Abstract: For the problem that qualitative analysis methods are mostly adopted for the study of spam information propagation and it is difficult to reveal intrinsic propagation rules of spam information propagation, an ILDR (Ignorant-Lurker-Disseminator-Removed) model of spam information was proposed based on the idea of virus propagation modeling by considering the actual factors such as different input rates and removal rates. Firstly, the equilibrium point and the propagation threshold were calculated, and the stability conditions of the equilibrium point were given. Secondly, the local stability of non-spam information and spam information was proved by the Routh-Hurwitz criterion, then the global stability of non-spam information was certified via the invariance principle of LaSlle, which was proved based on the Bendixson criterion. Theoretical research shows that the non-spam information equilibrium is global asymptotically stable when propagation threshold is less than 1; the spam information equilibrium is global asymptotically stable when propagation threshold is greater than 1. The numerical simulation validates that the value of the propagation threshold can be decreased when decreasing the transfer rate from the lurker to the disseminator, increasing the transfer rate from the ignorant to the remover and the transfer rate from the lurker to the remover; the value of the disseminator can be decreased via increasing the proportionality coefficient from the ignorant to the lurker, or increasing the transfer rate of the disseminator to the remover and the system removal rate.

Key words: spam information, propagation model, equilibrium point, propagation threshold, global stability

摘要： 针对网络垃圾信息传播研究多采用定性分析方法，难以揭示垃圾信息内在传播规律的问题，基于病毒传播的建模思想，考虑不同输入率和移出率等现实因素，提出了垃圾信息的ILDR传播模型。首先，计算了平衡点和传播阈值，并给出了平衡点的稳定性条件。其次，利用Routh-Hurwitz准则证明了无垃圾信息和垃圾信息的局部稳定性，并采用LaSlle的不变性原理证明了无垃圾信息全局稳定性，根据Bendixson判据证明了垃圾信息的全局稳定性。理论研究表明:当传播阈值小于1时，无垃圾信息平衡点全局渐进稳定；当传播阈值大于1时，垃圾信息平衡点全局渐进稳定。根据数值模拟可知:减小潜伏者到传播者的转化率，增加无知者到移出者的转化率和潜伏者到移出者的转化率，可以减小传播阈值的值；减小无知者到潜伏者的比例系数，增加传播者到移出者的转化率和系统的移出率，可以使传播者的取值减小。

关键词: 垃圾信息, 传播模型, 平衡点, 传播阈值, 全局稳定性