Differentially Private Opinion Dynamics of Influence Networks

In this article, a unified influence network model incorporating differential privacy mechanisms (DPMs), called the differentially private opinion dynamics (DPODs) model, is proposed. In this model, each individual uses protected opinions rather than the private opinions of his/her neighbors to update his/her private opinions, where the protected opinion of an individual is a blend of private opinions and random noise following Laplace distribution. Building on stochastic analysis techniques and matrix theory, we show that the influence network under consideration converges under specific conditions governing individual sensitivities and interaction weights. In addition, the statistical properties related to convergence accuracy are established by utilizing the Markov inequality to estimate a lower bound on the probability of all individuals' final opinions converging to a neighborhood formed by their initial opinions' convex hull. We further conduct a differential privacy analysis to validate the efficacy of the proposed DPMs in safeguarding the private opinions of all individuals. Finally, two examples, including one of the Karate-Club networks, are provided to shed new light on the effectiveness of the theoretical results.