  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.
