Spin-type model for interacting agents with memory

This paper presents a model and a solution method to investigate the emergence of self-organization in populations where interaction between members is combined with individual memory of previous behavior. The Markov model is second order to accommodate an extended memory range including the previous as well as the next-to-previous time instant. The proposed solution method leads to a set of recurrence relations between the present and previous macrostates of the system and the equilibrium distribution of the population is then obtained as the stable fixed points of these recurrences. The approach applies to a category of interaction mechanisms and is illustrated here on a simple example showing how memory allocation may affect the emergence of a phase transition in the collective behavior of the population.