Turing instability analysis of a rumor propagation model with time delay on non-network and complex networks

With the development of the Internet and social media, rumors can spread not only through word-of-mouth but also rapidly through the network. In this paper, a dynamic model of rumor propagation with time delay is proposed separately for non-network and network scenarios. Additionally, we analyze the equilibrium points and their existence conditions for rumor propagation. After the linear approximation of the model, the necessary conditions for Turing instability are derived. Furthermore, the amplitude equation corresponding to the model in this paper is derived. Finally, through numerical simulations in the non-network model, we validate the aforementioned theories and find that changing the removed coefficient and cross-diffusion coefficient have a significant impact on Turing patterns, while time delay and periodic diffusion have a smaller impact. In the network model, we compare the effects of two network structures, the WS network and the BA network, through numerical simulations, verifying the feasibility of the Monte Carlo simulation method.