In view of the existing bad information diffusion models do not consider the information diffusion between different social networks, the connectivity principle in graph theory was used to establish a dynamic model of the diffusion of bad information among multiple social networks, and the optimal control theory was applied to the model. Through the optimal control principle, the existence of the optimal control strategy was proved. Then the optimal control model of the diffusion of bad information was obtained. The experimental results show that the introduction of optimal control measures can effectively suppress the diffusion scale of bad information, and the strength of the control strategy can be dynamically adjusted as needed. In addition, by simulating the conditions with and without information transmitted between different social networks, it is found that the information transmission between social networks will increase the scale and speed of the diffusion of bad information.

Concerning that the existing networks virus propagation models do not consider the influence of interactive behaviors among the users in different social networks on network virus propagation, a dynamic model of differential equations was established. The stability theory was used to analyze the dynamical behaviors of the network virus propagation, and the accurate expression of the basic reproduction number was obtained, which is the threshold of controlling the network virus propagation. Furthermore, using Runge-Kutta numerical method, the correctness of theoretic analysis was verified by simulations. The results show that the basic reproduction number is the direct decisive factor of network virus prevalence situations. When the value of the basic reproduction number is less than or equal to one, the propagation of the network viruses will be controlled with the evolution of time. Additionally, the research reveals that it is helpful for distributing the users to different social networks to slow the prevalence of network viruses.

Internet Service Provider (ISP), as the convergers and distributors of network information, is the best network virus defender. However, ISP usually would like to detect the ingress information while ignoring the egress information according to the cost. The security measures for ISP in the whole network were analyzed and the strategy of egress detection was presented to provide reference for ISP's selection of strategies. First, the game model between ISP and attackers and the spreading model of network virus were proposed. Secondly, the impacts of the strategies selected by ISP on the spreading of virus were analyzed when the dynamical spreading of network virus was considered. The results show that ISP will face an increase in invasion risk when it does not take egress detection, however the adoption of egress detection can improve not only the ISP's own utility, but also the security of the whole network. The validity of the theoretical results was verified by Matlab simulation.

To effectively control large-scale outbreak, the propagation properties of the leeching P2P (Peer-to-Peer) botnet was studied using dynamics theory. Firstly, a delayed differential-equation model was proposed according to the formation of the botnet. Secondly, the threshold expression of controlling botnet was obtained by the explicit mathematical analysis. Finally, the numerical simulations verified the correctness of theoretical analysis. The theoretical analysis and experimental results show that the botnet can be completely eliminated if the basic reproduction number is less than 1. Otherwise, the defense measures can only reduce the scale of botnet. The simulation results show that decreasing the infection rate of bot programs or increasing the immune rate of nodes in the network can effectively inhibit the outbreak of botnet. In practice, the propagation of bot programs can be controlled by some measures, such as uneven distribution of nodes in the network, timely downloading patch and so on.