Propagation of stochastic noise in complex network

Stochastic noise is prevalent in real world and play a significant role in fields such as financial, telecommunications, and probability theory. Moreover, in the field of network science, this type of random noise will spread throughout the entire network according to its topology, exerting crucial and sometimes determinative effects on various properties of networks or dynamical systems. In this paper, we propose an alternative framework to traditional approaches used in complex network(such as ODES and PDEs) by employing SDEs(stochastic differential dynamics) and the Itô’s formula to investigate the impact of stochastic processes. Through this framework, we focused on exploring the effect of stochastic noise on signal propagation and dimension reduction. Interestingly, our theoretical and simulation results demonstrate that stochastic noise significantly impact both propagation time and reduced systems, showing exponential differences in their expected values. Furthermore, the framework we have developed reveals the fundamental pattern of how stochastic noise influences dynamical properties, and it introduces a basic analytical method in analyzing stochastic noise. Importantly, this framework can be directly applied to other domains of complex network, such as dynamical control and network stability.