Pareto-Zipf law in growing systems with multiplicative interactions

Numerical simulations of multiplicatively interacting stochastic processes with weighted selections were conducted. A feedback mechanism to control the weight w of selections was proposed. It becomes evident that when w is moderately controlled around 0, such systems spontaneously exhibit the Pareto-Zipf distribution. The simulation results are universal in the sense that microscopic details, such as parameter values and the type of control and weight, are irrelevant. The central ingredient of the Pareto-Zipf law is argued to be the mild control of interactions.