This paper introduces a way to measure and track how failures at one or a few agents raise the risk of large deviations across a team that is trying to agree. The authors study a common setup called a consensus network, where agents repeatedly share information to pick a common value (for example, a meeting time). They add two realistic complications: random disturbances (noise) and a uniform communication delay. To capture the severity of possible cascades of bad outcomes, they use a risk measure called Average Value‑at‑Risk (AV@R), which focuses on the average of worst‑case deviations rather than just their probability.

To make the problem concrete the paper models each agent’s belief about the rendezvous time as a state variable. Agents follow a simple linear consensus rule over a fixed communication graph, and each agent is driven by its own random disturbance modeled as a Brownian motion. The authors focus on steady‑state behavior of agent deviations from the network average. They define what it means for the group to reach a practical agreement (a “c‑consensus”) and what it means for an individual agent to be prone to failure (its steady‑state deviation exceeds a tolerance with nontrivial probability). The main theoretical results are closed‑form formulas for conditional AV@R: the risk that other agents show large deviations given that one or more agents are already in an unsafe zone.

Those closed‑form results make clear which factors matter. In particular, the risk expressions depend explicitly on the network’s Laplacian spectrum (a set of numbers that encodes how the network is connected), the amount of communication delay, and the statistics of the noise driving each agent. The paper also proves fundamental lower bounds on performance under time‑delay constraints. These bounds act as feasibility certificates: they tell a designer whether a desired safety or performance goal is achievable without having to try every possible network layout.