Single-Chord Augmentation of Weighted Cycles for Algebraic Connectivity and Network Coherence
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| Format: | Preprint |
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2026
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| _version_ | 1866911713095319552 |
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| author | Deng, Jiarong Chang, Liu Yang, Quanshun |
| author_facet | Deng, Jiarong Chang, Liu Yang, Quanshun |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_24479 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Single-Chord Augmentation of Weighted Cycles for Algebraic Connectivity and Network Coherence Deng, Jiarong Chang, Liu Yang, Quanshun Optimization and Control Systems and Control 05C50, 90C27, 93B52 G.3.2; I.2.11; G.1.6 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. |
| title | Single-Chord Augmentation of Weighted Cycles for Algebraic Connectivity and Network Coherence |
| topic | Optimization and Control Systems and Control 05C50, 90C27, 93B52 G.3.2; I.2.11; G.1.6 |
| url | https://arxiv.org/abs/2605.24479 |