Toward combating the malicious rumor spreading on online social networks, we discuss a hybrid control strategy that combines a continuous truth spreading method and an impulsive rumor blocking method in this article. In this strategy, the truth is introduced as an anti-rumor to compete with the rumor while the rumor-infected users would be quarantined with a limited isolation period. The stability of the rumor-free equilibrium of the rumor-truth propagation system is analyzed and conditions to reach uniform asymptotic stability as well as global exponential stability are obtained. For the endemic case, the conditions under which rumor would persist uniformly are obtained. An optimal hybrid control problem is formulated to address the tradeoff between restraining rumor and minimizing the cost of control. The necessary conditions for optimal system and the structure of optimal hybrid control are obtained. Numerical simulations are carried out to illustrate the theoretical results and evaluate the potential roles of the hybrid control strategy.

In the process of rumor spreading, controlling and killing rumor problem is of great importance on social networks. In this paper, we present a new SEIR (susceptible-exposed-infected-removed) rumor spreading model with hesitating mechanism. By using mean-field theory, the equilibrium of the model and the basic reproduction number \(R_{0}\) are obtained. The global stability of the rumor-free equilibrium and the permanence are proved in detail, and the global attractivity of the rumor-prevailing equilibrium is proved by using a monotone iterative technique. Furthermore, the modified model with feedback mechanism on social networks is introduced. The feedback mechanism cannot change the basic reproductive number but it can reduce the continuous level and the spread of rumor. Numerical simulations confirm the analytical results.

Nowadays, human behaviors are known to spread through social contact. Word-of-mouth (WOM) communication has also become a prevalent strategy for product sales on the internet and most consumers attribute a higher importance to WOM communication. Mathematical modeling is an important approach to study WOM communication in online social networks. In order to investigate the characteristics of WOM communication and heterogeneity of online social networks, a SIALS (susceptible-infected-acknowledged-loath) model is presented on scale-free networks. The spreading dynamics of WOM are analyzed in detail by using the theory of mean field. The basic reproductive number R ₀ is calculated by the next generation matrix method and two equilibriums are derived. The theoretical analysis indicates that the basic reproductive number depends mainly on the transmission of WOM and the topology of the underlying networks. Furthermore, the global stability of the WOM-disappearance equilibrium is proved. The permanence of WOM information spreading and the global attractivity of the WOM-prevailing equilibrium are also studied in detail. Finally, numerical simulations confirm the analytical results.

This paper investigates N versus N attack-defense game problem of heterogeneous nonlinear multi-agent systems under the condition of actuator faults on the defensive side. Specifically, N attackers need to approach the target area as closely as possible in order to attack it, while N defenders need to approach the attackers as closely as possible and intercept them. We decompose this problem into two N-player nonzero-sum game problems. Based on differential game method, the optimal attacking game strategies for the attackers are first presented. Then, considering that the defenders contains actuator faults, this paper designs a fault observer for each defending agent based on adaptive estimation theory to estimate unknown actuator fault and proposes a novel optimal defense game strategy applicable for actuator fault. To address the problem that it is difficult to directly solve the optimal control strategies in nonlinear systems, we propose an online learning algorithm based on adaptive dynamic programming to obtain approximate solutions for the optimal cost functions and the optimal control strategies. The effectiveness of the proposed method is verified through theoretical analysis and simulation examples.