Different Phases in a Supermarket Chain Network: An Application of an Ising Model on Soap Froth

The competition and cooperation between two supermarket chains in a given two-dimensional point pattern is investigated in the framework of a Multi-Agent System. Each point represents a shopping mall, which can contain only one of the two supermarket chains. The point can be associated with a cell via a Voronoi contruction, with the cell representing the domain of a customer base associated with the shopping mall. For scientific reference, we employ soap froth, a standard physical system that possessesscaling property in the steady state as our given two-dimensional cellular network. The area and perimeter of the cell are related to the preference of the customers based on the geometric location of the shopping malls. The color of the cell denotes the choice of the supermarket chain, while the competition and cooperation between the two chains are manifest in the evolution of the color on the static cellular network. This problem of color evolution corresponds to the statistical mechanics of an Ising model defined on a given point pattern and we explore the phases of this model by Monte Carlo simulation. We find evidence of two distinct phases: a random phase and a cluster phase. Verification of the phases and their interpretations are made using an analysis of the interior bubble distribution function. Copyright Kluwer Academic Publishers 2003