Journal on Policy and Complex Systems
exhibits a behavior that is neither completely stable nor completely chaotic , generating localized structures that appear and interact in various complicated-looking ways , which characterizes what is called Class 4 behavior .
Keeping that in mind , the main idea of this second experiment is to show that there is another interesting possible source of complexity in terms of the rise of long memory processes : large deviations from stationary states implied in their collective motion laws .
By treating cellular automata as stochastic dynamic systems , it is possible to show that individual transition rules lead to collective motion laws . On the other hand , depending on the initial conditions ( for simplicity ) large deviations from the equilibrium points may be induced , generating collective smooth regime switches ( which are not intuitive ). Hence , these switches toward their respective equilibrium may generate long-range dependency . Furthermore , as it is shown later , there are some rules that , despite regime switches induced by the configuration of initial conditions , have convergence rates with different speeds , affecting the rise of long-range dependency .
Therefore , in order to build such exercise , the first step is to aggregate all the black cells ( represented by ones ) in each time step , given the fact that they are randomly started according to a binomial probability function in two distinct cases :
• p 0
= p * ;
• and p 0
≠ p * ; where p * denotes a stable fixed point ( that is going to be introduced later ). Consequently , if p 0
= p *, no regime switches are expected ; if p 0
≠ p *, regime switches should be expected since , as the time passes , p t should converge to p * . Consequently , p t also denotes the expected value of the density of black cells at each time step .
Moreover , in order to establish this experiment as a tool to investigate possible sources of long memory properties , if one considers the following expression :
where x i , t represents the value of the cell at position i at time step t , S ( t ) represents the amount of black cells present at each time step and , over this number , long-range dependency statistics are calculated .
The reason to use such measurement is the fact that the Central Limit Theorem ensures that the variable S ( t ) is normally distributed for a sufficiently large N if x i , t is binomially distributed . One should also notice that if p t is a stable fixed point , in theory , S ( t ) is also stable normal distributed over time . Hence , it is expected that the sequence of normal variables is that they will compose a stable white noise process , i . e . a short memory process .
On the other hand , if p t is not a stable fixed point , following the assumptions made , a sequence of regime switches happens until the system reaches its equilibrium point . As it is known that regime switches may cause
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