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.

中文翻译：

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非网络和复杂网络上时滞谣言传播模型的图灵不稳定性分析

随着互联网和社交媒体的发展，谣言不仅可以通过口碑传播，还可以通过网络快速传播。本文分别针对非网络和网络场景提出了带有时滞的谣言传播动态模型。此外，我们还分析了谣言传播的平衡点及其存在条件。对模型进行线性逼近后，推导了图灵不稳定性的必要条件。进一步推导了本文模型对应的振幅方程。最后，通过在非网络模型中的数值模拟，我们验证了上述理论，发现改变去除系数和交叉扩散系数对图灵模式有显着影响，而时间延迟和周期性扩散影响较小。在网络模型中，我们通过数值模拟比较了WS网络和BA网络两种网络结构的效果，验证了蒙特卡罗模拟方法的可行性。