Distributed Differential Privacy Consensus of Multi-Agent Systems With Jointly Connected Topology

Abstract

With the development of social media, mobile computing, and edge intelligence, the communication topology frequently changes over time, making it difficult to maintain connectivity. Traditional average consensus relies on continuous state exchange, which can easily lead to state-evolution trajectory leakage. Most existing privacy-preserving consensus methods are designed for fixed or instantaneously connected switching topologies. However, under jointly connected topologies, the instantaneous graph may be disconnected at some time instants and contain isolated nodes, so the sensitivity does not necessarily contract at every time step. To address these issues, this paper proposes a Distributed Differential Privacy Consensus Model (DDPCM), where agents update their states based on their internal states and noisy states broadcast from neighbors, avoiding direct exposure of private states. A window-triggered noise-scale decay mechanism is designed to dynamically adjust the Laplace noise scale according to jointly connected windows and isolated-node conditions, thereby protecting complete state-evolution trajectories. Theoretical analysis proves that DDPCM satisfies differential privacy at each time step and achieves almost-sure consensus under jointly connected topologies. The final consensus value is a random variable whose expectation equals the average of the initial states and whose variance is bounded. An accuracy characterization is also provided to quantify the consensus error caused by privacy noise. Finally, numerical experiments and a real-world case study based on X.com mobile social network data related to the 2024 U.S. presidential election validate the effectiveness and robustness of the proposed model.