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Connectivity Preservation and Graph Stretching in Range-Only Swarm Dispersion

Ariel Barel

arXiv:2609.28190v1cs.MA

TL;DR

The paper asks how anonymous agents with range-only sensing can disperse while preserving visibility edges. It derives a maximal safe jump based on the farthest visible neighbor and proves two-agent convergence, while general-swarm experiments characterize topology-dependent expansion without establishing general convergence.

  • Problem

    Under range-only sensing, agents lack bearings and other information for reconstructing local geometry or determining whether a visible edge is redundant, although losing a critical edge can disconnect the swarm.

    Source: §I · p. 1

  • Method

    The rule uses the farthest visible-neighbor range to set a maximal safe isotropic jump, then moves each agent in a random direction within that certified radius.

    Source: §III · p. 2, §VIII · p. 7

  • Results

    For two agents, maximal safe jumps yield positive conditional squared-distance drift, almost-sure convergence to the visibility boundary, and finite expected time to any fixed boundary neighborhood; general-swarm experiments show topology-dependent stretching but not general convergence.

    Source: §VII · p. 6, §VIII · p. 7

  • Takeaways & Limitations

    The results establish edge preservation and structural guarantees for arbitrary swarm size, while limiting stochastic convergence guarantees to the two-agent system.

    Source: §VIII · p. 7

    Preserving every edge is conservative: complete graphs cannot exceed diameter V, and safely releasing redundant constraints from anonymous current ranges remains open.

    Source: §VII · p. 6

Abstract

from arXiv · show

We study connectivity-preserving finite-jump dispersion of anonymous, identical, and oblivious agents under an idealized range-only sensing model. Each agent measures only the distances to its visible neighbors, without bearings, identifiers, communication, memory, or a shared coordinate system. We derive the largest isotropic displacement certifiable as safe from these measurements alone. The resulting rule requires only the distance to the farthest visible neighbor: each agent selects a random direction and moves by half of its remaining visibility margin. The rule preserves every existing visibility edge under synchronous finite motion and therefore preserves connectivity. For two agents, we prove positive conditional drift in squared distance, almost-sure convergence to the visibility boundary, and finite expected time to reach any fixed neighborhood of that boundary. A one-million-run Monte Carlo experiment agrees with the exact first-round moments and estimates approximately 9.5 rounds to reach distance 0.97V from coincident initial positions; an independent Bellman-equation computation gives the same estimate. For general swarms, 1,000 runs across five initial-topology classes reproduce the deterministic safety guarantee at implementation level and reveal a consistent topology-dependent ordering of attainable diameter under the tested protocol. These results provide a theoretical foundation for connectivity-preserving multi-robot dispersion under minimal sensing, while isolating the guarantees achievable from anonymous range measurements alone.

I. INTRODUCTION

The paper asks how anonymous agents with range-only sensing can disperse while preserving visibility edges, and derives a maximal safe isotropic jump from current ranges. It establishes structural guarantees for arbitrary swarms, stochastic convergence results for two agents, and distinguishes this setting from prior approaches requiring richer observations.

I. INTRODUCTION

  • Each agent moves in a random direction by half the remaining visibility margin to its farthest visible neighbor, the largest bearing-independent isotropic displacement certified from current ranges.The rule preserves every current edge under synchronous finite motion, so an initially connected swarm remains connected.Source: §I · p. 1
  • Every absorbing configuration contains at least n/2 visibility-boundary edges, and an edge at distance V permanently immobilizes both endpoints.The paper also frames dispersion as increasing swarm extent, not maximizing minimum pairwise distance.Source: §I · p. 1
  • For two agents, squared distance has positive conditional drift below V, yielding almost-sure convergence to the visibility boundary and finite expected time to any fixed boundary neighborhood.For larger swarms, the paper reports topology-dependent empirical stretching rather than a convergence proof.Source: §I · p. 1

II. RELATED WORK

  • Prior connectivity-control and navigation approaches use relative positions, graph information, navigation functions, or trained policies unavailable under the stated range-only sensing model.Nearby scalar-distance approaches still use bearing, state estimation, target distances, landmarks, or communication.Source: §II · p. 2
  • The novelty is the maximality of the action certificate under anonymous range-only information, its reduction to one scalar statistic, and the stochastic behavior induced by repeatedly exhausting that certificate.The paper notes that preserving every edge is conservative because current anonymous ranges do not reveal which edges are globally redundant.Source: §I · p. 1, §II · p. 2

III. MODEL AND MOTION RULE

Agents observe the current ranges to visible neighbors and use the farthest range to determine a safe isotropic movement radius. The radius is maximal under the requirement that every current edge survive all feasible bearings and synchronous moves.

III. MODEL AND MOTION RULE

  • For a visible pair at distance d, the largest common isotropic displacement that guarantees edge preservation is half the remaining visibility margin.Any larger radius permits both agents to move directly apart and exceed V.Source: §III · p. 2
  • With multiple visible neighbors, the farthest neighbor imposes the smallest safe radius, so the full multiset of ranges reduces to its maximum.The corresponding safe disks are concentric, and the smallest disk is induced by the farthest current neighbor.Source: §III · p. 2
  • Agents sample independent uniform directions and move within the certified radius using only current ranges, without persistent neighbor identities or geometric reconstruction.The initial visibility graph is assumed connected.Source: §III · p. 2

A. Information Use and Design Rationale

The rule separates deterministic edge safety from exploration: a scalar range statistic sets the safe radius, while a random direction uses that radius. In the two-agent safe family, taking the full radius maximizes one-step expected squared-distance progress.

A. Information Use and Design Rationale

  • The full range multiset reduces to the farthest-neighbor distance because all pairwise safety disks are centered at the current agent.No neighborhood ordering, persistent measurement association, geometric reconstruction, or target formation is required.Source: A. Information Use and Design Rationale · p. 2
  • The radius certifies that every direction is safe, while a random direction explores the safe set without reconstructing bearings.Some neighbor distances can decrease on a move, but no current edge is lost.Source: A. Information Use and Design Rationale · p. 2
  • α = 1 maximizes one-step expected squared-distance progress in the two-agent safe isotropic family.Scaling both movement vectors by α multiplies the quadratic drift term by α^2.Source: A. Information Use and Design Rationale · pp. 2–3

IV. SAFETY AND STRUCTURAL GUARANTEES

Every existing visibility edge is preserved at each round, so an initially connected swarm remains connected. Structural guarantees also characterize boundary locking and the boundary edges present if the swarm becomes absorbing.

IV. SAFETY AND STRUCTURAL GUARANTEES

  • Every current visibility edge persists at the next round, for every realization of the selected directions.The edge-preservation guarantee is independent of the direction-selection distribution, provided each displacement respects the certified radius.Source: §IV · p. 3
  • An initially connected swarm remains connected because its original graph persists as a spanning subgraph.The result follows directly from preservation of every initial edge.Source: §IV · p. 3
  • A visibility-boundary edge locks both endpoints: neither moves thereafter, so their mutual distance remains V.Each endpoint has a visible neighbor at distance V and therefore has no remaining movement radius.Source: §IV · p. 3
  • Every absorbing configuration contains at least n/2 edges of length V.This lower bound is conditional on absorption and does not establish general-swarm convergence.Source: §IV · p. 3
  • Exact equality is an ideal-model property; numerical implementations must distinguish a tolerance band from exact locking.Finite precision can otherwise create apparent locking or unlocking events.Source: §IV · p. 3

V. TWO-AGENT STOCHASTIC STRETCHING

For two agents, the stochastic analysis establishes positive squared-distance drift and almost-sure approach to the visibility boundary, without finite-time exact arrival. It also bounds the expected time to enter any fixed neighborhood of that boundary.

V. TWO-AGENT STOCHASTIC STRETCHING

  • Positive conditional drift in squared distance yields almost-sure convergence to V for every initial distance D0 in [0, V].The process is reduced to the scalar inter-agent distance using rotational symmetry and fresh independent directions.Source: §V · p. 3
  • Agents starting below V almost surely approach the boundary without hitting it exactly in finite time.Exact arrival requires direction pairs in a measure-zero subset of possible pairs.Source: §V · pp. 3–4
  • The expected time to reach any fixed boundary neighborhood is finite when D0 < V − ε.Before entry, the agents retain the original maximal safe jump radius; the proof uses a drift lower bound of ε^2/2 and notes the resulting bound can be conservative.Source: §V · p. 4

VI. EXPERIMENTAL EVALUATION

The evaluation combines two-agent simulations and Bellman calculations with multi-run studies of larger swarms. It finds near-boundary hitting times of about 9.5 rounds for two agents and topology-dependent final diameters for general swarms under the tested protocol.

General swarms

  • No edge or connectivity loss was observed over five million rounds; complete graphs stayed within V, while paths expanded over many visibility ranges.The authors describe this as an implementation check of the deterministic safety guarantee, not independent evidence for the theorem.Source: §VI · p. 4
  • 1.65V, 2.2V, and 3.6V were the median final diameters for dense, medium, and sparse random-connected N = 10 groups.The final normalized diameter comparison covers five N = 10 topology groups, with 200 runs per group.Source: §VI · p. 4, Figure 1 · p. 4
  • Final diameter measures global expansion, while fixed-tree stretching measures deformation of an initially connectivity-sufficient structure; their variability differed.Within the random-connected family, greater initial density was associated with smaller final diameter under the tested initialization protocol.Source: §VI · pp. 4–5
  • Most observed expansion occurred in the first 500 rounds, followed by substantially slower variation through round 5,000; topology differences persisted across runs.Fixed-tree edges stretched in every topology group, with strongest normalized growth for paths and sparse random-connected graphs.Source: §VI · p. 5
  • Complete < Dense < Medium < Sparse < Path describes the median final-diameter ordering across tested swarm sizes.The N = 20 and N = 50 sensitivity experiments used 100 runs per group.Source: §VI · p. 5, Figure 3 · p. 5
  • 44.23V versus 17.22V were the path groups’ median final diameters at N = 50 and N = 20, respectively.The N = 20 and N = 50 diameter comparisons used 100 runs per group.Source: §VI · p. 5, Figure 3 · p. 5

Two agents

  • 9.5143 rounds was the mean first-hitting time for D_t ≥ 0.97V in one million two-agent runs starting at D_0 = 0.The median was 7 rounds, and only 1.2 × 10^-5 of runs had not entered by round 100.Source: §VI · p. 5, Figure 4 · p. 6
  • Approximately 9.50 rounds from the Bellman calculation independently corroborated the Monte Carlo estimate for the same first-hitting-time quantity.The computed linear-system residual was below 5 × 10^-14, which verifies the solve but does not bound discretization error.Source: §VI · p. 5, Table I · p. 5

VII. DISCUSSION AND LIMITATIONS

The policy preserves edges and connectivity, but its conservative retention of every edge constrains geometry; the paper’s claims assume exact ranges and synchronous motion.

VII. DISCUSSION AND LIMITATIONS

  • For general swarms, edge preservation, connectivity, boundary locking, and a boundary-edge lower bound conditional on absorption are guaranteed, but topology-dependent stretching is empirical rather than a convergence result.The policy separates its deterministic safety certificate from random exploration.Source: §VII · p. 6
  • Complete graphs cannot exceed diameter V, whereas sparse connected graphs can expand over several visibility ranges; safely releasing redundant constraints remains open.Preserving every edge retains every pairwise constraint in a complete graph.Source: §VII · p. 6
  • The analysis assumes exact ranges and synchronous motion, leaving noise, actuation errors, asynchrony, collisions, and robot dynamics outside scope.Robustification would require a noise-aware safety margin and analysis of delayed or asynchronous observations.Source: §VII · p. 6

A. Interpretation of the General-Swarm Results

In larger swarms, pairwise progress is coupled through each endpoint’s farthest-neighbor range. Experiments suggest initial topology constrains attainable geometry, but do not establish a density-based law or convergence.

A. Interpretation of the General-Swarm Results

  • Each edge endpoint’s movement radius depends on its own farthest visible neighbor, coupling pairwise progress through the rest of the graph.The two-agent scalar drift argument therefore does not directly extend to larger swarms.Source: A. Interpretation of the General-Swarm… · p. 6
  • Experiments indicate that preserving many initial edges imposes stronger geometric confinement, while sparse connected structures permit expansion over several visibility ranges.This is an observed topology-dependent pattern, not a theorem that edge density alone determines final diameter.Source: A. Interpretation of the General-Swarm… · p. 6
  • The observed ordering is specific to the stated initialization protocol, and convergence to an absorbing state is not established.A theoretical characterization of limiting configurations for coupled swarms remains open.Source: A. Interpretation of the General-Swarm… · p. 6

B. Limitations of Preserving Every Edge

Retaining every edge is conservative because anonymous range-only agents cannot identify redundant constraints. This limits complete-graph expansion while leaving safe release of redundant edges unresolved.

B. Limitations of Preserving Every Edge

C. Scope and Future Robustification

The assumptions of exact ranges and synchronous jumps isolate the range-only information question, but do not capture several physical-platform limitations. Robust implementation would need additional safety and execution analysis.

C. Scope and Future Robustification

VIII. CONCLUSION

The paper derives a maximal isotropic displacement rule from anonymous current ranges that preserves every visibility edge and connectivity. It establishes convergence results for two agents and reports topology-dependent diameter ordering in general-swarm experiments, while leaving coupled-swarm convergence open.

VIII. CONCLUSION

  • The farthest-neighbor rule uses the maximal certifiable isotropic displacement and preserves connectivity for arbitrary swarm size.It preserves every visibility edge under synchronous finite motion.Source: §VIII · p. 7
  • Every absorbing configuration contains at least n/2 boundary edges, which permanently anchor endpoints at the visibility limit.Edges at the visibility limit immobilize both endpoints.Source: §VIII · p. 7
  • For two agents, maximal safe random jumps yield positive conditional drift, almost-sure boundary convergence, and finite expected time to any fixed boundary neighborhood.Exact first-round moments, one million Monte Carlo runs, drift validation, and an independent Bellman calculation provide consistent characterizations.Source: §VIII · p. 7
  • General-swarm experiments reproduce the safety guarantee and show a consistent topology-dependent ordering of attainable diameter across tested swarm sizes.Convergence for coupled swarms, limiting configurations, and noise-aware variants remain open problems.Source: §VIII · p. 7
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