Threshold Models of Collective Behavior
The Java applet below visualizes the dynamics of a Threshold Model of Collective Behavior. The interaction dynamics is based on the model introduced by Mark Granovetter [1] . The Model The system consists of N agents placed at the nodes of a two dimensional network with periodic boundary conditions and Moore's neighborhood (k = 8). Each agent i can choose between two possible states, denoted by S_i=+1 and S_i=-1. Each agent is assigned a threshold 0 < T < 8 which determines the number of neighbors required for agent i to change her state. For T = 4 this dynamics coincides with the Spin Flip Kinetic Ising model at zero temperature. The system evolves by iterating the following time steps: 1) A agent i is randomly chosen in the lattice (called active agent). 2) The threshold value of the active node is assigned at random. This threshold follows a Gaussian distribution with "T" as mean value and standard deviation "S" 3) If the the number of neighbors of agent i in a state different that S_i is larger than the threshold "T", then the active agent changes its state (S_i) to (-S_i). Otherwise it keep the state. In the case that the fraction of neighbors in a different state is equal to the threshold "T", then with probability 0.5, the active agent changes its state.

Threshold Models of Collective Behavior

The Java applet below visualizes the dynamics of a Threshold Model of Collective Behavior. The interaction dynamics is based on the model introduced by Mark Granovetter [1] .

The Model

The system consists of N agents placed at the nodes of a two dimensional network with periodic boundary conditions and Moore's neighborhood (k = 8). Each agent i can choose between two possible states, denoted by S_i=+1 and S_i=-1. Each agent is assigned a threshold 0 < T < 8 which determines the number of neighbors required for agent i to change her state.
For T = 4 this dynamics coincides with the Spin Flip Kinetic Ising model at zero temperature.

The system evolves by iterating the following time steps:

1) A agent i is randomly chosen in the lattice (called active agent).

2) The threshold value of the active node is assigned at random. This threshold follows a Gaussian distribution with "T" as mean value and standard deviation "S"

3) If the the number of neighbors of agent i in a state different that S_i is larger than the threshold "T", then the active agent changes its state (S_i) to (-S_i). Otherwise it keep the state.

In the case that the fraction of neighbors in a different state is equal to the threshold "T", then with probability 0.5, the active agent changes its state.

A guide for using the applet

+ and - buttons increase/decrease the basic parameters of the model.

L is the linear size of the square lattice with periodic boundary conditions.

T (Threshold) is the number of neighbors in a different state required for the active agent to change her state.

Time is the number of full interactions divided by the system size: In one time unit each site performs, on the average, one update trial.

Start/stop allows you to toggle the state of the applet between "running" and "paused".

The fast option makes the dynamics run as fast as possible, slow options reduce the speed. Note that system time is not necessarily proportional to real time. In particular with the fast option this ratio varies immensely.

Initial conditions buttons:

Random: state (+1 white and -1 white) of each agent is randomly distributed. The density of state -1 is controled with the slide boton.

Strip-like: Half of the lattice is in state +1 (white), while the other half is in state -1 (black).

Bubble initial configuration consists of two zones with state (-1) and (+1) with a circle-shaped interface.

S: Standard Deviation of T.

OUTPUT.

(P+). Density of nodes with state +1.

(P_ ). Density of nodes with state -1.

(P). Density of links that connect nodes with different states.