Selecting the Minimal Multi-Hop Radius for Resilient Consensus: A Hybrid Robustness–Proxy Framework for MW-MSR

Achieving resilient consensus in adversarial environments often requires extending the W-MSR algorithm to multi-hop communication. While the robustness guarantees of multi-hop W-MSR are now well understood, the problem of how to determine the minimal hop radius h* that ensures these guarantees has remained largely unaddressed. Existing work typically assumes a fixed h, leaving practitioners without a systematic way to balance robustness requirements against communication and computational cost. This paper introduces a new hop-selection framework that identifies the smallest communication horizon capable of satisfying the robustness assumptions underlying MW-MSR consensus. The framework combines exact robustness verification—when tractable—with a hierarchy of computationally efficient proxy tests based on local feasibility, normalized algebraic connectivity, and adversary-dilution criteria. These components provide a practical and scalable mechanism for establishing h* in both synchronous and bounded-delay asynchronous settings. Design-time and runtime procedures, complexity analysis, and validation on IEEE 14-, 30-, and 57-bus networks demonstrate that the proposed approach reliably detects resilience thresholds and substantially improves consensus behavior under stealthy and burst-type adversaries. The results show that systematic hop selection is essential for avoiding failure at small h while preventing unnecessary communication overhead at large h. The framework thus offers an implementable and deployment-oriented strategy for resilient distributed coordination in sparse and adversarial multi-agent networks.