Random belief system dynamics in complex networks under time-varying logic constraints

People change their beliefs randomly such that both the structure and the properties of the belief system vary with time under several conditions, which is difficult to describe mathematically due to the complexity of this randomness. A precise belief system dynamics can not only help us to identify which properties the belief system has and forecast how it moves, but it can also help the belief system to move in a reasonable direction. To find this dynamics, a generalized Friedkin model with time-varying parameters was constructed to analyze the following events: (1) inter-dependent issues, (2) heterogeneous systems of issue dependency constraints, and (3) inter-dependent issues under three types of systems. These systems are characterized with: (1) parameter variations and distributions being only assumed to be bounded, (2) zero mean random parameter variations and disturbances being, in general, a correlated process, and (3) the parameter process {Δk} being a random walk. The dynamics is then obtained by invoking LMS algorithms due to the property of the time-varying parameter. Parameters of the dynamic statistical model produce large enough random matrices with random time-varying structure. The random average theorem was then introduced to transform these matrices to deterministic ones under four strict conditions such that the dynamics could be estimated accurately. The stability conditions corresponding to the belief system dynamics are then provided, in case the belief system is not complex. By contrast, if the belief system is more complex, a corresponding multi-agent computational model is constructed to explain how a more unstable belief system would move. The results show that, the parameter estimated via LMS is relatively deterministic if the issues are inter-independent and the parameter variation is smooth, and vice versa. It is thus possible to conclude that there exists a critical point of the parameter variation such that the analytic solution of the dynamics could be obtained if the complexity is simpler than the critical point. Otherwise, the belief system is unstable and uncontrollable. In the latter case, the dynamics just relies on the parameters statistical property if the beliefs or issues are inter-independent and the dynamics of belief system is more complex. Furthermore, there would exist a phase transition for the belief system such that the minority view would become the dominant view under certain conditions, if the beliefs or issues are inter-dependent. It is concluded that both the system structure and the belief configuration are important to the belief dynamics. The two most important findings of this work are: (1) the stochastic time-varying model matches the property of the belief system and can thus be used to discover more interesting results and (2) the estimation method of the LMS driven by the random average theorem can be generalized to almost all social systems, if the parameters changes are not too violent, while the multi-agent simulation can be used if the parameters change more strongly. These results reveal the law of the essence of economic and management complex adaptive systems.