Finite/Fixed-time synchronization for stochastic multilayer networks with pinning control

In recent years, there has been a strong focus on the stability and control of stochastic networks, particularly single-layer networks. However, with the emergence of multi-layer networks, the research in this area has expanded significantly. Multi-layer networks have gained much attention due to their relevance in network science. In this paper, we address the synchronization of stochastic multi-layer networks (SMLNs) using finite-time (FnT) and fixed-time (FxT) methods with pinning control. Firstly, we provide the conditions for achieving FxT synchronization in SMLNs. Secondly, we present the conditions for FnT synchronization in SMLNs. Additionally, two kinds of finite-time convergent values of the system are given, which can be any given time. Furthermore, we discuss the relationship between the coupling matrix and the network layers, particularly in the context of minimum convergence time. This analysis has practical implications for real-world applications. Finally, we present numerical simulations to demonstrate the effectiveness of our proposed method.