Distributionally Robust Cascading Risk in Multi-Agent Rendezvous: Extended Analysis of Parameter-Induced Ambiguity

TL;DR

This work develops a distributionally robust framework for cascading risk in time-delayed multi-agent rendezvous networks. By modeling observables as zero-mean Gaussians with covariance bounds induced by bounded parameter uncertainty, the authors derive a closed-form conditional DR risk and reformulate it as a tractable optimization over variances and correlations. They establish fundamental limits linking DR risk to Laplacian eigenvalues and provide concrete bounds and case studies across common graph topologies, showing that increased connectivity can nontrivially elevate risk in certain regimes. The methodology offers design insights for resilient rendezvous tasks under uncertainty and delays, with broad potential applicability to networked control systems.

Abstract

Ensuring safety in autonomous multi-agent systems during time-critical tasks such as rendezvous is a fundamental challenge, particularly under communication delays and uncertainty in system parameters. In this paper, we develop a theoretical framework to analyze the \emph{distributionally robust risk of cascading failures} in multi-agent rendezvous, where system parameters lie within bounded uncertainty sets around nominal values. Using a time-delayed dynamical network as a benchmark model, we quantify how small deviations in these parameters impact collective safety. We introduce a \emph{conditional distributionally robust functional}, grounded in a bivariate Gaussian model, to characterize risk propagation between agents. This yields a \emph{closed-form risk expression} that captures the complex interaction between time delays, network structure, noise statistics, and failure modes. These expressions expose key sensitivity patterns and provide actionable insight for the design of robust and resilient multi-agent networks. Extensive simulations validate the theoretical results and demonstrate the effectiveness of our framework.

Figures (6)

• Figure 1: Geometry of Ambiguity Set $\left(\tau \neq 0\right)$. The shaded region illustrates the feasible set of covariance matrices $\Sigma$ consistent with the assumed distributional ambiguity.
• Figure 2: Graph of $f_c(u) = \frac{\cos(cu)}{2c (1- \sin(cu))}$.
• Figure 3: Graph of $f_c(v) = \frac{\cos(cv)}{2v (1- \sin(cv))}$.
• Figure 4: Distributionally Robust Cascading Risk under parameter uncertainty (rows) for various (equal edge weights) graph topologies (columns). The markers indicate the type of risk: • DR cascading risk, • cascading risk without distributional robustness, • single-agent risk. Each row corresponds to a different level of parameter uncertainty, and each column corresponds to a graph topology (Complete, 14-cycle, 6-cycle, Path).
• Figure 5: Distributionally robust cascading risk for uncertainty in network parameters with zero time delay. The markers indicate the type of risk: • DR cascading risk, • cascading risk without distributional robustness, and • single-agent risk.

Theorems & Definitions (21)

• proof : Proof of Lemma \ref{['lem:sigma_y_steady']}
• proof : Proof of Lemma \ref{['lem:principle_covariance_invertibility']}
• proof : Proof of Lemma \ref{['lem:bivariate_normal']}
• proof : Proof of Proposition \ref{['prop:ambiguity_b']}
• proof : Proof of Proposition \ref{['prop:ambiguity_time_delay']}
• proof : Proof of Proposition \ref{['prop:ambiguity_edge_weight_tau_zero']}
• proof : Proof of Proposition \ref{['prop:ambiguity_edge_weight_tau_non_zero']}
• proof : Proof of Lemma \ref{['lem:conditional_expectation']}
• proof : Proof of Theorem \ref{['thm:DR_risk']}
• proof : Proof of Corollary \ref{['cor:single_risk_rho_0']}