Wang et al. achieve differential privacy for delayed multi-agent consensus

A new paper on arXiv (2606.15135) by Mingyu Wang, Xiaofeng Zong, Jimin Wang, and Ji-Feng Zhang addresses a critical gap in privacy-preserving distributed control: how to achieve differentially private consensus when communication delays exist. Multi-agent systems—like drone swarms or sensor networks—often rely on sharing state information to reach agreement (consensus). But sharing that data can leak private initial conditions, especially when delays obscure the timeline. The authors propose a Laplace-noise-based mechanism where the noise variance can vary over time and even increase, a departure from standard fixed-noise approaches. They define a novel adjacency relation that considers the entire delayed initial history of each agent, ensuring stronger privacy guarantees. To analyze convergence, they use a difference resolvent function method to derive decay estimates for fundamental solutions of delayed difference equations, then apply a backstepping technique to prove three types of consensus: mean square weak, mean square strong, and almost sure strong.

The paper's key technical contribution is an explicit sensitivity bound derived from those decay estimates, which directly links the noise level to the privacy budget. This allows a constructive parameter design that achieves a prescribed infinite-horizon ε*-differential privacy level—meaning the privacy guarantee holds over the entire lifetime of the system, not just at steady state. The 19-page paper includes 4 figures from numerical simulations that illustrate the trade-off between privacy and convergence speed. For practitioners, this work provides a theoretical foundation for deploying differential privacy in real-time distributed systems where delays are unavoidable, such as autonomous vehicle coordination or smart grid load balancing. The ability to protect entire delayed histories while still reaching consensus could make privacy-by-design feasible in safety-critical multi-agent applications.