Mexican wave: Crowd behaves as excitable medium

Background

People: an excitable medium performing the Mexican wave

The term "Mexican wave" was first 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 scientist, the interesting specific 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.

Simulation models

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 above "simulation: Mexican Wave").

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:

    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).

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 − Δc, c + Δc).

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 p₀₋₁. 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 p₁₋₂ at each simulation update, and the transition from the refractory to the inactive state with probability p_2-0.