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The spread of rumors in Online Social Networks (OSNs) poses great challenges to the social peace and public order. It is imperative to model propagation dynamics of rumors and develop corresponding countermeasures. Most of the existing works either overlook the heterogeneity of social networks or do not consider the cost of countermeasures. Motivated by these issues, this paper proposes a heterogeneous network based epidemic model that incorporates both the network heterogeneity and various countermeasures. Through analyzing the existence and stability of equilibrium solutions of the proposed ODE (Ordinary Differential Equation) system, the critical conditions that determine whether a rumor continuously propagates or becomes extinct are derived. Moreover, we concern about the cost of the main two types of countermeasures, i.e., Blocking rumors at influential users and spreading truth to clarify rumors. Employing the Pontryagin's maximum principle, we obtain the optimized countermeasures that ensures a rumor can become extinct at the end of an expected time period with lowest cost. Both the critical conditions and the optimized countermeasures provide a real-time decision reference to restrain the rumor spreading. Experiments based on Digg2009 dataset are conducted to evaluate the effectiveness of the proposed dynamic model and the efficiency of the optimized countermeasures.
He et al. (Mon,) studied this question.