Cluster fluctuation in two-dimensional lattices with local interactions

It is known that under the local interactions of agents, we find cluster formations of states of agents, but few works exist on the changes (fluctuations) of clusters. This paper deals with the analysis of fluctuations of clusters with respect to states of agents placed on two-dimensional lattices having local interactions. We use two models; first we use Model S that consists of agents producing goods, whose states denote the activity of agents, as well as the accompanying multiple states. An agent decides the states in the subsequent time period by monitoring the states of neighboring agents. Then, we introduce Model P to cope with cases that are more realistic. In Model P, we assume that there exists a group of agents comprising a pool of labor agents and several firm agents collaborating in production, where the labor agents are free to move neighborhoods seeking better incomes. We find cluster formation and equilibrium in both models after the passage of time, but these cluster formations deform and become other formations by changing certain states of agents into random states, or by changing parameters to be random. In simulation studies, we show the conditions for the fluctuations of clusters and possible real applications.