonalitieswith granular media. In fact, dense driven granular media may form densitywaves, while moving forward [96], and can display turbulent-like states [97,98]. Normalized time P o s i t i on x ( m ) LaminarStop and goTurbulent −0.02 0 0.02 0.04−0.06−0.04−0.0200.02 v x (m/s) v y ( m / s ) Fig. 7.
Pedestrian dynamics at different densities. Left: Representative trajec-tories (space-time plots) of pedestrians during the laminar, stop-and-go, andturbulent flow regime. Each trajectory extends over a range of 8 meters, whilethe time required for this stretch is normalized to 1. To indicate the differentspeeds, symbols are included in the curves every 5 seconds. While the laminarflow (top line) is fast and smooth, motion is temporarily interrupted in stop-and-go flow (medium line), and backward motion can occur in “turbulent” flows(bottom line). Right: Example of the temporal evolution of the velocity compo-nents v x ( t ) into the average direction of motion and v y ( t ) perpendicular to itin “turbulent flow”, which occurs when the crowd density is extreme. One canclearly see the irregular motion into all possible directions characterizing “crowdturbulence”. For details see Ref. [51]. Moreover, under quasi-static conditions [97], force chains [99] are building up,causing strong variations in the strengths and directions of local forces. As inearthquakes [100,101] this can lead to events of sudden, uncontrollable stressrelease with power-law distributed displacements. Such a power-law has alsobeen discovered by video-based crowd analysis [51].
Time (h) P r e ss u r e ( / s ) Start of turbulence Start of accident
Fig. 8.
Left: Snapshot of the on-line visualization of “crowd pressure”. Red colors(see the lower ellipses) indicate areas of critical crowd conditions. In fact, thesad crowd disaster during the Muslim pilgrimage on January 12, 2006, startedin this area. Right: The “crowd pressure” is a quantitative measure of the onsetof “crowd turbulence”. The crowd disaster started when the “crowd pressure”reached particularly high values. For details see Ref. [51].Turbulent waves are experienced in dozens of crowd-intensive events eachyear all over the world [102]. Therefore, it is necessary to understand why, whereand when potentially critical situations occur. Viewing real-time video recordingsis not very suited to identify critical crowd conditions: While the average densityrarely exceeds values of 6 persons per square meter, the local densities can reachalmost twice as large values [51]. It has been found, however, that even evaluatingthe local densities is not enough to identify the critical times and locationsprecisely, which also applies to an analysis of the velocity field [51]. The decisivequantity is rather the “crowd pressure”, i.e. the density, multiplied with thevariance of speeds. It allows one to identify critical locations and times (see Fig.8). There are even advance warning signs of critical crowd conditions: The crowdaccident on January 12, 2006 started about 10 minutes after “turbulent” crowdmotion set in, i.e. after the “pressure” exceeded a value of 0.02/s (see Fig. 8).Moreover, it occured more than 30 minutes after stop-and-go waves set in, whichcan be easily detected in accelerated surveillance videos. Such advance warning signs of critical crowd conditions can be evaluated on-line by an automated videoanalysis system. In many cases, this can help one to gain time for correctivemeasures like flow control, pressure-relief strategies, or the separation of crowdsinto blocks to stop the propagation of shockwaves [51]. Such anticipative crowdcontrol could increase the level of safety during future mass events. Having understood some of the main factors causing crowd disasters, it is inter-esting to ask how pedestrian facilities can be designed in a way that maximizesthe efficiency of pedestrian flows and the level of safety. One of the major goalsduring mass events must be to avoid extreme densities. These often result fromthe onset of congestion at bottlenecks, which is a consequence of the breakdownof free flow and causes an increasing degree of compression. When a certaincritical density is increased (which depends on the size distribution of people),this potentially implies high pressures in the crowd, particularly if people areimpatient due to long delays or panic.The danger of an onset of congestion can be minimized by avoiding bot-tlenecks. Notice, however, that jamming can also occur at widenings of escaperoutes [91]. This surprising fact results from disturbances due to pedestrians,who try to overtake each other and expand in the wider area because of their re-pulsive interactions. These squeeze into the main stream again at the end of thewidening, which acts like a bottleneck and leads to jamming. The correspondingdrop of efficiency E is more pronounced,1. if the corridor is narrow,2. if the pedestrians have different or high desired velocities, and3. if the pedestrian density in the corridor is high.Obviously, the emerging pedestrian flows decisively depend on the geometryof the boundaries. They can be simulated on a computer already in the planningphase of pedestrian facilities. Their configuration and shape can be systemati-cally varied, e.g. by means of evolutionary algorithms [28,103] and evaluated onthe basis of particular mathematical performance measures [7]. Apart from the efficiency E = 1 N (cid:88) α v α · e α v α (19)we can, for example, define the measure of comfort C = (1 − D ) via the discomfort D = 1 N (cid:88) α ( v α − v α ) ( v α ) = 1 N (cid:88) α (cid:32) − v α ( v α ) (cid:33) . (20)The latter is again between 0 and 1 and reflects the frequency and degree ofsudden velocity changes, i.e. the level of discontinuity of walking due to necessaryavoidance maneuvers. Hence, the optimal configuration regarding the pedestrianrequirements is the one with the highest values of efficiency and comfort.During the optimization procedure, some or all of the following can be varied: Fig. 9.
The evolutionary optimization based on Boolean grids uses a two-stagealgorithm (see Ref. [104] for details). Left: In the “randomization stage”, obsta-cles are distributed over the grid with some randomness, thereby allowing forthe generation and testing of new topologies (architectures). Right: In the “ag-glomeration stage”, small nearby obstacles are clustered to form larger objectswith smooth boundaries. After several iterations, the best performing designsare reasonably shaped. See Fig. 10 for examples of possible bottleneck designs.1. the location and form of planned buildings,2. the arrangement of walkways, entrances, exits, staircases, elevators, escala-tors, and corridors,3. the shape of rooms, corridors, entrances, and exits,4. the function and time schedule. (Recreation rooms or restaurants are oftencontinuously frequented, rooms for conferences or special events are mainlyvisited and left at peak periods, exhibition rooms or rooms for festivitiesrequire additional space for people standing around, and some areas areclaimed by queues or through traffic.)In contrast to early evolutionary optimization methods, recent approaches al-low to change not only the dimensions of the different elements of pedestrianfacilities, but also to vary their topology. The procedure of such algorithms isillustrated in Fig. 9. Highly performing designs are illustrated in Fig. 10. It turnsout that, for an emergency evacuation route, it is favorable if the crowd doesnot move completely straight towards a bottleneck. For example, a zigzag de-sign of the evacuation route can reduce the pressure on the crowd upstream ofa bottleneck (see Fig. 11). The proposed evolutionary optimization procedurecan, of course, not only be applied to the design of new pedestrian facilities,but also to a reduction of existing bottlenecks, when suitable modifications areimplemented. Fig. 10.
Two examples of improved designs for cases with a bottleneck along theescape route of a large crowd, obtained with an evolutionary algorithm based onBoolean grids (after Ref. [104]). People were assumed to move from left to rightonly. Left: Funnel-shaped escape route. Right: Zig-zag design.
Fig. 11.
Left: Conventional design of a stadium exit in an emergency scenario,where we assume that some pedestrians have fallen at the end of the downwardsstaircase to the left. The dark color indicates high pressures, since pedestriansare impatient and pushing from behind. Right: In the improved design, theincreasing diameter of corridors can reduce waiting times and impatience (evenwith the same number of seats), thereby accelerating evacuation. Moreover, thezigzag design of the downwards staircases changes the pushing direction in thecrowd. Computer simulations indicate that the zig-zag design can reduce theaverage pressure in the crowd at the location of the incident by a factor of two.(After Ref. [8].) In this contribution, we have presented a multi-agent approach to pedestrian andcrowd dynamics. Despite the great effort required, pedestrian interactions canbe well quantified by video tracking. Compared to other social interactions theyturn out to be quite simple. Nevertheless, they cause a surprisingly large varietyof self-organized patterns and short-lived social phenomena, where coordinationor cooperation emerges spontaneously. For this reason, they are interesting tostudy, particularly as one can expect new insights into coordination mechanismsof social beings beyond the scope of classical game theory. Examples for observedself-organization phenomena in normal situations are lane formation, stripe for-mation, oscillations and intermittent clogging effects at bottlenecks, and theevolution of behavioral conventions (such as the preference of the right-handside in continental Europe). Under extreme conditions (high densities or panic),however, coordination may break down, giving rise to “freezing-by-heating” or“faster-is-slower effects”, stop-and-go waves or “crowd turbulence”.Similar observations as in pedestrian crowds are made in other social systemsand settings. Therefore, we expect that realistic models of pedestrian dynam-ics will also promote the understanding of opinion formation and other kindsof collective behaviors. The hope is that, based on the discovered elementarymechanisms of emergence and self-organization, one can eventually also obtaina better understanding of the constituting principles of more complex social sys-tems. At least the same underlying factors are found in many social systems:non-linear interactions of individuals, time-dependence, heterogeneity, stochas-ticity, competition for scarce resources (here: space and time), decision-making,and learning. Future work will certainly also address issues of perception, antic-ipation, and communication.
Acknowledgments
The authors are grateful for partial financial support by the German ResearchFoundation (research projects He 2789/7-1, 8-1) and by the “Cooperative Cen-ter for Communication Networks Data Analysis”, a NAP project sponsored bythe Hungarian National Office of Research and Technology under grant No.KCKHA005.
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Nichtlineare Ph¨anomene in einem fluid-dynamischen Verkehrsmodell (Master’s thesis, University of Stuttgart, 1998).29. V. J. Blue and J. L. Adler, Transportation Research Records 1644, 29–36 (1998).30. M. Fukui and Y. Ishibashi, Journal of the Physical Society of Japan 68, 2861–2863(1999).31. M. Muramatsu, T. Irie, and T. Nagatani, Physica A 267, 487–498 (1999).32. H. Kl¨upfel, M. Meyer-K¨onig, J. Wahle, and M. Schreckenberg, in S. Bandini andT. Worsch (eds.)
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Spatial Choices and Processes (North-Holland, Amsterdam, 1990), pp. 169–200.35. C. W. Reynolds, in D. Cliff, P. Husbands, J.-A. Meyer, and S. Wilson (eds.) FromAnimals to Animats 3: Proceedings of the Third International Conference on Sim-ulation of Adaptive Behavior (MIT Press, Cambridge, Massachusetts, 1994), pp.402–410.036. D. Helbing, Behavioral Science 37, 190–214 (1992).37. D. Helbing, P. Moln´ar, I. Farkas, and K. Bolay, Environment and Planning B 28,361-383 (2001).38. J. Klockgether and H.-P. Schwefel, in D. G. Elliott (ed.) Proceedings of theEleventh Symposium on Engineering Aspects of Magnetohydrodynamics (Cali-fornia Institute of Technology, Pasadena, CA, 1970), pp. 141–148.39. D. Helbing, in G. Haag, U. Mueller, and K. G. Troitzsch (eds.) Economic Evolutionand Demographic Change. Formal Models in Social Sciences (Springer, Berlin,1992), pp. 330–348.40. N. E. Miller, in J. McV. Hunt (ed.) Personality and the behavior disorders, Vol. 1,(Ronald, New York, 1944).41. N. E. Miller, in S. Koch (ed.)
Psychology: A Study of Science , Vol. 2 (McGrawHill, New York, 1959).42. K. Lewin, Field Theory in Social Science (Harper & Brothers, New York, 1951).43. D. Helbing, Journal of Mathematical Sociology 19(3), 189–219 (1994).44. S. Hoogendoorn and P. H. L. Bovy, Optimal Control Applications and Methods24(3), 153–172 (2003).45. A. Johansson, D. Helbing, and P. K. Shukla, Advances in Complex Systems, inprint (2007).46. T. I. Lakoba, D. J. Kaup, and N. M. Finkelstein, Simulation 81(5), 339–352 (2005).47. A. Seyfried, B. Steffen, and T. Lippert, Physica A 368, 232–238 (2006).48. J. Kerridge and T. Chamberlain, in N. Waldau, P. Gattermann, H. Knoflacher,and M. Schreckenberg (eds.) Pedestrian and Evacuation Dynamics ’05 (Springer,Berlin, 2005).49. S. P. Hoogendoorn, W. Daamen, and P. H. L. Bovy, in Proceedings of the 82ndAnnual Meeting at the Transportation Research Board (CDROM, Mira DigitalPublishing, Washington D.C., 2003).50. K. Teknomo, Microscopic pedestrian flow characteristics: Development of an imageprocessing data collection and simulation model (PhD thesis, Tohoku UniversityJapan, Sendai, 2002).51. D. Helbing, A. Johansson and H. Z. Al-Abideen, Physical Review E 75, 046109(2007).52. L. P. Kadanoff, Journal of Statistical Physics 39, 267–283 (1985).53. H. E. Stanley and N. Ostrowsky (Eds.),
On Growth and Form (Martinus Nijhoff,Boston, 1986).54. T. Arns, Video films of pedestrian crowds (Stuttgart, 1993).55. H.-H. Stølum, Nature 271, 1710–1713 (1996).56. I. Rodr´ıguez-Iturbe and A. Rinaldo,
Fractal River Basins: Chance and Self-Organization (Cambridge University, Cambridge, England, 1997).57. D. Helbing, I. Farkas, and T. Vicsek, Physical Review Letters 84, 1240–1243 (2000).58. T. Schelling, Journal of Mathematical Sociology 1, 143–186 (1971).59. D. Helbing and T. Platkowski, International Journal of Chaos Theory and Appli-cations 5(4), 47–62 (2000).60. S. P. Hoogendoorn and W. Daamen, Transpn. Sci. 39(2), 147–159 (2005).61. T. Kretz, A. Gr¨unebohm, and M. Schreckenberg, J. Stat. Mech. P10014 (2006).62. K. Ando, H. Oto and T. Aoki, Railway Research Review 45 (8), 8-13 (1988).63. K. H. Drager, G. Løv˚as, J. Wiklund, H. Soma, D. Duong, A. Violas, and V. Lan`er`es,in the
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Engineering for Crowd Safety , pp. 361–370,R. A. Smith and J. F. Dickie (Eds.) (Elsevier, Amsterdam, 1993).66. S. Okazaki and S. Matsushita, in
Engineering for Crowd Safety , pp. 271–280, R.A. Smith and J. F. Dickie (Eds.) (Elsevier, Amsterdam, 1993).67. G. K. Still,
New computer system can predict human behaviour response to buildingfires , Fire 84, 40–41 (1993).68. G. K. Still,
Crowd Dynamics (Ph.D. thesis, University of Warwick, 2000).69. P. A. Thompson and E. W. Marchant,
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Hazard Management and Emergency Planning ,Chap. 10, D. J. Parker and J. W. Handmer (Eds.) (James & James Science, London,1992).74. D. Canter (Ed.),
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Friction, Arching, Contact Dynamics (WorldScientific, Singapore, 1997).94. D. Helbing, A. Johansson, J. Mathiesen, M.H. Jensen and A. Hansen PhysicalReview Letters 97, 168001 (2006).295. S. Ghashghaie, W. Breymann, J. Peinke, P. Talkner, and Y. Dodge, Nature ,767–770 (1996).96. G. Peng and H. J. Herrmann, Phys. Rev. E , R1796–R1799 (1994).97. F. Radjai and S. Roux, Phys. Rev. Lett. , 064302 (2002).98. K. R. Sreenivasan, Nature , 192–193 (1990).99. M. E. Cates, J. P. Wittmer, J.-P. Bouchaud, and P. Claudin, Phys. Rev. Lett. ,1841–1844 (1998).100. P. Bak, K. Christensen, L. Danon, and T. Scanlon, Phys. Rev. Lett. , 178501(2002).101. P. A. Johnson and X. Jia, Nature , 871–874 (2005).102. J. J. Fruin, in R. A. Smith and J. F. Dickie (eds.) Engineering for Crowd Safety ,(Elsevier, Amsterdam, 1993), pp. 99–108.103. T. Baeck,
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Books and Reviews
Pedestrian and Evacuation Dynamics 2003 (CMS Press, Lon-don, 2003).108. D. Helbing, P. Moln´ar, I. Farkas, and K. Bolay (2001) Self-organizing pedestrianmovement. Environment and Planning B 28, 361-383.109. D. Helbing (2001) Traffic and related self-driven many-particle systems. Reviewsof Modern Physics 73, 1067-1141.110. D. Helbing, L. Buzna, A. Johansson, and T. Werner (2005) Self-organized pedes-trian crowd dynamics: Experiments, simulations, and design solutions. Transporta-tion Science 39(1), 1-24.111. V. M. Predtechenskii and A. I. Milinskii,
Planning for Foot Traffic Flow in Build-ings (Amerind, New Delhi, 1978).112. M. Schreckenberg and S. D. Sharma (eds.) Pedestrian and Evacuation Dynamics(Springer, Berlin, 2002).113. R. A. Smith and J. F. Dickie (Eds.),
Engineering for Crowd Safety (Elsevier,Amsterdam, 1993).114. G. K. Still, Crowd Dynamics Ph.D. thesis (University of Warwick, 2000).115. J. Surowiecki, The Wisdom of Crowds (Anchor, 2005).116. N. Waldau, P. Gattermann, and H. Knoflacher (eds.), Pedestrian and EvacuationDynamics 2005 (Springer, 2006).117. J. Tubbs and B. Meacham, Egress Design Solutions: A Guide to Evacuation andCrowd Management Planning (Wiley, 2007).118. U. Weidmann,
Transporttechnik der Fußg¨anger (Schriftenreihe des Institut f¨urVerkehrsplanung, Transporttechnik, Straßen- und Eisenbahnbau90