System reduction: an approach based on probabilistic cellular automata

The goal of this paper it to explore the possibility to simplify a stochastic complex dynamical system by reducing the number of its degrees of freedom, through a coarse graining procedure. Our objective is to create reduced systems which requires less computational burden to control. In other words, the question is whether one can act on the simplified system to control its evolution towards a given target, so that the same strategy also applies to the full system. We analyze this problem in three stages. First we consider the general case of coarse-graining 1D cellular automata. We show that the procedure is expected to produce extra stochasticity in the system, while still retaining some characteristics of the original system. In a second step, we study a 1D voter model for which the coarse-graining procedure requires component weighting, with weights based on causal paths. Finally, in a third part, we consider the same voter model on a scale-free graph, and perform a reduction according to a partition of the graph in communities. The equivalence between the reduced and full systems, whether controlled or not, is investigated through a delayed mutual information analysis, a concept from information theory.