Dynamical toy model of interacting $N$ agents robustly exhibiting Zipf's law

We propose a dynamical toy model of agents which possess a quantity and have an interaction radius depending on the amount of the quantity. They exchange the quantity with agents existing within their interaction radii. It is shown in the paper that the distribution of the quantity of agents is robustly governed by Zipf's law for a small density of agents independent of the number of agents and the type of interaction, despite the simplicity of the rules. The model can exhibit other power laws with different exponents and the Gaussian distributions. The difference in the mechanism underlying Zipf's law and other power laws are studied by mapping the systems into graphs and investigating quantities characterizing the mapped graph. Thus, this model suggests one of the origins of Zipf's law, i.e., the most common fundamental characteristics necessary for Zipf's law to appear.