Differentially Private Consensus for Time-Delay Multi-agent Systems

Abstract: This paper is concerned with the differentially private consensus problem for discrete-time multi-agent systems with communication delays. The purpose of the paper is to achieve differentially private consensus for such systems while protecting the entire delayed initial histories of all agents. A novel adjacency relation for delayed histories is introduced, and a Laplace-noise-based privacy mechanism is developed, where the noise variance is allowed to vary with time and even increase. By using the difference resolvent function method, decay estimates for the fundamental solutions of the delayed difference equations are derived. Based on these estimates and a backstepping technique, mean square weak consensus, mean square strong consensus, and almost sure strong consensus are established. The estimates for the fundamental solutions are also used to derive an explicit sensitivity bound. Furthermore, a constructive parameter design is provided to achieve a prescribed infinite-horizon $ε<sup>\star$-differential privacy level. Numerical simulations illustrate the theoretical results.

• The paper introduces the first differential privacy framework for delayed discrete-time multi-agent consensus that protects each agent’s complete initial history rather than only its current state.
• The paper combines delayed-system fundamental-solution estimates, sensitivity analysis, and stochastic convergence tools to establish mean-square and almost-sure consensus over directed graphs with a spanning tree.
• The paper shows that privacy noise can remain constant or increase over time while preserving consensus, and provides constructive gain and noise-scale conditions for meeting a prescribed infinite-horizon privacy budget.

Problem and motivation

Consensus in multi-agent systems (MASs) is typically analyzed under the assumption that exchanged data can be broadcast openly, yet two practical obstacles coexist: communication delays and the risk of privacy leakage to honest-but-curious agents or eavesdroppers. Existing differentially private consensus results—spanning average consensus [(2606.15135) references such as Huang et al. 2012 and Nozari et al. 2017], event-triggered and resilient variants, and extensions to signed networks—almost uniformly assume delay-free communication and treat only a single initial state $x_i(0)$ as the private datum. For a delayed discrete-time system, the trajectory depends on the entire initial history $\{x_i(r)\}_{r=-d}^{0}$, so protecting a single state is insufficient.

This paper formulates the first, to the authors' knowledge, differentially private consensus framework for discrete-time MASs with communication delays, in which the entire delayed initial history is the protected dataset. The central technical obstacles are twofold: characterizing how differences between adjacent delayed histories propagate through the closed-loop dynamics (the sensitivity analysis), and establishing stochastic convergence when communication delays and injected Laplace privacy noises act jointly.

System model and privacy definition

The paper considers $N$ discrete-time first-order integrators with a delayed consensus protocol perturbed by Laplace noise. Agent $j$ broadcasts $\theta_j(k) = x_j(k-d) + w_j(k)$, where the entries of $w_j(k)$ are independent $\mathrm{Lap}(0, b_j(k))$ variables with a time-varying scale $b_j(k)$ that is permitted to grow. The control input is $u_i(k) = c(k)\sum_j a_{ij}(\theta_j(k) - x_i(k-d))$.

Two modeling choices are noteworthy. First, the adjacency relation between private datasets is defined over history segments: two global histories are $\Delta_H$-adjacent if they differ in a single agent's history with $\{x_i(r)\}_{r=-d}^{0}$0. This is the natural analogue of record-level adjacency for delayed systems and is, in itself, a contribution. Second, the privacy mechanism $\{x_i(r)\}_{r=-d}^{0}$1 maps histories to observation transcripts $\{x_i(r)\}_{r=-d}^{0}$2, and $\{x_i(r)\}_{r=-d}^{0}$3-differential privacy is required in the standard likelihood-ratio sense for all $\{x_i(r)\}_{r=-d}^{0}$4-adjacent pairs.

Fundamental solution estimates

The analytical backbone is a pair of decay estimates for the fundamental solutions of the scalar delayed difference equation $\{x_i(r)\}_{r=-d}^{0}$5, where $\{x_i(r)\}_{r=-d}^{0}$6. Using a difference resolvent function in the spirit of Diblík and Khusainov, combined with a weighted Lyapunov construction and a backstepping argument, the authors prove that under the gain condition

$\{x_i(r)\}_{r=-d}^{0}$7

both the two-parameter fundamental solution $\{x_i(r)\}_{r=-d}^{0}$8 and the history fundamental solution $\{x_i(r)\}_{r=-d}^{0}$9 decay exponentially in $N$0, i.e., $N$1. The choice of decay rate $N$2 is always feasible because the defining function $N$3 is continuous with $N$4. A companion Toeplitz-type kernel lemma shows that the convolution array $N$5 has uniformly bounded row sums and vanishing columns, enabling convergence arguments via the Toeplitz Lemma. These estimates serve a dual role: they underpin the convergence analysis and, as shown later, yield the explicit sensitivity bound for the privacy proof.

Consensus results

Convergence is established by decomposing the network dynamics through the Laplacian's Jordan form, reducing the problem to Jordan-chain coordinates of the scalar delayed stochastic recursion. The main results are:

• Mean square weak consensus holds under condition (C1): $N$6 together with a convolution-type summability condition on the noise-gain product. The proof proceeds by induction along Jordan chains, with the coupling term handled via the Toeplitz kernel lemma.
• Mean square and almost sure strong consensus hold under the stronger condition (C4): $N$7 and $N$8. The consensus component is shown to be an $N$9-bounded martingale (converging via the martingale convergence theorem), while the disagreement component vanishes by the weak-consensus argument and a delay-free comparison system with Robbins–Siegmund and backward-induction arguments.

Two consequences deserve emphasis. First, the framework strictly relaxes prior delay-free results: when $j$0, condition (C1) is weaker than the condition $j$1 required in the state-of-the-art signed-network result, the analysis covers general directed graphs containing a spanning tree rather than undirected graphs, and the gain bound $j$2 is weaker than the previously required $j$3. Second, a corollary with $j$4 and $j$5 shows that strong consensus is achievable even when the privacy noise variance increases over time, provided $j$6. This contradicts the prevailing design heuristic in the differential-privacy consensus literature, where noise variance must be constant or decaying; here, growing noise is compatible with privacy because the sensitivity of the delayed-state trajectory to the protected history decays exponentially in the accumulated gain.

Sensitivity bound and infinite-horizon privacy

For two adjacent histories held against a fixed transcript, the state difference obeys a decoupled delayed recursion $j$7, where $j$8 is agent $j$9's in-degree. Applying the fundamental-solution estimate with $\theta_j(k) = x_j(k-d) + w_j(k)$0 yields the explicit sensitivity bound

$\theta_j(k) = x_j(k-d) + w_j(k)$1

under the in-degree condition $\theta_j(k) = x_j(k-d) + w_j(k)$2. The Laplace densities then give $\theta_j(k) = x_j(k-d) + w_j(k)$3, where $\theta_j(k) = x_j(k-d) + w_j(k)$4 is the smallest noise scale across agents.

The constructive design theorem is the paper's most actionable result: with $\theta_j(k) = x_j(k-d) + w_j(k)$5 and $\theta_j(k) = x_j(k-d) + w_j(k)$6, closed-form upper bounds on the infinite-horizon privacy loss $\theta_j(k) = x_j(k-d) + w_j(k)$7 are derived for $\theta_j(k) = x_j(k-d) + w_j(k)$8 (involving the condition $\theta_j(k) = x_j(k-d) + w_j(k)$9) and $w_j(k)$0 (involving the upper incomplete gamma function). Since these bounds are monotone in $w_j(k)$1, a designer can select the noise scale to meet any prescribed budget $w_j(k)$2 over the infinite horizon while retaining consensus. Notably, this is compatible with $w_j(k)$3, i.e., non-decaying or growing noise, because the exponentially decaying sensitivity dominates the noise-scale growth.

Numerical illustration

A five-agent, strongly connected digraph with delay $w_j(k)$4, in-degrees $w_j(k)$5, gain $w_j(k)$6, and noise scale $w_j(k)$7 is simulated. The verified quantities are concrete: the minimum modal decay parameter $w_j(k)$8, sensitivity-decay parameter $w_j(k)$9, admissible $\mathrm{Lap}(0, b_j(k))$0 (with $\mathrm{Lap}(0, b_j(k))$1), and $\mathrm{Lap}(0, b_j(k))$2 with $\mathrm{Lap}(0, b_j(k))$3, $\mathrm{Lap}(0, b_j(k))$4. The simulations confirm simultaneous mean square weak consensus, mean square strong consensus, and almost sure consensus, while the transmitted signals remain persistently perturbed by the growing Laplace noise.

Limitations and open questions

The framework is confined to first-order integrator dynamics; extension to higher-order or general linear MASs with delays is left open, and the authors note it as nontrivial. The topology is assumed fixed and contains a spanning tree; time-varying or randomly switching graphs are not covered. Communication is synchronous with a uniform delay $\mathrm{Lap}(0, b_j(k))$5; asynchronous updates and heterogeneous delays fall outside the model. The privacy guarantee is against honest-but-curious observers of the broadcast channel under the stated adjacency relation, and the sensitivity bound requires the in-degree condition $\mathrm{Lap}(0, b_j(k))$6, which constrains the admissible gain range relative to in-degrees and delay. Finally, event-triggered communication, which would reduce bandwidth while preserving history-level privacy, is identified as an unaddressed direction.

Conclusion

This paper establishes differentially private consensus for discrete-time MASs with communication delays, treating the full delayed initial history as the protected dataset. Its technical core—exponential decay estimates for fundamental solutions of delayed difference equations—simultaneously enables a new convergence analysis for delayed stochastic consensus over general directed graphs and an explicit sensitivity bound that yields constructive infinite-horizon $\mathrm{Lap}(0, b_j(k))$7-differential privacy. The demonstration that consensus persists under time-increasing privacy noise, and the resulting relaxation of gain conditions relative to delay-free prior work, are the results most likely to influence subsequent design of private distributed algorithms.