Long-range coherent transmission of informations for collective dynamics in discrete-time models

The conditions on the stationary and the state-varying transition probabilities, which lead to the asymptotic occurrence of uniform collective behaviors in the corresponding discrete-time Markov chains, are established and analyzed. The conditions imply the existence of long-range coherent transmission of informations arising from the local communications. Based on the conditions, a measure, called the minimal time-interval of coherent transmission of informations, is introduced to characterize the speed of convergence towards a uniform collective behavior qualitatively. The numerical calculations on the proposed models indicate that the measure is consistent with the geometric distance of the corresponding network of dynamical system.

Automated summary

This paper establishes and analyzes the conditions on the stationary and state-varying transition probabilities in the corresponding discrete-time Markov chains, which lead to the occurrence of uniform collective behaviors. It introduces a new measure called the minimal time-interval of coherent transmission of informations to qualitatively characterize the convergence speed towards a uniform collective behavior. The numerical calculations on the proposed models show that this measure is consistent with the geometric distance of the corresponding network of dynamical system.