The spread of fake news, especially on online social networks, has become a matter of concern in the last few years. These platforms are also used for propagating other important authentic information. Thus, there is a need for mitigating fake news without significantly influencing the spread of real news. We leverage users inherent capabilities of identifying fake news and propose a warning-based control mechanism to curb this spread. Warnings are based on previous users responses that indicate the authenticity of the news. We use population-size dependent continuous-time multi-type branching processes to describe the spreading under the warning mechanism. We also have new results towards these branching processes. The (time) asymptotic proportions of the individual populations are derived. These results are instrumental in deriving relevant type-1, type-2 performance measures, and formulating an optimization problem to design optimal warning parameters. The fraction of copies tagged as real (fake) are considered for the type-1 (type-2) performance. We derive structural properties of the performance, which help simplify the optimization problem. We finally demonstrate that the optimal warning mechanism effectively mitigates fake news, with negligible influences on the propagation of authentic news. We validate performance measures using Monte Carlo simulations on ego-network database related to Twitter.
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