Abstract:Ring-like communication graphs appear in UAV formations, cyclic patrols, perimeter monitoring, and other multi-agent tasks in which agents exchange information mainly with neighboring vehicles along a closed route. When measurement and actuation noise are persistent, a useful augmentation should improve both the convergence rate of consensus and the steady-state disagreement level. This paper studies the addition of a single weighted chord to a connected weighted cycle. The central observation is that a chord is not just a generic rank-one edge update: it splits the cycle into two complementary resistance arcs, and this resistance split governs both the algebraic-connectivity gain and the Kirchhoff-index reduction. We first derive exact chord-induced effective-resistance and Kirchhoff-index update formulas, giving a closed-form coherence objective. We then prove that, under bounded conductances and small resistance discrepancy, near-antipodal resistance-balanced chords are near-optimal for algebraic-connectivity improvement; an i.i.d. bounded-conductance model yields the same conclusion with high probability. Finally, because the best convergence-rate chord and the best coherence chord need not coincide, we formulate the design as a finite Pareto problem and introduce RBAPS and AW-RBAPS, two resistance-balanced screening rules that retain only linear or near-linear candidate sets. Numerical experiments show that AW-RBAPS remains effective beyond the formal moderate-heterogeneity regime and approximates the exhaustive Pareto front with mean hypervolume ratio $0.9987$ while evaluating about $10.1\%$ of admissible chords.
| Subjects: | Optimization and Control (math.OC); Systems and Control (eess.SY) |
| MSC classes: | 05C50, 90C27, 93B52 |
| ACM classes: | G.3.2; I.2.11; G.1.6 |
| Cite as: | arXiv:2605.24479 [math.OC] |
| (or arXiv:2605.24479v1 [math.OC] for this version) | |
| https://doi.org/10.48550/arXiv.2605.24479 arXiv-issued DOI via DataCite (pending registration) |
From: Chang Liu [view email]
[v1]
Sat, 23 May 2026 09:03:38 UTC (124 KB)
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