1.  The simulation models

To describe the simulation models, we show here a small section of the tribune during an actual simulation (10 rows out of 80 and 16 "columns" of seats out of 800). The wave was triggered left from this section. By the time it has reached the section shown, the wave has already grown to the width of the entire tribune. The simulation shown here uses version A of the Detailed n-state model (see below).

 
inactive
(sitting)
     
active
(moving up)
     
refracter
(moving down or sitting)

The mexican wave can be decomposed into two phenomena: (i) the formation of the wave and (ii) 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 neighbors — may quickly spread out. 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 amplified, 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., neighbors jumping to their feet, are explained below.

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 3 out of 4 mexican waves move in the clockwise direction, while 1 out 4 waves moves in the anti-clockwise direction. Being aware that numerous psychological and physical phenomena influence 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.

We have used three mexican wave models:

  • Detailed n-state model / version A
  • Detailed n-state model / version B
  • Minimal 3-state model

2.  Identical features of the models

2.1  General description

At time t person i can be in one of the following states: inactive, active, 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 influenced 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 influenced (activated) only in the inactive state, and they influence other people only if they are in the active state:

state can be
activated ?
can activate
others ?
inactive yes no
active no yes
refracter no no

During a simulation update first the weighted concentration of active people within a fixed radius around each person is compared to the 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 ` \tau ` 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 influenced inactive person to the influencing active person. People exactly to the left from a person have an influence ` w_0 ` times as strong as those exactly to the right (` w_0<1 `). Weights decay exponentially with distance: the decay length is R, and there is a cutoff at 3R. The sum of weights is 1 for each person.

2.2  Detailed description

We are using a rectangular simulation area with 80 rows and 800 "columns" of seats (units, or people). The simulation is visualized 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 fixed 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).

The angle of the vector ` \vec{r}_{ij} `, pointing from person ` i ` to person ` j `, is ` \varphi_{ij} `. The angle of the vector (1,0) is ` \varphi_{ij}=0 `, and the angle of the vector $(0,1)$ is ` \varphi_{ij}=\pi/2 `. The interaction decay length is ` R `, and there is a cutoff at ` 3R `. If person ` j_{Left} ` is to the left from person ` i ` and person ` j_{Right} ` to the right at the same distance, then the ratio of influence strengths from ` j_{Left} ` vs. from ` j_{Right} ` is ` w_{0} `. If one compares all possible directions, the strongest influence comes from the right (the direction ` \varphi_{ij}=\pi `), and the weakest influence from the left (the direction ` \varphi_{ij}=0 `), and between them the strength changes with the cosine of ` \pi-\varphi_{ij} `.

In summary, the weight of the `j`th person's influence on the `i`th person is
` w_{ij}=K_i^{-1} \exp( -|\vec{r}_{ij}| / R) [ 1 + w_0 + (1-w_0)\cos(\pi-\varphi_{ij})] `, if ` 0<|\vec{r}_{ij}|<3R ` and ` w_{ij}=0 ` otherwise. The normalizing constant, ` K_i ` is defined such that the sum of all influence weights on person ` i ` is ` \sum_j w_{ij} = 1 `. Finally, the weighted concentration of active people "felt" by person ` i ` is a sum for all active ` j ` other people: ` W_{i}=\sum_(j_{Active}) w_{ij} `.

In all three models, the activation of person ` i ` depends on whether ` W_{i} ` exceeds the activation threshold of this person, ` c_i `. The reaction time of a person is ` \tau `. This is the time difference between two updates of ` W_i `.

3.  Differences between the models

3.1  The detailed n-state model

This model divides the active state into ` n_a ` substates and the refractory state into ` n_r ` substates. Once activated, a person will deterministically step through the $n_a$ active states, then through the ` n_r ` refractory states, and last it will return to the inactive state.

Activation in version A of the detailed n-state model

If the weighted concentration of neighboring 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-\Delta c `, `c + \Delta c` ).

Activation 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 neighboring active people (` W_i `) is below ` c `, then the person can be activated spontaenously with probability ` p_{Spont} `. If `W_i>c`, then the activation of the person will be induced with probability `p_{Ind}`.

3.2  The minimal 3-state model

If the weighted concentration of neighboring active people around a person is below ` c `, then the person is activated spontaneously with probability `p_{0-1}`. If `W_i>c`, then it is activated deterministically (with probability 1). The transition from the active to the refractory state (i.e., from state 1 to state 2) takes place with probability `p_{1-2}` at each simulation update, and the transition from the refractory to the inactive state with probability `p_{2-0}`.