A Geometric Approach to Resilient Distributed Consensus Accounting for State Imprecision and Adversarial Agents

TL;DR

The paper tackles resilient distributed consensus when agent states are imprecise and some agents may be Byzantine. It introduces a geometry-based framework that replaces precise neighbor states with imprecision regions $B_v$ and defines invariant hulls IHull($B_V$) to guarantee safety despite uncertainty. The core method CPIH computes safe points as intersections of IHulls over $k$-tuples of potential regions, with $k = \frac{dN_v}{d+1} + 1$, and updates states toward these safe points. Theoretical characterization and a practical computation approach are provided, complemented by simulations showing that normal agents remain inside the initial convex hull and cluster within a bounded region whose size depends on the imprecision. Overall, the work extends resilient consensus to uncertain observations, preserving resilience while quantifying the safety–accuracy trade-off in noisy environments.

Abstract

This paper presents a novel approach for resilient distributed consensus in multiagent networks when dealing with adversarial agents imprecision in states observed by normal agents. Traditional resilient distributed consensus algorithms often presume that agents have exact knowledge of their neighbors' states, which is unrealistic in practical scenarios. We show that such existing methods are inadequate when agents only have access to imprecise states of their neighbors. To overcome this challenge, we adapt a geometric approach and model an agent's state by an `imprecision region' rather than a point in $\mathbb{R}^d$. From a given set of imprecision regions, we first present an efficient way to compute a region that is guaranteed to lie in the convex hull of true, albeit unknown, states of agents. We call this region the \emph{invariant hull} of imprecision regions and provide its geometric characterization. Next, we use these invariant hulls to identify a \emph{safe point} for each normal agent. The safe point of an agent lies within the convex hull of its \emph{normal} neighbors' states and hence is used by the agent to update it's state. This leads to the aggregation of normal agents' states to safe points inside the convex hull of their initial states, or an approximation of consensus. We also illustrate our results through simulations. Our contributions enhance the robustness of resilient distributed consensus algorithms by accommodating state imprecision without compromising resilience against adversarial agents.

Figures (7)

• Figure 1: (a) An exact point $x_v$ (no imprecision). (b) A disk imprecision region in $\mathbb{R}^2$. The observed point is $r_v$, which is the imprecise version of the exact point $x_v$ that lies somewhere inside the disk. (c) A square imprecision region in $\mathbb{R}^2$.
• Figure 2: Illustration of centerpoint. In (a) and (b), centerpoint is denoted by '$\times$' and lines are passing through the centerpoint. The green shaded region in (c) is the centerpoint region.
• Figure 3: Centerpoint region based on the observed states (due to imprecision) is not contained entirely in the convex hull of normal agents' initial states.
• Figure 4: (a) Normal agents achieve resilient consensus with no imprecision. (b) Normal agents do not converge and move outside the convex hull of normal agents' initial states due to imprecision.
• Figure 5: (a) Invariant hull of a set of potential regions. (b) Invariant hull is a subset of the convex hull of an arbitrary potential configuration.

Theorems & Definitions (11)

• Definition 3.1
• Definition 4.1
• Definition 4.2
• Lemma 4.1
• proof
• Corollary 4.2
• Lemma 4.3
• proof
• Theorem 4.4
• Theorem 4.5