Optimal control of stochastic fractional rumor propagation model in activity-driven networks

Abstract

Activity-driven networks have become a key paradigm for studying the time evolution of stochastic networked systems. Consider the fact that individuals in activity-driven networks have some degree of memory, and they assess the credibility of current information based on their prior knowledge. In addition, the number of potential participants in rumor propagation dynamically changes, and the actual network topology driven by activity is affected by environmental noise. Previous studies have overlooked the stochastic nature of activities and instead generalized the system dynamics with constant parameters. To fill this gap, we propose a stochastic fractional-order rumor propagation model that takes into account the variation of node activity rates and conduct an optimal control study. We utilize Caputo fractional-order derivatives to depict memorability, model stochasticity using standard Wiener processes, and formulate fractional-order stochastic differential equations to describe the dynamics of rumor propagation on activity-driven networks. Firstly, stochastic fractional-order equations with consistent unit dimensions are derived. Secondly, the existence and uniqueness of the solution to the stochastic fractional-order model are proven using Picard's iterative method, and the issue of rumor extinction is investigated. Then, Pontryagin's minimum principle and optimal control theory are applied to derive the necessary conditions for fractional-order optimal control. Finally, the theoretical results are validated using two real datasets, and the optimal fractional order is determined through a real-world case. The simulation results show that the smaller the fractional order, the more significant the suppression effect on rumor outbreaks, but the extinction time of the rumor is prolonged accordingly. In addition, the appropriate level of noise intensity can effectively suppress rumor outbreaks, suggesting the presence of stochastic resonance. Furthermore, the implementation of optimal control can effectively curb rumor outbreaks and shorten the life cycle of rumors.