Human waves in stadiums

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Physica A 330 (2003) 18 – 24
Human waves in stadiums
I. Farkasa , D. Helbingb , T. Vicseka;∗
a Department
of Biological Physics and Biological Physics Research Group of HAS, Eotvos University,
PÃazmÃany Peter Setany 1A, Budapest H-1117, Hungary
b Institute for Economics and Tra c, Dresden University of Technology, Dresden D-01062, Germany
Mexican wave ÿrst widely broadcasted during the 1986 World Cup held in Mexico, is a
human wave moving along the stands of stadiums as one section of spectators stands up, arms
lifting, then sits down as the next section does the same. Here we use variants of models
originally developed for the description of excitable media to demonstrate that this collective
human behaviour can be quantitatively interpreted by methods of statistical physics. Adequate
modelling of reactions to triggering attempts provides a deeper insight into the mechanisms by
which a crowd can be stimulated to execute a particular pattern of behaviour and represents a
possible tool of control during events involving excited groups of people.
Interactive simulations, video recordings and further images are available at the webpage
dedicated to this work:
c 2003 Elsevier B.V. All rights reserved.
PACS: 05.40.−a; 87.23.Ge; 05.45.Yv; 05.65.+b; 82.40.Ck
Keywords: Excitable media; Collective motion; Human behaviour
1. Introduction
The application of ideas, methods and results of statistical physics to a wide range of
phenomena occurring outside of the realm of the non-living world has gained a great
momentum recently. Among many others, examples include the studies of various
group activities of people from the physicist’s viewpoint. Here, we give an account of
an investigation in this direction, involving the interpretation of a wave spontaneously
produced by a very large number of people.
Corresponding author.
E-mail address: [email protected] (T. Vicsek).
0378-4371/$ - see front matter c 2003 Elsevier B.V. All rights reserved.
I. Farkas et al. / Physica A 330 (2003) 18 – 24
Our motivation originates in the observation that there has been a growing interest
in a better, more exact understanding of the mechanisms underlying the main processes
in societies. There is a clear need for the kind of ÿrm, reliable results produced by
natural sciences in the context of the studies of human behaviour [1]. The revolution in
information and transportation technology brings together larger and larger masses of
people (either physically or through electronic communication). New kinds of communities are formed, including, among many others, internet chat groups or huge crowds
showing up at various performances, transportation terminals or demonstrations. The
way people behave in stadiums represents a well deÿned and relatively simple example
for related collective human behaviour.
The term “Mexican wave” was ÿrst used during the 1986 World Cup held in Mexico
and since then it has become a paradigm for a variety of phenomena in which an initial
perturbation propagates in the form of a single “planar” wave. The increasing popularity
of this phrase, also known as La Ola, is due to its unique origin; it means a human
wave moving along the stands of stadiums as one section of spectators stands up, arms
lifting, then sits down as the next section does the same. The corresponding wave at a
given instance involves a coordinated behaviour of hundreds, and all together tens of
thousands of people as the wave makes its turn around the stadium.
For a physicist, the interesting speciÿc feature of this spectacular phenomenon is
that it represents perhaps the simplest spontaneous and reproducible behaviour of a
huge crowd with a surprisingly high degree of coherence and level of cooperation. In
addition, La Ola raises the exciting question of the ways by which a crowd can be
stimulated to execute a particular pattern of behaviour.
2. Observations
We have collected video recordings of waves in stadia holding over 50,000 people.
We found that the Mexican wave can be decomposed into two phenomena: the formation of the wave and the propagation of the wave. The wave is usually formed by
the simultaneous standing up of a small number of spectators. The spectators standing up in this group represent a small perturbation (excitation), which—depending on
their neighbours—may quickly spread out. However, this spreading will occur only,
if the audience is in the “appropriate mood” for a wave: not too bored, but not too
excited either. In other words: the audience should be sitting, but “excitable”. If the
perturbation is ampliÿed, then—according to our observations (14 videos of Mexican
waves)—during the following short time interval (less than a second) the wave will
die out on one side, and survive on the other. The parameters describing the reactions
of spectators to nearby excitations, e.g., neighbours jumping to their feet, is described
The spontaneous symmetry breaking observed during the formation of the wave is
probably due not only to the asymmetrical nature of human perception, but also to
the expectations of spectators with previous experience on Mexican waves and other
forms of collective human behaviour. According to video recordings we have analysed,
approximately three out of four Mexican waves move in the clockwise direction, while
I. Farkas et al. / Physica A 330 (2003) 18 – 24
distance [seats]
t [s]
Fig. 1. Speed of the Mexican wave. Each point shows the distance between the ÿnal and starting position of
a wave on one of the video recordings versus the time needed by the wave to cover that distance. A linear
ÿt, s = vt + s0 , to the data points is shown with a solid line. The ÿt parameters are v = (22 ± 3) seats/s and
s0 = (−25 ± 37) seats.
one out of four waves moves in the anti-clockwise direction. Being aware that numerous
psychological and physical phenomena in uence this symmetry breaking, we decided
that for our simulations we will use the simplest model that can reproduce the (i)
formation of the wave by triggering and spontaneous symmetry breaking and (ii) the
stable propagation of the wave.
To measure the speed of the waves, we did the following. For each wave, we noted
the wave’s starting and ÿnal position on the recording and measured the time that the
wave needed to cover this distance. In Fig. 1, the distance travelled by each wave is
plotted against the time di erence between the starting and ÿnal position of the wave
on the video. Out of the 14 recorded waves available to us nine were of a quality
that allowed for the evaluation of their speed. (The remaining ÿve waves were either
recorded for a very short time or had irregular shapes.)
As shown in Fig. 1, we have found that the number of seats travelled by a wave is—
within experimental error—a linear function of the elapsed time, and the speed is
22±3 seats=s. In addition to measuring the speed of the waves and ÿnding the direction
of propagation, we recorded that the typical width of a wave was 6 –12 m (equivalent
to an average width of approximately 15 seats).
3. Simulation models
The relative simplicity of the phenomenon allowed us to develop microscopical models of the motion of inidividuals, and to reproduce the observed macroscopical quantities with these models. We have found that well-established approaches to excitable
media [2–5] such as forest ÿres or wave propagation in heart tissue can be generalized
I. Farkas et al. / Physica A 330 (2003) 18 – 24
Table 1
States of a spectator: inactive, active or refracter
Can be activated?
Can activate?
to include the behaviour of human crowds, and for this reason we started building
our models from the descriptions used in these ÿelds. Interactive simulations, video
recordings and further images are available in Refs. [6,7].
3.1. General description of the models
At time t person i can be in one of the following states: inactive, active or refracter.
In the inactive (ground) state a person is sitting, and can be activated. An active person
is moving upwards (standing up) and is not in uenced by others. In the refractory state
a person is sitting down or already sitting while passively recovering from the previous
activity. In summary, people can be in uenced (activated) only in the inactive state,
and they in uence other people only if they are in the active state (see Table 1).
During one simulation update ÿrst the weighted concentration of active people within
a ÿxed radius around each person is compared to that particular person’s activation
threshold. Next, if a person is inactive, then he/she can be activated using the result of
this comparison and the respective model’s activation rule. Each person needs a reaction
time to update the weighted concentration of active people around him/herself.
Weights are proportional to the cosine of the direction angle of the vector pointing
from the in uenced inactive person to the in uencing active person. People exactly to
the left from a person have an in uence w0 times (w0 ¡ 1) as strong as those exactly
to the right. Weights decay exponentially with distance: the decay length is R, and
there is a cuto at 3R. The sum of weights is one for each person.
3.2. Detailed description of the models
We are using a rectangular simulation area with 80 rows and 800 “columns” of
seats (units, or people). In the x direction (the tangential direction in the stadium)
boundary conditions are periodic, and in the y (radial) direction boundary conditions
are non-periodic. The simulation is visualised by arranging the spectators evenly on
the stadium-shaped tribune.
Each person is facing the inside of the stadium. The coordinate system is always
local, and it is ÿxed to a person such that the vector (1; 0) points towards the left of the
person (i.e., parallel to the rows of the stadium in the clockwise direction, if viewed
from above), and the vector (0; 1) points towards the back of the person (radially out
from the stadium, if viewed from above).
I. Farkas et al. / Physica A 330 (2003) 18 – 24
The angle of the vector ˜r ij , pointing from person i to person j, is ’ij . The angle
of the vector (1; 0) is ’ij = 0, and the angle of the vector (0; 1) is ’ij = =2. The
interaction decay length is R, and there is a cuto at 3R. If person jleft is to the left
from person i and person jright to the right at the same distance, then the ratio of
in uence strengths from jleft vs. jright is w0 . If one compares all possible directions,
the strongest in uence comes from the right (the direction ’ij = ), and the weakest
in uence from the left (the direction ’ij = 0), and between them the strength changes
with the cosine of − ’ij .
In summary, the weight of person j’s in uence on person i is
wij = Si−1 exp(−|˜r ij |=R)[1 + w0 + (1 − w0 ) cos( − ’ij )] ;
if 0 ¡ |˜r ij | ¡ 3R; and wij = 0 otherwise :
The normalizing constant, Si is deÿned such that the sum of all in uence weights on
person i is 1:
wij :
The weighted concentration of active people felt by person i is
Wi =
wij :
j active
In all three models, the activation of person i depends on whether Wi exceeds the
activation threshold of this person, ci . The reaction time of a person is . This is the
time di erence between two updates of the sum of weights, Wi , felt by the ith person.
3.3. Variants of the model
We have used three variants of the Mexican wave model: (i) version A of the
detailed n-state model, (ii) version B of the detailed n-state model and (iii) the minimal
three-state model.
The detailed n-state model divides the active state into na substates and the refractory
state into nr substates. Once activated, a person will deterministically step through the
na active states, then through the nr refractory states, and last it will return to the
inactive state.
In version A of the detailed n-state model a person will be activated under the
following conditions. If the weighted concentration of neighbouring active people is
above a person’s threshold, then the person is activated. Individual activation thresholds
are used for each person. The threshold values are randomly and uniformly selected
from the interval (c− c; c+ c). (In the parameter window of the simulation program
c is written as “delta c”.)
In version B of the detailed n-state model the activation threshold of person i is c (a
constant between 0 and 1). If the weighted concentration of neighbouring active people
(Wi ) is below c, then the person can be activated spontaneously with probability pspont .
If Wi ¿ c, then the activation of the person will be induced with probability pind .
I. Farkas et al. / Physica A 330 (2003) 18 – 24
Fig. 2. Simulation model of the Mexican wave. The interactive simulations that were used to produce this
image are available in color at The simulation is shown 1, 2, 3, 5 and
15 s after the triggering event. The darkest color shows inactive people. The color of active people turns
gradually lighter as they stand up, and the color of inactive people turns again darker, as they sit down. The
simulation model used for this image was obtained with version A of the detailed n-state model. Parameter
values were na = nr = 5, c = 0:25, c = 0:05, R = 3, w0 = 0:5 and = 0:2s. See the detailed n-state model
in Section 3.3 below for the explanation.
In the minimal three-state model, if the weighted concentration of neighbouring
active people around a person is below c, then the person is activated spontaneously
with probability p01 . If Wi ¿ c, then activation will be deterministic (with probability
1). The transition from the active to the refractory state takes place with probability
p12 at each simulation update, and the transition from the refractory to the inactive
state with probability p20 (see Fig. 2).
4. Results
As an application, we computed the conditions necessary for the triggering of a
Mexican wave in a stadium. The size of the stadium we analyzed was 80 rows of
seats with 800 seats in each row. For the simulation, we used a square lattice of size
80 × 800 instead of the stadium-shaped tribune. The stadium shape was used only for
visualization purposes. Each ÿeld of the square lattice contained one seat (representing
one spectator in the stadium) and the simulation was updated with a parallel update
We computed the dependence of the probability of triggering a wave as a function
of two di erent pairs of parameters. (In both cases, we used version A of the detailed
n-state model.) First, we changed the size of the triggering group (N ) and the activation
threshold (c) and found that at N = 30; c = 0:25 a wave could be triggered with almost
unit probability [6]. Other parameters were na = nr = 5; c = 0:05; R = 3; w0 = 0:5 and
= 0:2. During the second analysis simulations were started with the same parameters
as in the ÿrst one except for the two variables: the activation threshold (c) and the
variations of the activation threshold ( c). The N = 30; c = 0:25; c = 0:05 point
is shown again as the threshold for wave triggering and a higher variability in the
thresholds can maintain waves at lower average thresholds (see Fig. 3). The explanation
for this might be quite simple: even when the average activation threshold is low, a
small number of people that are easy to activate can strongly increase the chances for
the survival of a perturbation in the form of a full-grown wave.
I. Farkas et al. / Physica A 330 (2003) 18 – 24
Fig. 3. The ratio of successful triggering events, P(c; c), where P is the probability that a wave can be
induced; c and c are the average activation threshold and the di erences in activation thresholds among
people (for a more detailed deÿnition, see version A of the detailed n-state model in Section 3.3). Other
parameters were identical to those of Fig. 2. Our (numerical) deÿnition for the necessary condition of the
survival of a wave was the following: 20 s after the triggering event there has to be a column of spectators
(a line of spectators sitting behind each other) out of whom at least 50% are either active or refracter (i.e.,
not passive). Interactive simulations are available at
This research has been supported by the Hungarian Scientiÿc Research Fund under
Grant No. OTKA 034995.
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[7] The webpage dedicated to the present work o ers further data and
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