Traffic and Crowd Dynamics: The Physics of the City

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Meyers: Encyclopedia of Complexity and Systems Science — Entry 641 — 2008/9/8 — 11:48 — page 1 — le-tex��

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Traffic and Crowd Dynamics: The Physics of the City 1

Traffic and Crowd Dynamics:The Physics of the CityARMANDO BAZZANI, BRUNO GIORGINI,1

SANDRO RAMBALDI2

“Fisica della Città” Laboratory, Center L. Galvani3

for Biocomplexity, Physics Department and INFN4

Sezione di Bologna, Bologna, Italy5

Article Outline6

Glossary7

Definition of the Subject8

Introduction9

Automata Gas and Crowd Behavior10

Automata Gas and Traffic Behavior11

Simulation Results and Empirical Observations12

Future Directions13

Bibliography14

15

TS2

Glossary16

Automaton A physical particle with internal cognitive17

dynamics able to process information and to make de-18

cisions in order to achieve an aim.19

Cognitive dynamics A dynamical system that mimics the20

individual decisionmechanisms pointed out by neuro-21

science and behavioral studies.22

Emergent properties Macroscopic properties of a statis-23

tical systems that are not the result of the superposition24

of the microscopic states of individual particles.25

Global positioning system A system to detect the geo-26

graphical coordinates of a receiver on the earth using27

a satellite network.28

Mean field theory a technique introduced in statistical29

physics that studies the macroscopic dynamics using30

the dynamics of averaged quantities.31

Self-organized dynamics Stationary macroscopic states32

that are the result of collective interactions among the33

microscopic components.34

Utility function A function that measures the expected35

advantages or the disadvantages of future decisions.36

Definition of the Subject37

The physics of the city CE3 has been proposed as an in-38

terdisciplinary research field that applies the techniques39

of exact sciences to describe and understand the dynam-40

ical states of the new metropolis. The continuous growth41

and the social changes outlined by urban planners and 42

sociologists give to the actual cities a complex structure 43

whose study requires a new generation of dynamical mod- 44

els. From this point of view, the traffic and crowd dynam- 45

ics have a fundamental importance due to their impact on 46

individual freedom and of life quality. The study of the ob- 47

served macroscopic phenomena using a complex systems 48

approach, the search for the control parameters and the 49

understanding of the individual mobility demand are key 50

points for future contributions of the complexity science 51

to real society problems. 52

Introduction 53

Crowd and traffic dynamics are a crucial issue for model- 54

ing the complex urban mobility [5,8,11]. The idea that the 55

city itself can be considered a whole laboratory to under- 56

stand the mobility is at the base of a new research field: 57

the physics of the city [13]. “Aggregate of beings that hold 58

their biological history into its borders andmodel it within 59

all their intentions proper to thinking creatures, the city 60

results at the same time by the biological generation, the 61

organic evolution, and the aesthetics creativity. The city is 62

a contemporary a natural object and a subject of culture”. 63

From these words of Claude Levi Strauss clearly emerges 64

the complex nature of every urban system that today it is 65

generally accepted and considered in order to study the 66

city, its development and evolution [8]. The city shows 67

itself to be polymorphous, polysemic and polyglot [22], 68

stratified in time and crossed by actors and objects whose 69

dynamics are extremely different and can be conflictive to 70

the point to engendering a feeling of chaos, i. e., one can 71

say that cities live at the edge of chaos and the problem is 72

exactly to govern the system emerging complexity, and not 73

real chaos situations. To try to express the quality of this 74

complexity in quantitative terms, using the instruments of 75

the exact sciences, without losing its texture, one has to re- 76

duce the semantic, logical, syntactic and phenomenologi- 77

cal field in which to articulate the possibilities of construct- 78

ing models able to be descriptive, explanatory and, at least 79

to some extent, predictive [78]. These models will never be 80

completely isomorphic to reality but can simulate some of 81

its aspects and characteristics considered salient by the ob- 82

server, thus playing a fundamental role in the choice of the 83

significant observables and control parameters, which ob- 84

viously depend from what one wants to control and gov- 85

ern. CE4 A simple consideration can help us in the reduc- 86

tion process: regardless of the variety and complexity of 87

flows, forms and information, an urban system exists in- 88

sofar as it is inhabited. Therefore our physics of the city 89

will be essentially physics of an inhabited city, and given 90

Please note that the pagination is not final; in the print version an entry will in general not start on a new page.TS2 Your Section “Article Outline” was deleted. In this encyclopedia abstracts are not allowed, please confirm.CE3 Please confirm change.CE4 Please confirm change.CE5 Please confirm change.

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2 Traffic and Crowd Dynamics: The Physics of the City

the large number of elementary components, this means91

non-equilibrium statistical physics [61,77], because a town92

is an open system. Moreover, given that the elementary93

components move in urban space-time, this also means94

the physics of dynamical systems, and since the individ-95

uals in the system have free will, probabilistic physics has96

to be considered (Pascal firstly modeled the human free97

will by the probability function, using the game of chance98

as paradigm). To finish, since the elementary components99

have memory and are capable of drawing information100

from the environment, processing it according to inten-101

tions, choices and decisions, the physics of the city must102

also be intentional, cognitive and decisional. As everyone103

knows, another basic ingredient of physics, besides the el-104

ementary components, is a space-time where the dynam-105

ics can develop (ubi materia, ibi geometria – Kepler). So106

one can ask if it is possible to identify a space-time struc-107

ture proper to a generic urban system. Obviously we can108

describe the street network, and the different morpholo-109

gies with a spatial metric that usually is not Euclidean,110

but is not sufficient to develop an urban dynamics. One111

needs also of a clock, a time structure scale invariant. CE5112

This structure can be modeled by the chronotopoi (liter-113

ally, places of time), the primal agents of urban tempo-114

ral dynamics able to generate time correlations that would115

not exist without them [10]. In the city planning language,116

they are defined as areas where are implanted, temporal117

scheduled activities, for example a hospital, the univer-118

sity, a shopping center and so on, that generates/attract119

mobility. Urban topology thus becomes chronotopic120

and the interaction of the individual’s agenda with the121

pulsation of the chronotopos produces complex urban122

mobility [36].123

Assuming the mobility as a property of the individu-124

als able to process information, the microscopic dynam-125

ics one generates both by properly physical interactions126

and by some decisional/cognitive mechanism [1]. More-127

over a mesoscopic dynamics for individuals aggregate has128

been developed to study self-organization behaviors in the129

context of a non-equilibrium and non-linear evolution130

of the system (being the urban mobility a complex phe-131

nomenon that develops itself in an open complex system,132

the city) [39,45]. Using a computer different mobility sit-133

uations one simulates and analyzes, in order to study the134

control parameters, phase transitions and shift from or-135

der to chaos, criticalities and the whole range of possible136

orbits, also those that are very difficult to reach in mate-137

rial experiments [24,25]. Obviously there is the problem138

of realism, because the simulations can be consistent and139

reasonable but far, sometime very far, from the effective140

mobility.141

In general social systems, like urban traffic and crowd 142

dynamics, consist of many autonomous intelligent enti- 143

ties, which are distributed over a structured space and in- 144

teract each other to achieve certain goals [33,68]. A power- 145

ful method to study such systems are multi-agent models, 146

which allow one introduce a cognitive behavior in the dy- 147

namics of individual entities. The concept of “agent” has 148

been introduced in many research areas [17,67,87], from 149

computer science and artificial life research to economy 150

and sociology, so that its definition is quite generic: agent 151

is anything able to act in a given environment. To intro- 152

duce a more specific agent definition in the simulation 153

of traffic and crowd dynamics [6,7,94], the crucial issue 154

concerning the meaning and the utility of a social system 155

model has to be first discussed [19,20,33]. Complexity sci- 156

ence faces the problem of studying systems whose dynam- 157

ics cannot be reduced to physical fundamental principles. 158

The different scales of description turn out to be intercon- 159

nected not only by a bottom-up interaction structure typ- 160

ical of physical systems (the dynamics at microscopic lev- 161

els determines the evolution of macroscopic variables), but 162

also by top-down interactions so that the microscopic be- 163

havior is influenced by the macroscopic states of the sys- 164

tem [30]. A classical statistical physics approach is ques- 165

tionable in such a case. A further difficulty is that even the 166

interactions among the elementary components of a com- 167

plex system are unknown, and they can only be described 168

in a qualitative way according to some hypotheses, con- 169

sequences of direct observations of real phenomena [78]. 170

Usually a model is a simplified representation of the re- 171

ality, but not too simple in order to describe the inter- 172

ested phenomenon; moreover any model should have suf- 173

ficiently generic features to be able to describe analogous 174

phenomena observed in different systems from social to 175

physical or biological ones [76,90]. In complexity science 176

the role of the models is modified becoming not a specific 177

realization of a general theory (think for example to dif- 178

ferent cosmological models which correspond to different 179

boundary conditions and solutions for the Einstein equa- 180

tions), but rather an instrument of knowledge to perform 181

in silico experiments that allows one to verify the consis- 182

tency of the assumptions with the considered phenom- 183

ena. In other words the models realize a “virtual reality” 184

where the simulations point out the relevance of different 185

hypotheses on the appearance and evolution of emergent 186

properties identifying the control parameters. These mod- 187

els cannot be validated in the usual Galilean sense, since 188

the models are themselves experiments. Moreover there is 189

an intrinsic difficulty in defining Galilean experiments on 190

complex systems due to the great information needed for 191

the reproducibility [9,34]. From a physical point of view, 192

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one has to look for statistical laws that consider the min-193

imal conditions for the existence of phase transitions to-194

ward particular self-organized states.195

In our opinion, a paradigmatic physical model for196

complex systems can be the “automata gas” [95]: a statis-197

tical system whose elementary components perform both198

a physical dynamics and a cognitive behavior based on in-199

formation exchanges [35,92]. In the sequel an automata200

gas model to describe the crowd and traffic dynamics is201

described. As a general convention the capital letters will202

denote scalar variables, whereas the lower letters will indi-203

cate vectors.204

Automata Gas and Crowd Behavior205

The pedestrian mobility is a basic human activity strictly206

linked with the detailed structure of urban environment207

and the social life [14,32,81]. Indeed due to the human be-208

ing social nature, the pedestrian mobility will continue to209

play a fundamental role for the life quality of metropo-210

lis. The development of fast and comfortable transporta-211

tion has reduced the spatial scale of walking activity to212

an average dimension of '500m. The direct observations213

on video films turns out to be the most useful tool to ex-214

tract detailed information on pedestrian dynamics [50,99].215

Various research group have pointed out the following216

items [39]:217

I Pedestrians tend to walk at a desired velocity toward218

a local destination. The desired walking speed depends219

both on social characters (for example age, sex, job, . . . )220

and on the cognitive internal state.Moreover individu-221

als develop strategies in order to walk in themore com-222

fortable way (for example avoiding collisions and least223

energy consuming), and following a best path accord-224

ing to their propensities and knowledge of the urban225

space.226

II Pedestrians like to keep a certain distance from other227

pedestrians and borders as walls or obstacles. This dis-228

tance can be interpreted as a social space preserving229

individuality and it depends on various factors like230

crowding conditions, individual hurry and space ge-231

ometry. Usually individuals don’t want CE6 their pace232

violated by extraneous people. Moreover individu-233

als who know each other, tend to form clusters that234

move as single entities performing a flocking dynam-235

ics [24,25].236

III To avoid physical contact, individuals reduce the walk-237

ing velocity and take local detours, but they feel aver-238

sion to move in opposite to their desired direction. In239

case of physical collisions, the incompressible nature240

of bodies play an important role and pedestrians move241

their body, so that they can slide on each other (if it is 242

possible). 243

IV The best pedestrian paths usually are not the mini- 244

mal time or space geodetic, becausemany other factors 245

contribute to the choice of the trajectory, so a minimal 246

action principle cannot model dynamics [15,54]. 247

V The pedestrian paths seem to optimize some utility 248

functions time or energy consuming [64], but depend 249

on the information level of the individual on the urban 250

space and single’s propensities. They may be dynami- 251

cally modified by decision mechanisms [40,46] in the 252

presence of multiple choices, particular chronotopoi 253

and geometrical or crowding conditions [11]. In nor- 254

mal crowding condition, there are strategies automat- 255

ically applied at particular points, like at a bottleneck, 256

where the priority convention avoid criticalities. The 257

decision mechanism reflects the individual free will, 258

which is an unpredictable relevant character of pedes- 259

trian dynamics, but there are also some average behav- 260

iors, such as the herding effect or habit [40,49]. 261

VI When panic phenomena occur, pedestrians lose their 262

rationality and the dynamics are dominated by hard 263

physical interactions [39]. People try to move consid- 264

erably faster than normal and start pushing. As a con- 265

sequence the pressure become extremely dangerous in 266

jammed crowds, arching and clogging are observed at 267

the exits so that even escape become very difficult or 268

impossible [37]. 269

In order to define an automata gas model for pedes- 270

trian dynamics that takes into account the previous obser- 271

vations, physical and information-based interactions are 272

treated separately. Let us consider the following axioms, 273

that define a common framework for several pedestrian 274

dynamical models: 275

A1) each automaton i has an inertial mass mi, an incom- 276

pressible body-space of fixed radius Rb and a social 277

space of radius Rs around its center ri. 278

A2) Each automaton i has a desired velocity vi , and it con- 279

tinuously modifies its actual velocity vi according to 280

vi D ��1(vi � vi) (1) 281

A3) When two automata i; j have distance Di j D kr j � 282

rik D 2R and their velocities satisfy the condition 283

(v j � vi ) � (r j � ri ) < 0, then a physical collision and 284

the change of momenta pi D mivi are computed ac- 285

cording to 286

p0i D ˛

�pi � (R(�) � I)�(v j � vi)

�

p0j D ˛

�p j C (R(�) � I)�(v j � vi )

� (2) 287

CE6 Please confirm change.


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Traffic and Crowd Dynamics: The Physics of the City, Figure 1Sketch of automata collision in the center of a mass system:p D �(vj � vi) is the center of mass momentum and � the col-lision angle with respect to the normal to the collision directionrj � ri

where � D mimj/(mi C mj) is the reduced mass,288

˛ � 1 the dissipation coefficient andR(�) is a coun-289

terclockwise rotation of an angle � > � (see Fig. 1),290

whereas I is the identity transformation.291

A4) Each automaton has a local visual space given by292

a semicircle of radius Rv, symmetric with respect to293

the desired velocity direction. At a given time inter-294

val�t, the automaton obtains information on the po-295

sitions and the velocities of any other automaton or296

obstacle in its visual space.297

A5) Each automaton i rotates and reduces its velocity298

vi D ˙! � vi � ˇvi ˇ > 0 (3)299

when it realizes that another automaton or an obsta-300

cle will enter into its social space, within a reaction301

time interval � .302

The axiom A1 concerns the physical properties of the au-303

tomata gas: the incompressible body is usually considered304

by models for simulation of crowd dynamics and pedes-305

trian panic [44]. In principle one has to consider differ-306

ent body radius for different automata to take into ac-307

count the diversity among individuals in a crowd; this can308

be relevant studying the dynamics at bottlenecks [43,52].309

The “inertial mass” m is also introduced to differentiate310

the automata behavior during collisions or interactions:311

an automaton with a great mass behaves more aggres-312

sively by forcing other automata to deviate from its tra-313

jectory. The existence of a social space has been proposed314

by sociologists who related this space with the distance be-315

tween two individuals shaking hands with each other (in316

the West CE7 ). However this is not a fixed space, and it317

depends on the single characteristics (for example philan-318

thropic or misanthropic one), and on crowding conditions319

and presence of attractive points in the space where peo- 320

ple accept to be closer. One should also consider the pos- 321

sibility that individuals form compact group when there is 322

empathy among them. 323

The existence of a desired velocity (axiom A2) is al- 324

most obvious, but of course its determination is the result 325

of a cognitive process where both determinism and indi- 326

vidual free will have an important role. Eq. (1) is the most 327

simple dynamics to simulate the automaton tendency to 328

move at the desired velocity and the same choice has been 329

proposed by other models, where the only relevant param- 330

eter is � : i. e., the “relaxation time scale” [45]. 331

The collision dynamics defined by axiom A3 is in- 332

spired by granular flows interactions [18,38], and it allows 333

one to simulate both the body repulsion and a sliding ef- 334

fect that mimics the real effect due to the possibility of 335

rotating the body when two pedestrians collide. The rel- 336

evant parameters are the rotation angle � and the dissipa- 337

tion parameter ˛ 2 [0; 1]: when � D 2� (see Fig. 1) and 338

˛ D 1 one has a perfect elastic collision, whereas � D � 339

and ˛ D cos � gives an anelastic collision between two 340

spheres. The direct observation of video films suggests the 341

existence of a ˛ dependence on � , which has a maximal 342

value at � D �/2 (i. e., two pedestrians tend to reduce sud- 343

denly their velocity in a head on collision). Conversely the 344

choice of the � within the interval [�; 2�] seems to be suit- 345

able for the simulations. 346

The physical automata dynamics, which is the combi- 347

nation of Eqs. (1) and (2), can be defined in an equiva- 348

lent way by using the social force model, where a second- 349

order dynamics is used to simulate pedestrian movements 350

with interaction “social forces”, which depend both on the 351

distance and on the reciprocal orientation of pedestrians 352

(non-Newtonian character) [41]. The collisional approach 353

used in the automata gas model gives an advantage us- 354

ing essentially a first-order dynamics in the simulations, 355

so that the numerical instability due to high force gradi- 356

ents are automatically solved and a longer integration time 357

step can be used. Conversely the cellular automata mod- 358

els [3,48,70] solve the collisions problem by discretizing 359

the space and are extremely convenient from a compu- 360

tational point of view, but they can introduce some bias 361

in the microscopic dynamics whose macroscopic effect 362

should be carefully considered [60]. 363

The axioms A4 and A5 introduce a local vision mech- 364

anism: each automaton sees the others in front of it whose 365

actual position is inside its visual space. For each of them, 366

it computes the forecast position (according to the actual 367

velocity) after a time � . Then it selects the automata whose 368

forecast positions are inside its social space, and it rotates 369

and reduces its velocity, in order to avoid future collisions 370

CE7 Please confirm change.


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choosing the suitable sign in Eq. (3). The vision has some371

typical features of an information based interactions [84]:372

a) It depends on the forecast positions of other automata373

according to the available information;374

b) It has a dichotomic nature since only the automata375

that will eventually enter the social space are consid-376

ered;377

c) The sign choice of the rotation velocity is the result of378

an utility function evaluation.379

Finally the interval �t defines the decision timescale,380

when new information is achieved; by assumption, the fol-381

lowing inequality holds382

�tvi � Rb � �vi ' Rv : (4)383

Therefore the automaton dynamics is the result of many384

decision processes, and the vision mechanism determines385

a collective interaction dynamics in the automata model,386

which is an essential feature for the appearance of emer-387

gent properties. According to the inequality (4), the aver-388

age effect of local vision is to rotate the velocity in the di-389

rection where the “density” of counteracting automata is390

minimal and to reduce the velocity proportionally to the391

surrounding automata density. From this point of view,392

the local vision is equivalent to a non-Newtonian repul-393

sive long-range force with a spatial decaying scale ' Rv,394

as proposed in the social force model. Indeed both the395

approaches give rise to the same macroscopical states in396

many cases of study. But the individual character of in-397

formation-based interactions allows one to introduce the398

free will in the model, using the concept of cognitive state,399

without the artifact of external random perturbations in400

the dynamics. Using the idea of cognitive state to define401

the automata desired velocity and the choice among mul-402

tiple possibilities permits one to state two other axioms:403

A6) Each automaton has a cognitive network representa-404

tion of the physical surrounding space dependent on405

its knowledge level. Each network node is related to406

a possible decision, whereas the arcs introduce con-407

nections between successive decisions (see for exam-408

ple Fig. 2a).409

A7) The decision mechanism is defined by a subjective410

probability evaluation [27] based on the existence of411

an utility function and an internal cognitive dynam-412

ics [16,28]; the probability are computed taking into413

account the available information and the previous414

history.415

The axiom A6 implies an internal dynamical structure for416

the particle automaton, which acquires a knowledge capa-417

bility of urban space. The nodes are identified with partic- 418

ular regions of interest in the urban space (for example in 419

Fig. 2 the nodes correspond to doors neighborhoods). The 420

decision process is discontinuous, since the automaton 421

gets new information at the node points. One can compare 422

the cognitive network structure with a best paths origin- 423

destination algorithm, based on the deterministic mini- 424

mal action principle, which has been proposed to study 425

the accessibility of urban spaces [15,54]. However, this ap- 426

proach gives a global and rigid description of the space that 427

turns out to be unrealistic and, sometimes, useless, when 428

one studies crowd dynamics where the automata paths are 429

the results of a continuously evolving process dominated 430

by the collective interactions and the individual free-will. 431

On the contrary, the space representation using a cognitive 432

network that focuses the attention only on some particu- 433

lar points, it is both robust and flexible to allow a single 434

automaton to creat its own individual path with optimiza- 435

tion criteria based on local information. 436

The axiom A7 establishes a relation between an in- 437

ternal cognitive dynamics [93] and a probabilistic choice 438

among different possibilities [62]. Let us consider the ele- 439

mentary dichotomic decision between the choicesA and B, 440

by associatingA or B to the values of the variable xi (for ex- 441

ample A when xi > 0) that defines the “cognitive state” of 442

the i automaton. One assumes the existence of a stochas- 443

tic dynamical system that represents the neuronal activi- 444

ties [83] 445

xi D @U@x

(xi ; I) Cp2Ti�t (5) 446

where U(x; I) is an utility function with two local max- 447

imaUA andUB. The “utility concept”, introduced in econ- 448

omy [31,96], is a measure of the perceived advantages of 449

a certain decision with respect to the automaton CE8 final 450

purpose, whereas the information I is any external variable 451

that may influence the decision. Under the very general 452

regularity hypotheses on U CE9 , the dynamics (5) evolves 453

towards one of the local maxima in a stochastic way, due to 454

the presence of the white noise � t (unpredictable factors in 455

the decision mechanism) and the “social temperature” Ti, 456

which is a measure of the “irrational behavior” of the au- 457

tomaton. As a consequence even if all the automata have 458

the same decisionmechanism, the effective decision is sub- 459

jective due to the different information I and the different 460

temperature Ti. According to the Arrhenius’ law, the tran- 461

sition probabilities from the choice A to B and vice versa 462

read 463

PAB / exp�

�UA(I)Ti

�PBA / exp

��UB(I)

Ti

�(6) 464

CE8 Please confirm change.CE9 Please check sentence.


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Traffic and Crowd Dynamics: The Physics of the City, Figure 2a Cognitive network representation of a space with two parallel walls and four connecting doors; the automata enter from the leftside and have to reach the exit on the right. The squares are the decision nodewhere automata obtain new information and changethe desired velocity. b Snapshot of the automata dynamics in the space shown in left picture. The cognitive network causes a fluxorganization along the shortest paths, but the subjective probabilistic decision implies the existence of “irrational” automata whichfollow longer paths

By introducing an average balance equation for the sta-465

tionary probabilities to choose A or B466

PAPAB � PBPBA D 0467

one obtains [16,46]468

PA Dexp

�UATi

�

exp�UATi

�C exp

�UBTi

�

PB Dexp

�UBTi

�

exp�UATi

�C exp

�UBTi

�

(7)469

The non-linear relation between utility and probability has470

two limit behaviors: for low temperature values, Eq. (7) de-471

fines a sigmoidal like function with a sharp transition be-472

tween the two choices, whereas for high temperature val-473

ues the relation is almost linear. These two regimes corre-474

spond to two different social behaviors of the automaton:475

in the first case there is a sudden change in the decision476

as utility overcomes a certain threshold (rational automa-477

ton), in the second case, the automaton choice becomes478

insensitive to utility changes (irrational behavior).479

Automata Gas and Traffic Behavior480

The traffic behavior in a urban network is the consequence481

of citizens mobility demand driven by the chronotopic482

activity of modern cities [21]. The application of physics483

to traffic modeling dates back more than 40 years [80].484

From one hand the microscopic traffic dynamics is forced485

on a one-dimensional space (the road), from the other 486

hand its spatial scale is extended to the whole city. For 487

this reason various researches have proposed to simplify 488

the microscopic physical dynamics and to focus the atten- 489

tion on the cognitive behavior at the base of the individ- 490

ual mobility agenda and on the fluxes optimization in the 491

road network [2,42,86,94]. Nevertheless the microscopic 492

drivers behavior is responsible of local congestions forma- 493

tion that may propagate in the road network and cause 494

a macroscopic phase transition of the whole system. The 495

proposed models are divided in two main classes: the cel- 496

lular automata models which discretize the road dynam- 497

ics using discrete stochastic maps [63,75] and the continu- 498

ous models based on stochastic differential equations [39]. 499

The auto-mobilis model [12] is based on a simple micro- 500

scopic continuous dynamics, but able to describe different 501

driving styles and behaviors at the crossing points (traffic 502

lights, roundabouts and stop signals at insertions). The au- 503

tomata in the models move on a road using alternatively 504

the two main behaviors that has been proposed by vari- 505

ous researchers [51]: the so called “car-following” [4] and 506

the “optimal velocity” behavior [74]. In the first case each 507

driver adjusts his velocity to the velocity of the successive 508

vehicle; in the second case each driver chooses his velocity 509

according to the safety distance with the successive vehicle. 510

Both models can be described by the differential equation 511

s j D v jv j D a(s j�1 � s j ; v j; v j�1(t � �);)

(8) 512

where (s j; v j) are the position and the velocity of the j-au- 513

tomaton on a road (by convention the j-driver follows the 514

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j � 1-driver), is the parameters set that characterizes515

the individual behavior, � is a reaction time to the velocity516

changes of the successive vehicle. The effect of line changes517

can be neglected since they are not frequent along a urban518

street (of course this is not true at the intersections points519

that will be consider separately). To specify the accelera-520

tion in Eq. (8), the main ideas at the base of variousmodels521

are summarized, by means of the following assumptions:522

B1) An automaton-driver has a finite dimension d0, it523

keeps a minimal distance dmin from the preceding524

automaton, it has a reaction time � to the velocity525

changes of the other vehicles and it has a desired526

velocity vmax (usually the maximal allowed velocity527

along the considered road, assuming that people re-528

spect the traffic rules)529

v j D ˛(v j � vmax) (9)530

where ˛ is the acceleration capacity in a comfortable531

driving.532

By measuring sj at the center position the distance be-533

tween two successive drivers is �s j D s j�1 � s j � d0.534

B2) Each automaton has a safety distance Ds and an atten-535

tion distance Datt (i. e., a distance at which the driver536

considers the presence of the other drivers)537

Ds D dmin C 2�v j Datt D dmin C 2:5�vmax (10)538

Each time �s j � Ds it reduces the velocity according539

to540

v j D �ˇ(v j � vs) (11)541

where ˇ is the breaking capacity.542

Eq. (10) TS10 established a linear relation between543

reaction time � and the safety distance Ds; this is544

with direct observations for the typical urban velocity545

ranges. But to avoid accidents one requires a condi-546

tion on the breaking capacity ˇ� � 1/2.547

B3) When the distance �s j is less than the attention dis-548

tance Datt, the automaton adjusts its CE11 velocity to549

the velocity of the preceding vehicle550

v j D �˛(v j � v j�1(t � �) � � ) (12)551

where the parameter � � 0 simulates the behavior of552

impatient drivers.553

To be more realistic one takes also into account that the554

automata cannot correctly evaluate small velocities so that555

when v j�1 � vmin one set v j�1 D 0 in Eq. (12).556

The Eqs. (9), (11), (12) define the microscopic dynam-557

ics along a road. All the parameters are individual charac-558

ters and have a random distribution among the automata559

populations. Indeed the driver behavior can be correlated 560

to citizens social characters (sex, age, birth place etc.), so 561

that several classes of drivers can be identified. In some re- 562

alistic situations the presence of different driver classes is 563

crucial to understand urban traffic. The model can simu- 564

late different traffic regimes: at low density the free flow 565

(Eq. (9)) is described, when the density increases the au- 566

tomata perform a synchronized flow, which becomes un- 567

stable when the local density overcome 1/Ds and gives rise 568

to congestion effects. Indeed the model has infinite equi- 569

libria with automata moving at the same velocity v � vs. 570

However a linear stability analysis using the Fourier trans- 571

form shows that the dynamics (12) is always stable with 572

a relaxation time 573

T ' 2(1 C ˛�)3

˛(2�/�)2(13) 574

in the limit � � 1 where is the density and � the wave- 575

length of the perturbation, whereas the dynamics (11) be- 576

come unstable when ˇ� < 1 [37,69]. Therefore when the 577

local density satisfies � 1/Ds, a congestion regime may 578

rise; the instability can be driven by small percentage of 579

“bad drivers” in the automata population, as shown in 580

Fig. 3. This reflects the possibility that the different driv- 581

ing styles may cause local instability due to microscopic 582

interactions, as suggested by direct observations [91]. 583

To complete the dynamical model the urban traffic 584

model, one has to set the behavior at the crossing points 585

considering different situations: 586

B4) At a traffic light in case of yellow colors, the automata 587

compute the time tc required to pass through the 588

cross at the actual velocity: if tc is longer than the yel- 589

low time-interval, they stop. 590

B5) At a connection of two streets with priority rules, the 591

vehicles in the secondary street are forced to reduce 592

their velocity at a given value vc (vc D 0 in case of 593

a stop signal), and they compute both the distance D 594

from the crossing point and the velocity v of the in- 595

coming vehicle in the main street: if the distance D 596

is greater than (v � vc)/ˇ, the other vehicles can en- 597

ter in the main street; otherwise it has to stop. In 598

Fig. 4 a snapshot of a simulation in a crossing point 599

with priority rules is shown. As a matter of fact, the 600

emergent global properties of urban traffic cannot be 601

understood, described, and eventually predicted, only 602

using the automata micro-dynamics on the road net- 603

work [86,94]. A global cognitive time-dependent field 604

interacting with individuals is needed: 605

TS10 Please check equation labels and references. There were some discrepancies in the input.CE11 Please note that automaton is neither masculine nor feminine: thus its pronouns are "it" and "its".


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Traffic and Crowd Dynamics: The Physics of the City, Figure 3a Stability study of the urban traffic micro-dynamics for an homogeneous automata population in a periodic road: the four curves(in the order: black, red, green blues) give the vehicle velocity (m/s) as a function of the position (m) on the road at different timeseparated by 5 minutes. The model uses “realistic” parameters: d0 D 4m, dmin D 1m, vmax D 16m/s, vmin D 1m/s, ˇ D 1:1 s�1,� D :75 s and˛:5 s�1. The initial equilibriumdensity 1/16m�1 is locally increased by a 10%.b the same as picture a, but the simula-tion considers a 20%of “bad drivers” randomly distributed on the road with reduced breaking capabilityˇ D :9 s�1; the blue curveshows as a stop-and-go instability has been developed by the microscopic dynamics

Traffic and Crowd Dynamics: The Physics of the City, Figure 4Snapshot of vehicles dynamics simulation in crossing point; theroad in a lighter color at the cross has the priority. The differentcolors are related to the different vehicle directions

B6) In the urban space-time there exist the “chronotopoi”,606

which are adaptive agents generating the individual607

mobility demand.608

Examples of chronotopoi are historical centers, shop-609

ping centers, university campus, great social events610

as concerts or football matches etc.; in our sense611

a chronotopos is not a geometrical or morphological612

property, but an entity that introduces timing correla-613

tions between the city macroscopic level (social time)614

and the automata microscopic dynamics (individual615

time). The chronotopoi provide a cognitive map [79]616

to the automata by means of cognitive fields. A sketch617

of chronotopic cognitive fields overlapped to a Man-618

Traffic and Crowd Dynamics: The Physics of the City, Figure 5Landscape of a abstract cognitive field superimposed to a roadnetwork

hattan-like road network is plotted in Fig. 5, where 619

the intensity of the attraction is measured by the well 620

amplitudes. 621

B7) Every automaton constructs a daily mobility agenda 622

based on its propensities and the available chrono- 623

topic information. 624

The decision mechanism is modeled by de Finetti 625

subjective probability [27], taking free will into ac- 626

count. 627

B8) The automaton trajectory is a path realization on 628

probabilistic a network, where the choice probabili- 629

ties at the nodes depend on chronotopic attraction. 630

The decision mechanism at a crossing point is out- 631

lined in Fig. 6, where the probabilities pj depend on 632

a best path algorithms to reach the chronotopos and 633

the weightswj measure the street quality (typically ac- 634

cessibility, safety and appealing). The final probability 635

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Traffic and Crowd Dynamics: The Physics of the City, Figure 6Snapshot of vehicles dynamics simulation in crossing point; theroad in a lighter color at the cross has the priority. The differentcolors are related to the different vehicle directions

choice is defined by636

Pk D pkwkPj p jw j

: (14)637

Simulation Results and Empirical Observations638

In complexity sciences, the simulations and the empirical639

observations play the role respectively of theory and lab-640

oratory experiments in the traditional physics framework.641

The main focus is to observe the emergent properties of642

the system and to understand the class of microscopic in-643

teractions that can explain such properties in order to de-644

tect the control parameters. Simulations and empirical ob-645

servations have to be compared in this optic [9]. In the646

physics of the city, the real urban systems are the natural647

laboratories to observe and study crowd and traffic flow.648

Crowd Behavior in Urban Spaces649

An occasion to illustrate the crowd behavior in a whole650

pedestrian city, is Venice’s Carnival, since about 100,000651

tourists visit Venice each day during the last week. The652

observations are mainly based on video film analysis of653

particular regions of interest, like San Marco square and654

Riva degli Schiavoni. In principle the videos allow one to655

follow the single individual dynamics; however, the di-656

mension of the considered regions is limited and a com-657

puter assisted analysis is difficult in crowded situations.658

Some quantitative information can be obtained by using659

software following the individual trajectories. The trajec-660

tories permit one to extract information both on the mi-661

croscopic pedestrian dynamics and on the individual cog-662

nitive behavior even if the noise level is still a serious prob-663

lem [50]. Using a sample of '600 trajectories in a mod- 664

erately crowed area, we have computed for each individ- 665

ual the density of surroundings coming from the oppo- 666

site direction and moving in a transverse direction within 667

a distance of '2m. Then after averaging all the local den- 668

sities, the results are plotted in the Fig. 7. The observa- 669

tions indicate the existence of a low density area along the 670

desired velocity direction, only if one focuses the atten- 671

tion on individuals moving in a counteracting way; con- 672

versely the individuals that move in other directions seems 673

not to be considered. This is the sign for the existence of 674

strategies to avoid collisions and to keep a certain area 675

free along the direction of motion, based on a local vi- 676

sion mechanism. The automata gas dynamics can be il- 677

lustrated performing some virtual experiments, where the 678

emergent properties can be analyzed. The model parame- 679

ters (cf. Eqs. (1), (2), (3), have been fixed using Rb as space 680

unit and measuring time in seconds: the desired velocity is 681

set kvk D 2Rb with a spread of 10% among the automata 682

and the relaxation time � D 1, whereas the collision (2) is 683

defined by � D � C �/10 (see Fig. 1) and ˛ D :9; finally 684

the local vision has a radius Rv D 2v, the social space is 685

2Rb and Eq. (3) uses! D 3� and ˇ D :5. Moreover all the 686

automata have the same mass. 687

Let us consider two automata populations moving 688

along opposite direction in a open space (i. e., a square). 689

The automata enter in the space from two sources lo- 690

cated at the borders: the red ones move from the left to 691

the right, whereas the yellow ones move from right to 692

left. This system has two equilibrium regimes: the entropic 693

regime where both the populations are uniformly spread 694

in the space, and the organized regime where automata 695

move along separated lines [47,89,98]. In the simulations, 696

at low densities the automata relax to the entropic regime, 697

whereas when the density increases one observes the lines 698

formation in the space as shown in Fig. 8a. The automata 699

form narrow straight non-interacting lines, separated by 700

empty spaces: sometimes the lines are continuous, in other 701

cases they interrupt or join together. Moreover there are 702

always empty regions, so that the space distribution is not 703

uniform. This phenomenon can be interpreted by a mean 704

field argument: every time two automata meet, they devi- 705

ate to avoid a collision and then they recover the desired 706

velocity according to Eq. (1). The final result is a trans- 707

verse displacement with respect to the desired velocity for 708

both the automata. As far as binary interactions are con- 709

sidered, the transverse displacements are symmetric and 710

only the entropic regime is stable. But the local vision 711

introduces long-range collective interactions, and aligned 712

automata moving in the same direction behave as a sim- 713

ple entity [57]: in other words when an automaton meets 714

CE12 Please check sentence.


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Traffic and Crowd Dynamics: The Physics of the City, Figure 7The scale color from red to yellow defines the average local density arounda pedestrian thatmove in the vertical up direction asmea-sured from the film video data during the Venice’s Carnival. The pedestrians where moving in San Marco’s Square in a moderatelycrowded condition. The considered radius around the pedestrian is 2m. The picture a refers to the density of peoplemoving into theopposite direction with respect to the considered individual. The picture b refers to the density of people moving into a transversedirection

Traffic and Crowd Dynamics: The Physics of the City, Figure 8a Simulation of two automata populations moving into opposite direction in a square. The parameters used in the model are (cf.previous section): Rb D 1 (the space unit), kvk D 2, � D 1, � D � C �/10, Rv D 4, ! D 3� and ˇ D :5. The square dimensions are120 � 120 and the automata enter from two sources located at the left border (redautomata) and the right border (yellowautomata).The appearance of an organized regimewith automatamoving along lines is clear.b the same as the picture a, but the incoming fluxfrom the sources has been increased by a factor of 2. The lines of people start to interact and a turbulent regime appears

a line of people moving in the opposite direction, it per-715

forms a larger transverse displacement than the automata716

in the line as sketched in Fig. 9. Therefore the line struc-717

ture becomes stable and the system relaxes towards an or-718

ganized regime. This behavior has been observed by var-719

ious research groups in real situations [47,55,65]: an ac-720

celerated video film in the crowding conditions shown 721

in Fig. 10, points out clearly the existence of stable lines 722

of people moving in opposite directions, and a non-ho- 723

mogenous distribution of pedestrians in the space. The 724

lines survive also when the density increases due proba- 725

bly to the slow down of the pedestrian desired velocity 726

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and the reduction of the social space. The simulations con-727

firm this parametric dependence, and they also point out728

as to maintain the organized regime stability (for a fixed729

desired velocity and social space) it is essential to have730

some space among the automata lines, otherwise the in-731

teraction among the lines themselves will destroy the lami-732

nar character of the dynamics, and a new turbulent regime733

appears [45,71,72]. This implies the existence of a critical734

density value as it is shown in Fig. 8, where the simulation735

are repeated with a twice flux from the external sources.736

This turbulent regime has been observed in critical situa-737

tions (like crowd stampede)when under effect of panic, in-738

dividuals behave irrationally, the velocity is increasing and739

the dynamics is mainly physical [50,59]. This was not the740

case during the Venetian Carnival and in the absence of741

panic, pedestrians take strategies that prevent critical situ-742

ations. The automata gas model allows the study of the dy-743

namics at a bottleneck, reducing the desired velocity when744

the density overcomes 1/R2b (i. e., when automata have to745

share the social space). A bottleneck is a narrow passage746

at which pedestrian interactions do not allow the laminar747

flow formation. The dynamics at bottlenecks is a crucial748

task for understanding the crowd behavior in urban spaces749

and it has been studied by various researchers [43,52,88].750

The simulation of the passage of two counteracting flows751

at a bottleneck is shown in Fig. 11a: the automata reduce752

the desired velocity when they see a counteracting flux in-753

side the bottleneck area. As a consequence there appears754

a self-organized oscillatory regime for the flux across the755

bottleneck (see Fig. 11b), as it has been observed in other756

models and in laboratory experiments [66]. A mean field757

theory approach suggests a possible interpretation of the758

oscillating fluxes regime. Let nr and ng the density of au-759

tomata moving in opposite directions in the bottleneck760

area; each automaton can be in two states: waiting state761

Pw and crossing state Pc. The balance equations according762

to the following transition probabilities read763

Prc;w / ngPg

c�1 C c

�nrPr

c � ngPgc��

Prw;c / �

1 � c�nrPr

c � ngPgc�� (15)764

based on the hypothesis that the probability to cross the765

bottleneck in the positive direction (red automata) in-766

creases if the flux nrPrc � ngPg

c in the bottleneck is pos-767

itive, and, on the contrary, the probability of returning in768

the waiting state increases when the flux is negative.More-769

over if the flux is positive, be the density nr is decreased770

by a quantity �n(nr Prc � ngPg

c )), due to people passing771

trough the bottleneck772

nr D �n(nr Prc � ngPg

c ) if (nrPrc � ngPg

c ) > 0 (16)773

Traffic and Crowd Dynamics: The Physics of the City, Figure 9Sketch of automata interaction: the difference between the bi-nary interaction dynamics and a collective interaction dynamicsis outlined

Traffic and Crowd Dynamics: The Physics of the City, Figure 10Snapshot of a video film of San Marco’s square during the Vene-tian Carnival 2007: the video analysis shows the existence oflines of people moving in opposite directions

The balance equations (analogous equations hold for the 774

yellow automata) read 775

Prw D �Pr

w;cPrw C Pr

c;wPrc

Prc D Pr

w;cPrw � Pr

c;wPrc

(17) 776

and the solution is plotted in Fig. 12 for a suitable set of pa- 777

rameters (the time unit is arbitrary), where the existence of 778

a stationary oscillation regime in the crossing probability 779

is evident. The parameter c (cf. Eq. (15)), which weights 780

the flux effect on the transition probabilities, is a control 781

parameter for the dynamics: if c is below a critical value, 782

no oscillation is observed and the stationary solution is 783

constant. Finally another interesting situation is illustrated 784

CE13 Please check sentence.


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Traffic and Crowd Dynamics: The Physics of the City, Figure 11aDynamics of two counteracting flows at a bottleneck in automata gas simulation; the transverse bottleneck dimension is 2 time Rband the incoming fluxes are one automaton each three times CE12 units.bNumber of “red” automata that reach the right destinationafter crossing the bottleneck versus time: it is clear the appearance of an oscillating regime at the bottleneck

Traffic and Crowd Dynamics: The Physics of the City, Figure 12Solution of the balance Eqs. (17) showing an oscillating behaviorfor the crossing probabilities at a bottleneck under the hypothe-ses (15); the parameter c is the control parameter and the oscil-lations disappear below a critical value

by Fig. 13, where people had to cross a narrow bridge785

during the Carnival: in the movie the people stop nearby786

the bridge forming a funnel-like queue, whereas the flux787

across the bridge is constant. As a result in the queue the788

people move in a discontinuous way (stop and go), using789

the empty spaces left by individuals that enter the bridge;790

in such a situation there are waves back-propagating along791

the queue [50]. The measure of the flux across a given area792

can be performed by simulating the situation plotted in793

Fig. 14a; the results (see Fig. 14b) show a periodic struc-794

ture that can be interpreted as short length back-propa-795

gating velocity waves. The finite automaton dimension has796

an essential role in the queue dynamics. As discussed pre-797

viously, a quantitative comparison between empirical ob-798

Traffic and Crowd Dynamics: The Physics of the City, Figure 13Snapshot of a video film of high density crowd crossing a narrowbridge during the Carnival; there is a long queue in front of thebridge whereas the flux across the bridge is constant

servations and virtual simulations have to face the intrinsic 799

difficulty of reproducing the complexity of crowd behav- 800

ior. However current computers can be used virtual simu- 801

lators that can study the crowd behavior in various realistic 802

situations CE13 , becoming a helpful instrument for mobil- 803

ity planning [56]. Fig. 15 shows an example of crowd be- 804

havior in a real environment and in a virtual experiment 805

by using the “Campus software” developed at the Physics 806

of the City Laboratory of Bologna [99]. 807

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Traffic and Crowd Dynamics: The Physics of the City, Figure 14a scheme of the bridge crossing geometry used in the gas automata simulation.b automata flux in across small area inside the queue('10 automata are considered simultaneously): fast periodic oscillations are detected due to short length back-propagatingvelocitywaves

Traffic and Crowd Dynamics: The Physics of the City, Figure 15aMobility at the entrance of San Marco’s square (snapshot from video movie). b Automata virtual experiment to study the mobilityin a virtual space reproducing the urban space of the left picture

Cognitive Behavior and Experimental Laws808

in Traffic Dynamics809

As discussed in the previous sections, the cognitive behav-810

ior of individuals due to information-based interactions811

plays a fundamental role to determine the emergent prop-812

erties of urban traffic and crowd dynamics [100]. A cru-813

cial theoretical issue is to understand when it is possible814

to apply a statistical physics approach to derive the evo-815

lution of macroscopic variables from the single automata816

behavior [23]. The main difficulty is that from a cognitive817

point of view, the automata are all “different particles” and818

the information-based interaction introduces long-range819

correlation among the automata dynamics, so that it is820

very hard to justify a mean field approach. To illustrate821

a heuristic mean field approach, one can use the simple822

example of a dichotomic choice A and B for an automata 823

gas, where two counteracting populations (red and yellow) 824

share the same spatial environment [40,62]. An automa- 825

ton i has an utility function U(Nr ; Ny) in the decision 826

mechanisms, depending on the number of automata of the 827

two populations that have made the same choice. Accord- 828

ing to Eq. (7), the probability of the choice A is given by 829

Pi;rA D 1

1 C exp��U(Nr

B ; NyB) � U(Nr

A; NyA�/Ti )

�

if i is red

Pi;yA D 1

1 C exp��U(Ny

B ; NrB) � U(Nr

A; NyA)

�/Ti

�

if i is yellow :

(18) 830

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By using a statistical approach, one set NrA D NrhPi

Air and831

NyA D NyhPi

Aiy where h ir;y indicates the the average832

value on the population. Then the Eqs. (18) give a self-con-833

sistent system834

NrA D

rX

i

11 C exp

��U(Nr � Nr

A; Ny � NyA) � U(Nr

A; NyA)

�/Ti

�

NyA D

yX

i

11 C exp

��U(Ny � Ny

A; Nr � NrA) � U(Nr

A; NyA)

�/Ti

�

(19)835

wherePr;y indicates that the sum consider only the red or836

yellow population. In a symmetric condition between the837

two populations, the solutions of (19) have the property838

NrA D Ny

B ; therefore one reduces to the single equation839

NA DX

i

11 C exp

�(U(N � NA; NA) � U(NA; N � NA))/Ti

� :

(20)840

The obvious solution NA D NB D N/2 exists, but if the841

following condition holds842

14T

@V@NA

ˇˇˇNADN/2

(P) > 1 (21)843

where V(NA) D U(N � NA; NA) � U(NA; N � NA) and844

T is the harmonic mean social temperature of the au-845

tomata population, the symmetry is broken and two new846

solutions appear where the two populations make oppo-847

site choices. Eq. (21) defines a critical temperature T� un-848

der which the phase transition occurs. This phenomenon849

allows one to simulate the herding cooperative effect, ob-850

served in crowd dynamics. In the Fig. 16, Eq. (20) is solved851

graphically for an utility function U(P) D P. The simu-852

lation results show that the statistical self-consistent ap-853

proach (20), allowing one to compute the correct value of854

critical temperature T�. In the simulation the automata855

use a utility function856

U D�M C � � v

N C 1

�; (22)857

where M is a variable depending on the previous choice858

(M D 0 if there is not a previous choice), � � v the flux at859

the door in the direction v of the desired velocity and N860

is the number of automata in a neighborhood of the door, 861

and the social temperature is uniformly spread in the same 862

interval for both the population. The memory M is intro- 863

duced to enforce the choice of two successive doors along 864

the same horizontal line. It is remarkable that there are al- 865

ways automata that move according to a minority choice, 866

simulating the existence of free-will. The effect of a finite 867

population size gives a critical value for N, that has to be 868

overcome for the formation of a self-organized dynamics. 869

An experimental observation of statistical laws in 870

crowd dynamics is very difficult due to the limited space 871

extension of the video movie, whereas most of the data for 872

traffic dynamics are on the road fluxes and not on the in- 873

dividual vehicles trajectories [53,58]. Recently a GPS sys- 874

tem has been set up on '1% of the vehicle population 875

in Italy for insurance reasons, measuring position, veloc- 876

ity and signal quality. The data registration starts every 877

time the engine is switched on, and ends when the en- 878

gine is switched off. The GPS system repetition rate is one 879

second, but the data are recorded each time a vehicle has 880

covered '2 km. The accuracy of GPS data has a standard 881

uncertainty in the position of '10m. The GPS data al- 882

low a reconstruction of the individual dynamics so that 883

it is possible to look for the existence of statistical macro- 884

scopic laws [82]. Different size cities have been considered: 885

a small town, (Senigallia, '5 � 104 inhabitants, near the 886

Adriatic see), a medium city (Bologna, '4 � 105 inhab- 887

itants) and a metropolis (Rome, '3 � 106 inhabitants). 888

The three cities have a very different topology: the Senigal- 889

lia topology is substantially a strip of 2 � 12 km2 bounded 890

by the sea, with few accesses, whereas Bologna has a radial 891

configuration around an historical center with a radius of 892

'6 km. The urban area of Rome is one order of magni- 893

tude larger. To understand the mobility demand [16,29], 894

the distribution of inner vehicle trajectories has been stud- 895

ied: i. e., the trajectories fully contained in the considered 896

urban area. The area of the considered regions are 130 km2897

for Senigallia, 160 km2 for Bologna and 400 km2 (the area 898

containing the “Grande Raccordo Anulare”) for Rome. In 899

Fig. 17 the plot the recorded data in the Bologna urban 900

area for the whole month June 2006 is shown. In Fig. 18 901

the trajectories length distribution (s) is plotted for one 902

month of data recording: the lengths are computed using 903

the polygonal metric in the three considered cases: Rome 904

(February 2007), Bologna (June 2006) and Senigallia (June 905

2006). Even if there are almost two orders of magnitude 906

between Senigallia and Rome’s statistical samples, the path 907

lengths distribution has the same behavior in the three 908

cases. Moreover, it is really remarkable that the same ex- 909

ponential decay for the “long distance” trajectories is de- 910

tected in all the considered cases, indicating a possible uni- 911

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Traffic and Crowd Dynamics: The Physics of the City, Figure 16a Graphical solution of the self-consistent Eq. (20) using an utility function U D NA/N for both choices and a temperature below thecritical threshold 0.5. The solutions are indicated by red circles; beyond the trivial solution P D NA/N D 1/2, two further solutionsexist that breaks the system symmetry. b Self-organized dynamics by simulating two populations moving into opposite direction inthe space sketched in Fig. 2

Traffic and Crowd Dynamics: The Physics of the City, Figure 17The dots are the GPS recorded data in the Bologna urban area during the month of June 2006; the dots correspond to more than1:5 � 106 data. The colors label the recorded velocity: red means v < 30km/h, yellow 30 < v < 60km/h, green 60 < v < 90 andblue 90 < v km/h

versal character at least in the class of Italian (or European)912

cities [82]. The exponential distribution law for individual913

trajectories has a statistical mechanics interpretation as the914

most probable distribution if each citizen has an indepen-915

dent mobility demand and there exists an average length916

for the urban trajectories. This means that the system be- 917

haves statistically as the citizens choose among randomly 918

distributed destinations in the urban space according to an 919

“utility function” (proximity function) that increases lin- 920

early as a function of the trip length [16,29]. Let flkg be 921

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Traffic and Crowd Dynamics: The Physics of the City, Figure 18Monthly length distribution of inner trajectories for the city ofRome (top) during February 2007, and for the cities of Bologna(center) and Senigallia (bottom) during June 2006. The verticalscale is logarithmic, and a polygonal metric for the lengths isused. The total number of trajectories is 1,176,763 for Rome,79,154 for Bologna and 13,838 for Senigallia

the set of all possible lengths of origin-destination trajec-922

tories in the city, and let us suppose that N citizens choose923

randomly the trajectories with the constraint that the total924

covered length is finite925

L DX

k

nk lk ;X

k

nk D N ; (23)926

where nk is the number of citizens choosing a trajectory of927

length lk. A simple combinatorial argument show that the928

statistical weight of the distribution fnkg is929

P(fnkg) / 1n1! � � � � � nk ! � : : :

: (24)930

In the limit nk � 1 using the Stirling’s approximation,931

one obtains932

log P(fnkg) D �X

k

(nk log nk � nk) C const : (25)933

The maximum of the logarithm (25) is computed using934

the Lagrangian multipliers method to take into account935

the constraints:936

@

@nj

"X

k

(nk log nk � nk) C ˛nk C ˇnk lk

#

D 0 (26)937

which implies938

nj D exp(�˛ � ˇ l j) (27)939

where the parameters ˛ and ˇ are determined by the con- 940

ditions (23). In the continuous limit, a statistical distribu- 941

tion (s) / exp(�ˇs) is achieved, consistently with exper- 942

imental data. It is really remarkable that ˇ�1 (i.e the aver- 943

age length of urban trajectories) is the same for very dif- 944

ferent cities (see Fig. 18) [73]; this can be an indication for 945

the existence of an individual mobility demand that can be 946

related to a universal energy law in human travel behavior, 947

according to the law proposed by D. Helbing and R. Kölbl 948

in [64], based on the data of the UK National Travel Sur- 949

veys during the years 1972–1998. Moreover it allows the 950

conjecture that the universal behavior of the length distri- 951

bution Fig. 18 can be explained as the result of a specific 952

cognitive behavior of the populations. In other words, the 953

spatial distance is the norm for the individual driving car 954

decision and the “activities sprawling” inmodern cities ob- 955

served by urban planners [7], introducing a decorrelation 956

in the system, allowing to apply the statistical physics tech- 957

niques. 958

In the case of Rome, the lengths distribution has been 959

also computed using the arc-length metric. The experi- 960

mental results are plotted in Fig. 9 confirming the expo- 961

nential behavior shown in Fig. 18, even if the decay slope 962

is slightly different. This is a remarkable fact since one ex- 963

pects a decreasing of the path-lengths distribution when 964

the length approaches to zero: it should be not convenient 965

to use the car for short travels (<1 km). A possible ex- 966

planation is the presence of a relevant stochastic compo- 967

nents in urban mobility (the so-called “asystematic mobil- 968

ity”), so that the individual mobility becomes the realiza- 969

tion of a sequence of local destinations spread in the ur- 970

ban space. The experimental data in Fig. 9 suggests that 971

the citizens use apparently two linear proximity functions, 972

for the short and for the long length paths. Then the length 973

distribution can be interpreted as an overlap of two expo- 974

nential distributions 975

(s) D A (exp(�ˇls) C c exp(�ˇss)) : (28) 976

The continuous curve in Fig. 9 shows the interpolation re- 977

sults using the parameters: ˇl D :15 km�1, ˇs D :6 km�1978

and c D :7. 979

However the existence of two exponentials curves for 980

the length distribution should be studied more carefully, 981

since the very short trajectories can be interpreted in var- 982

ious ways and the GPS signals have a low quality in such 983

a case. 984

Future Directions 985

From a theoretical point of view, we think that the gas of 986

automata can be a good paradigmatic framework to inves- 987

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Traffic and Crowd Dynamics: The Physics of the City, Figure 19Squares: length distribution of the inner trajectories in Romeduring February 2007, computed using the arc-length metric.Continuous curve: interpolation of the experimental data withthe analytical distribution (28). The interpolated formula hasthe following parameters: ˇl D :15 km�1, ˇs D :6 km�1, andc D :7. The vertical scale is logarithmic

tigate complexity in a general sense. For crowd and traf-988

fic flow, the previous model simulation and empirical re-989

sults permit one to tackle as a non-utopian the problem of990

predictability and governance of the mobility system [26].991

Last but not least the prediction, even if partial or prob-992

abilistic, is relevant not only for the pleasure knowledge,993

but also for the application in order to minimize the im-994

pact of provisional city users on the town and in particular995

on the mobility of local populations; to prevent critical and996

potentially dangerous crowding phenomena; to prefigure997

and plan safety and security saving trajectories.998

Bibliography999

1. Ball P (2003) The PhysicalModelling of Human Social Systems.1000

Complexus 1:1901001

2. Balmer M, Nagel K, Raney B (2004) Large-scale multi-agent1002

simulations for transportation applications. J Intell Transp1003

Syst 8(4):2051004

3. Bandini S, Manzoni S, Vizzari G (2006) Crowd Modeling and1005

Simulation. The role of multi-agent simulation in design sup-1006

port systems. In: Van Leeuwen JP, Timmermans HJP (eds) In-1007

novations in Design & Decision Support Systems in Architec-1008

ture and Urban Planning. Springer, Netherlands, pp 1611009

4. Bando M et al (1995) Dynamical model of traffic congestion1010

and numerical simulation. Phys Rev E 51(2):10351011

5. Batty M (1971) Modelling Cities as Dynamic Systems. Nature1012

231:425–4281013

6. Batty M (2003) Agent-Based Pedestrian Modeling. In: Longley1014

PA, Batty M (eds) Advanced Spatial Analysis. ESRI Press, Red-1015

lands, pp 81–1051016

7. BattyM (2005) Agents, Cells and Cities: New Representational 1017

Models for Simulating Multi-Scale Urban Dynamics. Environ 1018

Plan A 37(8):1373–1394 1019

8. Batty M (2005) Cities and Complexity. Understanding Cities 1020

with Cellular Automata, Agent-Based Models, and Fractals. 1021

MIT Press, Cambridge 1022

9. Batty M, Torrens P (2001) Modeling Complexity: The Limits to 1023

Prediction CASA. Working Paper Series, vol 36. Available on- 1024

line at www.casa.ucl.ac.uk 1025

10. Bazzani A et al (2003) A Chronotopic Model of Mobility in Ur- 1026

ban Spaces. Physica A 325:517–530 1027

11. Bazzani A et al (2007) A Model for Asystematic Mobility in Ur- 1028

ban Space. In: Albeverio S, Andrey D, Giordano P, Vancheri A 1029

(eds) The Dynamics of Complex Urban Systems: An Interdis- 1030

ciplinary Approach, vol 59. Physica, Heidelberg 1031

12. Bazzani A, Rambaldi S, Giorgini B, Turchetti G (2007) Complex- 1032

city: modeling urbanmobility. Adv Complex Syst 10(2):255 1033

13. Bazzani A, Giorgini B, Rambaldi S (eds) (2008) Physics and the 1034

City. Adv Complex Syst 10(2) TS14 1035

14. Bechtel R (1970) Human movement in architecture. In: 1036

Proshansky HM et al (eds) Environmental Psychology. Rine- 1037

hart andWinston, New York 1038

15. Bellman R (1957) Dynamics Programming. Princeton Uni- 1039

vesity Press, Princeton 1040

16. Ben-Akiva M, Lerman SR (1985) Discrete choice analysis the- 1041

ory and application to travel demand. The MIT University 1042

Press, Cambridge 1043

17. Benenson I, Torrens PM (2004) Geosimulation: Automata- 1044

based modeling of urban phenomena. Wiley, London 1045

18. Biroli G (2007) Jamming: A new kind of phase transition? Nat 1046

Phys 3:222 1047

19. Brian WA, Durlauf NS, Lane AD (eds) (1997) The Economy as 1048

an Evolving Complex System II, Proceedings, vol XXVII. Addi- 1049

son-Wesley, Reading 1050

20. Byrne DS (1998) Complexity Theory and the Social Sciences: 1051

An Introduction. Routledge, London 1052

21. Cascetta E (2001) Transportation Systems Engineering: The- 1053

ory and Methods. Kluwer, Boston 1054

22. Chomsky N (2000) New Horizons in the Study of Language 1055

and Mind. Cambridge University Press, Cambridge 1056

23. Chowdhury D, Santen L, Schadschneider A (2000) Statistical 1057

physics of vehicular traffic and some related systems. Phys 1058

Rep 329:199 1059

24. ChowdhuryD, Nishinari K, Schadschneider K (2004) Self-orga- 1060

nized patterns and traffic flow in colonies of organisms. Phase 1061

Transit 77:601624 1062

25. Cuckeqr F, Smale S (2005) Emergent Behavior in Flocks http:// 1063

ttic.uchicago.edu/smale/papers.html 1064

26. Daganzo FC (2007) Urban gridlock: Macroscopic modeling 1065

and mitigation approaches. Transp Res B 41:49 1066

27. de Finetti B (1972) Probability, Induction and Statistics. Wiley, 1067

New York 1068

28. De Martino B et al (2006) Frames, biases and rational decision 1069

making in the human being. Science 313:684–687 1070

29. Domencich TA, Domencich VD (1975) Urban Travel Demand. 1071

A Behavioral Analysis. North Holland, Amsterdam 1072

30. Ellis GRF (2006) Physics and the real world. Found Phys 36– 1073

2:227 1074

31. Fishburn PC (1983) Transitive Measurable Utility. J Econ The- 1075

ory 31:293 1076

32. Fruin JJ (1971) Pedestrian planning and design. Metropolitan 1077

TS14 Please provide page range.


http://www.casa.ucl.ac.uk

http://ttic.uchicago.edu/smale/papers.html

http://ttic.uchicago.edu/smale/papers.html

utente

Evidenziato

Uncor

recte

d Pro

of

2008

-09-

08

��


��


Association of Urban Designers and Environmental Planners,1078

New York1079

33. Garnsey E, McGlade J (eds) (2006) Complexity and Co-evolu-1080

tion Continuity and Change in Socio-Economic Systems. Ed-1081

ward Elgard, Cheetenham1082

34. Giorgini B (2007) Philosophie naturelle de la causalité et du1083

hasard dans un modèle de mobilité urbaine. In: Franceschelli1084

S, Paty M, Roque T (eds) Chaos et systèmes dynamiques. Col-1085

lection Visions des Sciences, vol 259. Editions Hermann, Paris1086

35. Giorgini B, Turchetti G (2005) From Newton-Boltzmann1087

paradigms to complexity: a bridge to biosystems. In: The Sci-1088

ence of Complexity: chimera or reality? Milan Research Cen-1089

tre for Industrial and AppliedMathematics, Esculapio Editore,1090

Bologna, p 181091

36. Giorgini B et al (2008) The Physics of the City: Modeling Com-1092

plex Mobility. Proceedings of the workshop: Urban civiliza-1093

tion: from yesterday to the next day, Milano 24 September1094

2004 (in publication) TS151095

37. Hayakawa H, Nakanishi K (1998) Universal behaviors in gran-1096

ular flows and traffic flows. Prog Theor Phys Suppl 130:571097

38. Heinrich MJ, Nagel RS, Behringer RP (1996) Granular solids,1098

liquids, and gases. Rev Mod Phys 68:12591099

39. Helbing D (2001) Traffic and Related Self-Driven Many-Parti-1100

cle Systems. Rev Mod Phys 73:10671101

40. Helbing D (2004) Dynamic decision behavior and optimal1102

guidance through information services. In: SchreckenbergM,1103

Selten R (eds) Models and experiments in Human Behaviour1104

and Traffic Networks. Springer, Berlin, p 471105

41. Helbing D, Molnár P (1995) Social force model for pedestrian1106

dynamics. Phys Rev E 51:42821107

42. Helbing D, Nagel K (2004) The physica of Traffic and Regional1108

Development TS161109

43. Helbing D et al (2006) Analytical approach to continuous and1110

intermittent bottleneck flows. Phys Rev Lett 97:1680011111

44. Helbing D, Farkas I, Vicsek T (2000) Simulating dynamical fea-1112

tures of escape panic. Nature 407:4871113

45. Helbing D, Farkas IJ, Vicsek T (2000) Freezing by Heating in1114

a Driven Mesoscopic System. Phys Rev Lett 84:12401115

46. Helbing D, Schönhof M, Kern D (2002) Volatile decision dy-1116

namics: experiments, stochastic description, intermittency1117

control and traffic optimization. New J Phys 4:331118

47. Helbing D, Molnár P, Farkas I, Bolay K (2003) Self-organizing1119

pedestrian movement. Environ Plan B 28:3611120

48. Helbing D, IsobeM, Nagatani T, Takimoto K (2003) Lattice gas1121

simulation of experimentally studied evacuation dynamics.1122

Phys Rev E 67:0671011123

49. Helbing D, Buzna L, Johansson A, Werner T (2005) Self-orga-1124

nized pedestrian crowd dynamics: Experiments, simulations,1125

and design solutions. Transp Sci 39–1:11126

50. Helbing D, Johansson A, Al-Abideen HZ (2007) Dynamics of1127

crowd disasters: an empirical study. Phys Rev E 75:0461091128

51. Hoogendoorn SP, Bovy PHL (2001) State-of-the-art of vehicu-1129

lar traffic flow modeling. In: Proceedings of the: Institution of1130

Mechanical Engineers. Part I. J Syst Control Eng 215–4:2831131

52. Hoogendoorn SP, Daamen W (2005) Pedestrian Behavior at1132

Bottlenecks. Transp Sci 39(2):1471133

53. Hoogendoorn SP et al (2003) Microscopic Traffic Data Collec-1134

tion by Remote Sensing. In: Transportation Research Board1135

(TRB) 82nd Annual Meeting. Mira Digital Publishing, St. Louis,1136

p 111137

54. Hoogendoorn SP, Bovy PHL, Daamen W (2002) Microscopic1138

Pedestrian Wayfinding and Dynamics Modelling. In: Schreck- 1139

enberg M, Sharma SD (eds) Pedestrian and Evacuation Dy- 1140

namics. Springer, Berlin, p 123 1141

55. Isobe M, Adachi T, Nagatani T (2004) Experiment and simula- 1142

tion of pedestrian counter flow. Physica A 336:638 1143

56. Jiang B (1998) SimPed: Simulating Pedestrian Flows in a Vir- 1144

tual Urban Environment. J Geogr Inf Decis Anal 3–1:21 1145

57. John A, Schadschneider A, Chowdhury D, Nishinari K (2004) 1146

Collective effects in traffic on bi-directional ant trails. J Theor 1147

Biol 231:279 1148

58. Kerner BS, Klenov SL (2006) Deterministic microscopic three- 1149

phase traffic flow models. J Phys A: Math Gen 39:1775 1150

59. Kirchner A, Nishinari K, Schadschneider A (2003) Friction Ef- 1151

fects and Clogging in a Cellular Automaton Model for Pedes- 1152

trian Dynamics. Phys Rev E 67:056122 1153

60. Kirchner A, Klüpfel H, Nishinari K, Schadschneider A, Schreck- 1154

enberg M (2004) Discretization Effects and the Influence of 1155

Walking Speed in Cellular Automata Models for Pedestrian 1156

Dynamics. J Stat Mech TS17 P10011 1157

61. Klimontovich YL (1999) Entropy, information, and criteria of 1158

order in open systems. Nonlinear Phenom Complex Syst 2– 1159

4:1 1160

62. Kluegl F, Bazzan ALC (2004) Route Decision Behaviour in 1161

a Commuting Scenario: Simple Heuristics Adaptation. In: 1162

SchreckenbergM, Selten R (eds) Human Behaviour and Traffic 1163

Networks. Springer, Berlin, p 285 1164

63. Knospe W et al (2000) CA Models for traffic flow: Comparison 1165

with empirical single-vehicle data. In: Helbing D, Herrmann 1166

HJ, SchreckenbergM,Wolf DE (eds) Traffic andGranular 1999: 1167

Social, Traffic, and Granular Dynamics. Springer, Berlin 1168

64. Kölbl RR, Helbing D (2003) Energy laws in human travel be- 1169

havior. New J Phys 5:48.1 1170

65. Kretz T et al (2006) Experimental study of pedestrian counter- 1171

flow in a corridor. J Stat Mech TS17 P10001 1172

66. Kretz T, Wölki M, Schreckenberg M (2006) Characterizing cor- 1173

relations of flow oscillations at bottlenecks. J Stat Mech The- 1174

ory Exp TS17 P02005 1175

67. Miller JH, Page SE (2007) Complex Adaptive Systems. Prince- 1176

ton Studies in Complexity TS16 1177

68. Millonas MM (1994) Swarms, phase transitions, and collec- 1178

tive intelligence. In: Langton CG (ed) Artificial Life III. Addison 1179

Wesley, Reading, p 418 1180

69. Mitarai N, Nakanishi H (1999) Stability Analysis of Optimal Ve- 1181

locityModel for Traffic andGranular Flowunder Open Bound- 1182

ary Condition. J Phys Soc Jpn 68–8:2475 1183

70. Mizar LF (2007) Agent-Based Modelling and Micro-Simula- 1184

tions of Pedestrian Dynamics and Crowd Phenomena in Pub- 1185

lic Enviroments. PHD Thesis, Università degli Studi, Milano 1186

Bicocca 1187

71. Muramatsu M, Nagatani T (2000) Jamming transition in two- 1188

dimensional pedestrian traffic. Phys A: Statist Mech Appl 1189

275:281 1190

72. Muramatsu M, Irie T, Nagatani T (1999) Jamming transition in 1191

pedestrian counter flow. Physica A 267:487 1192

73. Naess P (2006) Are short daily trips compensated by higher 1193

leisure mobility? Environ Plan B: Plan Des 33:197 1194

74. Nagatami T (2002) The physics of traffic jams. Report Prog 1195

Phys 65:1331 1196

75. Nagel K, SchreckenbergM (1992) A cellular automatonmodel 1197

for freeway traffic. J Phys I 2:2221 1198

TS15 Please update if possible.TS16 Please provide more details.TS17 Please provide volume number.


Uncor

recte

d Pro

of

2008

-09-

08

��


��


76. Nicolis G, Prigogine I (1989) Exploring Complexity: An Intro-1199

duction. Freeman, New York1200

77. Parisi G (1993) Statistical Physics and Biology. PhysicsWorld1201

Arch 6:421202

78. Parisi G (1999) Complex Systems: a Physicist’s Viewpoint. Phys1203

A 263–1:5571204

79. Portugali J (2005) Cognitive Maps Are over 60. COSIT 2511205

80. Prigogine I, Andrews FC (1960) A Boltzmann like approach for1206

traffic flow. Oper Res 8:7891207

81. Pushkarev B, Zupan JM (1975) Urban space for pedestrians.1208

The MIT Press, Cambridge1209

82. Rambaldi S et al (2007) Mobility in modern cities: looking for1210

physical laws. Proceeding of the ECCS07 Conference, Dres-1211

den, 1–3 October 2007, paper n. 1321212

83. Rizzolati G (1997) From Complexity to Creativity. Plenum1213

Press, New York1214

84. Roederer GJ (2003) On the concept of information and its role1215

in Nature. Entropy 5:31216

85. Schadschneider A, Pöschel T, Kühne R, Schreckenberg M,1217

Wolf DE (eds) (2007) Traffic and Granular Flow ’05. Springer,1218

Heidelberg1219

86. Schreckenberg M, Neubert L, Wahle J (2001) Simulation of1220

traffic in large road networks. Futur Gener Comput Syst 17–1221

5:6491222

87. Schweitzer F (2003) Brownian Agents and Active Particles.1223

Springer, Berlin1224

88. Tajima Y, Takimoto K, Nagatani T (2001) Scaling of pedestrian1225

channel flow with a bottleneck. Physica A 294:2571226

89. Tajima Y, Takimoto K, Nagatani T (2002) Pattern formation1227

and jamming transition in pedestrian counter flow. Physica1228

A 313:7091229

90. Taylor M (2003) The Moment of Complexity. University of1230

Chicago Press, Chicago1231

91. Treiber M, Hennecke A, Helbing D (2000) Congested Traf-1232

fic States in Empirical Observations and Microscopic Simula-1233

tions. Phys Rev E 62:18051234

92. Turchetti G, Zanlungo F, Giorgini B (2007) Dynamics and ther-1235

modynamics of a gas of automata. Europhys Lett 78–5:580031236

93. Van Gelder T (1998) The dynamical hypothesis in cognitive1237

science. Behav Brain Sci 21:6151238

94. Vogel A, Nagel K (2005) Multi-agent based simulation of in-1239

dividual traffic in Berlin. Comput Urban Plan Urban Manag1240

(CUPUM2005) 176 TS161241

95. Von Neumann J (1963) The general and logical theory of au-1242

tomata. In: Collected works, vol V. Pergamon Press, Oxford,1243

p 2881244

96. Von Neumann J, Morgenstern O (1947) Theory of Games and1245

Economic Behavior. Princeton University Press, Princeton1246

97. Wahle J et al (2000) Decision Dynamics in a Traffic Scenario.1247

Adv Complex Syst 2:11248

98. Weufeng F, Lizhong Y, Weicheng F (2003) Simulation of bi-1249

directional pedestrian movement using a celluar automata1250

model. Physica A 321:6331251

99. www.physicsofthecitylab.unibo.it1252

100. Yamori K (1998) Goingwith the flow:Micro-nacro dynamics in1253

the macrobehavioral patterns of pedestrian crowds. Psychol1254

Rev 105–3:5301255

http://www.physicsofthecitylab.unibo.it

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