Phase diagram of a model of self-organizing hierarchies

We introduce a simple model of self-organizing hierarchies in animal societies which relies on a basic positive feedback mechanism reinforcing the ability of a given individual to win or lose in a hierarchical interaction, depending on how many times it won or lost in previous interactions. If a forgetting strength is included, which determines the rate at which events in the past are forgotten and no longer influence the force of an individual, subcritical or supercritical bifurcations in the formation of the hierarchical structure are observed as the density p of individuals is varied. The nature of the transition is shown to depend on a parameter ~/, analogous to the inverse of a temperature, defining the amount of determinism in the outcomes of the fights. We therefore observe a dynamical tricritical point in the p-r/plane.