The Big Picture
Each agent can use just its farthest visible distance to move up to half the remaining visibility gap in any direction and guarantee no current link is lost, so a connected swarm stays connected while it spreads.
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The Evidence
A single scalar — the farthest visible neighbor distance — certifies the largest safe isotropic move: (visibility − farthest_distance)/2. Using that radius and picking a random direction preserves every current edge and therefore overall connectivity for any swarm size. Repeating maximal safe jumps makes the swarm expand and stretch preserved edges in a way that depends strongly on the initial connection pattern; for two agents the distance almost surely approaches the visibility limit and reaches near-boundary neighborhoods in finite expected time (~9.5 rounds to get within 3%). Cascading reliability failures are a relevant concern for larger deployments, and this work provides a basis for understanding resilience under simple local rules Cascading Reliability Failures.
Data Highlights
1Maximal safe step size per agent = (V − d_max)/2 where V is visibility and d_max is the farthest observed neighbor.
2Every fully immobilized (absorbing) configuration contains at least n/2 edges at the visibility limit (exactly V).
3In N=10 experiments, median final diameters were about 1.65·V (dense), 2.2·V (medium), and 3.6·V (sparse); for two agents mean time to reach 0.97·V ≈ 9.5 rounds.
What This Means
Engineers building decentralized robot swarms or minimal sensor agents can use this simple, provably safe local rule as a safety layer to prevent disconnection. Technical leads evaluating agent orchestration patterns can separate safety (range-only certificate) from task-specific direction policies to simplify design and verification. Researchers exploring minimal-information multi-agent control will find a clean theoretical baseline and open questions on noise and asynchronous updates Model Context Protocol.
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Key Figures
Fig 4: Fig. 4: Finite-round behavior in one million two-agent runs for D t ≥ 0.97 V D_{t}\geq 0.97V .
Fig 5: Fig. 5: Monte Carlo validation of Lemma 1 .
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Learn MoreKeep in Mind
The model assumes exact distance measurements, synchronous round updates, no collisions, and perfect movement — real systems need noise-aware margins and asynchronous analysis. Preserving every current edge is conservative: it keeps connectivity but can prevent removing redundant links that would allow more expansion. The paper proves full stochastic convergence only for two agents; for larger swarms the reported expansion trends are empirical, not formal convergence results. For coordination considerations, see the Orchestrator-Worker pattern Orchestrator-Worker Pattern.
Methodology & More
Model and rule: Agents are identical, anonymous, and only measure current distances to visible neighbors (no directions, IDs, memory, or communication). Each agent computes its farthest visible neighbor distance d_max and may move anywhere inside a disk of radius (V − d_max)/2 (V is visibility). Picking the disk boundary and choosing a direction uniformly at random exhausts the certified displacement while avoiding any possible loss of existing visibility edges. This single-scalar rule reduces the local observation set to one number and cleanly separates a deterministic safety certificate from the choice of exploration direction. Market-based coordination ideas are relevant to understanding scaling and resource allocation in larger swarms Market-Based Coordination Pattern. This single-scalar rule also has implications for evaluation and verification across deployments Evaluation. Findings and implications: The radius (V − d_max)/2 is the largest isotropic, bearing-independent move that always preserves every current edge; hence an initially connected graph remains connected forever. If an edge reaches exactly length V, its endpoints become permanently immobile and any absorbing state must include at least n/2 such boundary edges. For two agents there is positive expected progress in squared distance and almost-sure approach to the visibility boundary; numerics and a Bellman calculation show about 9.5 rounds on average to get within 3% of visibility. Large-scale simulations (millions of rounds across many initial topologies) reproduce the safety guarantee and reveal topology-dependent stretching: dense initial graphs stay compact, while sparse graphs can expand several times the visibility range. Open problems include handling noise, asynchrony, collisions, and proving convergence for coupled multi-agent swarms.
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Credibility Assessment:
Single author with no listed affiliation, arXiv preprint and zero citations — no recognizable institutional or author reputation signals.