In virtue of identifying the influence of nodes, the spatial distance of rumor propagation is defined with the partition and clustering in the network. Considering the temporal and spatial propagation characteristics of rumors in online social networks, we establish a delayed rumor propagation model based on the graph theory and partial functional differential equations. Firstly, the unique existence and uniform boundedness of the nonnegative solution are explored. Secondly, we discuss the existence of positive equilibrium points sufficiently. Thirdly, stabilities of the rumor-free and rumor-spreading equilibrium points are investigated according to the linearization approach and Lyapunov function. Finally, we perform several numerical simulations to validate theoretical results and show the influence of time delay on rumor propagation. Experimental results further illustrate that taking forceful actions such as increasing the time delay in the rumor-spreading process can control rumor propagation due to the timely effectiveness of the information.

中文翻译：

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复杂网络中信息传播模型的数学分析

通过识别节点的影响，通过网络中的划分和聚类来定义谣言传播的空间距离。考虑在线社交网络中谣言的时空传播特性，我们基于图论和偏泛函微分方程建立了延迟谣言传播模型。首先，探讨了非负解的唯一存在性和一致有界性。其次，我们充分讨论了正平衡点的存在。第三，根据线性化方法和Lyapunov函数研究了无谣言和谣言传播平衡点的稳定性。最后，我们进行了几次数值模拟来验证理论结果并显示时间延迟对谣言传播的影响。