Differentially Private Bipartite Consensus over Signed Networks with Time-Varying Noises

This paper investigates the differentially private bipartite consensus problem over signed networks. To solve this problem, a new algorithm is proposed by adding noises with time-varying variances to the cooperative-competitive interactive information. In order to achieve the privacy protection, the variances of the added noises are allowed to increase, which are substantially different from the existing works. In addition, the variances of the added noises can be either decaying or constant. By using a time-varying step-size based on the stochastic approximation method, we show that the algorithm converges in mean-square and almost-surely even with increasing privacy noises. We further develop a method to design the step-size and the noise parameter, affording the algorithm to achieve the average bipartite consensus with the desired accuracy and the predefined differential privacy level. Moreover, we give the mean-square and almost-sure convergence rates of the algorithm, and the privacy level with different forms of the privacy noises. We also reveal the trade-off between the accuracy and the privacy, and extend the results to local differential privacy. Finally, a numerical example verifies the theoretical results and demonstrates the algorithm's superiority against existing methods.