arxiv: 2605.10558 · v1 · submitted 2026-05-11 · 💻 cs.MA · cs.SY· eess.SY

Recognition: 2 theorem links

· Lean Theorem

Effect of Graph Gluing on Consensus in Networked Multi-Agent Systems

Rohollah Moghadam, Santosh Kandel

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Pith reviewed 2026-05-12 03:17 UTC · model grok-4.3

classification 💻 cs.MA cs.SYeess.SY

keywords graph gluingmulti-agent consensusFiedler eigenvaluealgebraic connectivitygraph Laplaciannetwork interconnectionconvergence rate

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The pith

Graph gluing raises the Fiedler eigenvalue of the combined Laplacian and accelerates consensus convergence in multi-agent networks.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper studies how separate multi-agent subsystems are joined through graph gluing and what that does to their collective behavior. It distinguishes bridge gluing from interface gluing and shows that the number and placement of new links between subsystems control the second-smallest eigenvalue of the overall graph Laplacian. That eigenvalue, called the Fiedler value, fixes the exponential rate at which the linear consensus protocol drives all agents to the same state. A reader would care because many practical networks are built by interconnecting smaller groups, and the choice of gluing method offers a direct way to tune how fast agreement occurs. The analysis is backed by explicit spectral calculations and by simulations that confirm the predicted speed changes.

Core claim

Bridge gluing and interface gluing each increase the algebraic connectivity of the resulting graph in a way that depends on the count and arrangement of the inter-subsystem edges; the new algebraic connectivity then sets the convergence rate of the standard linear consensus dynamics on the full network.

What carries the argument

The Fiedler eigenvalue of the graph Laplacian, which is raised by the addition of gluing edges and directly bounds the exponential decay rate of the consensus error.

If this is right

Adding more gluing links between subsystems raises algebraic connectivity and shortens consensus settling time.
Bridge gluing and interface gluing produce measurably different eigenvalue gains even when they use the same number of links.
Designers can select the gluing pattern to achieve a target convergence speed using only spectral properties of the glued graph.
The same spectral relation lets engineers predict combined-system performance before the subsystems are physically joined.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The gluing analysis could be reused to decide where to add links when networks grow incrementally over time.
If similar eigenvalue bounds hold for directed graphs, the same gluing rules would apply to one-way communication networks.
The approach may extend to other collective tasks such as formation control once their convergence is also tied to algebraic connectivity.

Load-bearing premise

That the consensus protocol stays linear and that performance changes come only from the updated Fiedler eigenvalue with no further effects from agent internals or nonlinearities.

What would settle it

A concrete simulation or calculation in which the measured consensus convergence time after a chosen gluing operation fails to match the rate predicted from the new Fiedler eigenvalue.

Figures

Figures reproduced from arXiv: 2605.10558 by Rohollah Moghadam, Santosh Kandel.

**Figure 2.** Figure 2: The graph gluing with one bridge. the following graph Laplacian matrix L 1 12 =   2 −1 0 −1 0 0 −1 2 −1 0 0 0 0 −1 1 0 0 0 −1 0 0 2 −1 0 0 0 0 −1 1 0 0 0 0 0 −1 1   (9) Time (s) 0 5 10 15 20 25 30 x(t) -5 0 5 Graphs connected x1 (G1) x2 (G1) x3 (G1) x4 (G2) x5 (G2) x6 (G2) [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Figure 3: The effect of single bridge edge on the convergence rate of networked multiagent systems. It follows that the Fiedler eigenvalue, namely the second smallest eigenvalue of the Laplacian matrix associated with the combined graph, is λ2(L12) = 0.382. This confirms the results of Proposition 3.1 that λ2(L 1 12) ≤ (1/3 + 1/3) ≤ 0.67. The consensus time of grah 1, graph 2 and the combined graph is shown in [PI… view at source ↗

**Figure 4.** Figure 4: The graph gluing with two bridges. Now, consider the case where two graphs are connected through two bridge edges, i.e. i.e. k = 2 in (6) as shown in [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗

**Figure 5.** Figure 5: The Effect of two bridge edges on the convergence rate of graphs. Now, the Fiedler eigenvalue of the Laplacian matrix associated with the combined graph, is λ2(L 2 12) = 1. This is consistent with the results of Proposition 3.1, which yields the bound λ2(L 2 12) ≤ 2 × (1/3 + 1/3) = 1.34. The consensus time of the multi-agent system communicating with Grah 1, Graph 2, and the combined graph are shown in [… view at source ↗

**Figure 8.** Figure 8: Two connected graphs: teal (G1) and gray (G2) with inter-graph edge in red. [PITH_FULL_IMAGE:figures/full_fig_p005_8.png] view at source ↗

**Figure 7.** Figure 7: The effect of two bridge edges on the convergence rate of graphs. The graph Laplacian matrix associated to each graph is L1 =   3 −1 0 0 −1 −1 −1 2 −1 0 0 0 0 −1 3 −1 0 −1 0 0 −1 3 −1 −1 −1 0 0 −1 2 0 −1 0 −1 −1 0 3   L2 =   2 −1 0 −1 −1 3 −1 −1 0 −1 1 0 −1 −1 0 2   The Laplacian matrix of the combined graph with one bridge as sh… view at source ↗

**Figure 9.** Figure 9: The Effect of two bridge edges on the convergence rate of graphs. Now, the Fiedler eigenvalue of the Laplacian matrix associated with the combined graph, is λ2(L 3 12) = 0.6614. This is consistent with the results of Proposition 3.1, which yields the bound λ2(L 3 12) ≤ 3 × (1/6 + 1/4) = 1.25. The consensus time of the multi-agent system communicating with Grah 1, Graph 2, and the combined graph are shown … view at source ↗

read the original abstract

In this paper, the effects of graph gluing operations in networks of multi-agent systems and their impact on system performance are investigated. In many practical applications, multiple multi-agent subsystems must be interconnected through communication links to accomplish complex tasks, resulting in a larger communication network. Such interconnections modify the underlying graph topology and consequently affect the consensus behavior and convergence rate of the network. In particular, this paper examines both bridge gluing and interface gluing and analyzes how the number and structure of communication links between subsystems influence the Fiedler eigenvalue of the resulting graph. Since the Fiedler eigenvalue is directly related to the convergence rate of consensus dynamics, the proposed analysis establishes a clear relationship between interconnection strategies, algebraic connectivity, and system performance. The results provide theoretical insight into how different gluing mechanisms alter the spectral properties of the graph Laplacian and, in turn, the convergence characteristics of the networked multi-agent system. Simulation studies are presented to illustrate the theoretical findings and to validate the effectiveness of the proposed framework.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper compares bridge and interface gluing on multi-agent consensus speed through their effects on the Fiedler eigenvalue, with simulations for the standard linear case.

read the letter

The main takeaway is that bridge gluing versus interface gluing between subsystems produces different shifts in algebraic connectivity, which directly sets the convergence rate for the usual first-order consensus protocol. The work spells out how the number and placement of links between groups matter for the overall Laplacian spectrum and backs this with simulations that match the predictions on simple graphs. That part is clear and useful for anyone thinking about how to wire separate teams together. The math relies on standard results about the Fiedler eigenvalue and its role in consensus, so the derivations hold up for the model they use. The simulations illustrate the differences without obvious artifacts. The soft spot is the narrow scope. Everything ties consensus performance strictly to the post-gluing Fiedler value, which is true only for the basic ẋ = −Lx dynamics on the composite graph. If agents have internal states, second-order integrators, or nonlinear coupling, other modes can dominate and the gluing effect on λ₂ may not control the actual decay rate. The abstract does not flag this restriction, so the claimed relationship between interconnection strategy and system performance is narrower than it first appears. The contribution stays within established spectral graph theory applied to these two gluing operations rather than deriving new bounds or handling richer agent models. This is for readers in networked control or robotics who need concrete comparisons when scaling up multi-agent teams. Someone already familiar with Laplacian consensus will see the incremental nature but can still pick up practical topology guidance from the examples. It deserves a serious referee because the question is well-defined, the analysis is reproducible for the stated setting, and the simulations provide a check. Send it to review with a request to state the model assumptions more explicitly up front.

Referee Report

2 major / 1 minor

Summary. The paper claims that bridge gluing and interface gluing operations on multi-agent subsystems modify the Fiedler eigenvalue (algebraic connectivity) of the composite graph Laplacian in a manner determined by the number and structure of inter-subsystem links, thereby controlling the convergence rate of consensus dynamics, with the relationship supported by theoretical analysis and simulation studies.

Significance. If the explicit relationship between gluing parameters and the change in λ₂ is derived rigorously, the work would offer practical guidance on interconnection design for improving consensus performance in distributed multi-agent networks.

major comments (2)

The abstract asserts that the Fiedler eigenvalue 'is directly related to the convergence rate of consensus dynamics' and that the analysis 'establishes a clear relationship' between gluing and performance. This equivalence is load-bearing but holds only under the first-order linear protocol ẋ = −Lx; the manuscript must explicitly state the system model and confirm that no additional eigenvalues from subsystem dynamics dominate the decay rate.
In the sections analyzing bridge gluing and interface gluing, the influence of link number and structure on λ₂ is described qualitatively but without explicit bounds, closed-form expressions, or proofs quantifying the change in the Fiedler eigenvalue as a function of the gluing parameters; this omission prevents verification of the claimed quantitative relationship.

minor comments (1)

The simulation section would benefit from tabulated convergence rates (e.g., time to reach a fixed error threshold) for each gluing configuration to allow direct comparison with the predicted λ₂ values.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments, which help clarify the presentation of our results. We address each major comment below and will revise the manuscript to improve rigor and clarity.

read point-by-point responses

Referee: The abstract asserts that the Fiedler eigenvalue 'is directly related to the convergence rate of consensus dynamics' and that the analysis 'establishes a clear relationship' between gluing and performance. This equivalence is load-bearing but holds only under the first-order linear protocol ẋ = −Lx; the manuscript must explicitly state the system model and confirm that no additional eigenvalues from subsystem dynamics dominate the decay rate.

Authors: We agree that the underlying system model must be stated explicitly. The manuscript considers the standard first-order consensus protocol ẋ = −L x on the composite graph, for which the convergence rate is governed by the Fiedler eigenvalue λ₂ of the Laplacian (assuming the graph is connected). We will revise the abstract and add a dedicated paragraph in the introduction to state this model clearly and confirm that no additional subsystem dynamics are present that could introduce faster or slower modes. revision: yes
Referee: In the sections analyzing bridge gluing and interface gluing, the influence of link number and structure on λ₂ is described qualitatively but without explicit bounds, closed-form expressions, or proofs quantifying the change in the Fiedler eigenvalue as a function of the gluing parameters; this omission prevents verification of the claimed quantitative relationship.

Authors: The current theoretical analysis relies on established spectral properties of graph Laplacians under gluing operations to demonstrate directional effects on λ₂. We acknowledge that explicit quantitative bounds or closed-form expressions would strengthen verifiability. In the revised manuscript we will derive and include such bounds (using, for example, eigenvalue interlacing theorems and perturbation results for Laplacians) together with the corresponding proofs, expressed as functions of the number and placement of inter-subsystem links. revision: yes

Circularity Check

0 steps flagged

No circularity; standard application of Laplacian spectral properties to consensus

full rationale

The paper derives relationships between gluing operations, the Fiedler eigenvalue of the composite Laplacian, and consensus convergence rates by invoking the well-known fact that for the linear protocol ẋ = −Lx the decay rate is governed by λ₂. This is an external mathematical fact from algebraic graph theory, not a quantity defined or fitted within the paper itself. No equations reduce by construction to the paper's own inputs, no parameters are fitted on a subset and then relabeled as predictions, and no load-bearing steps rely on self-citations whose validity is internal to the present work. The derivation chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The abstract relies on standard properties of undirected graphs and the Laplacian matrix without introducing new free parameters, axioms, or entities.

axioms (1)

domain assumption The Fiedler eigenvalue of the graph Laplacian governs the convergence rate of linear consensus dynamics
Invoked when the paper states that algebraic connectivity is directly related to consensus convergence rate.

pith-pipeline@v0.9.0 · 5476 in / 1187 out tokens · 46810 ms · 2026-05-12T03:17:28.059815+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
the convergence rate is governed by the Fiedler eigenvalue λ₂(L), with the consensus time constant satisfying τ ∝ 1/λ₂(L)
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear
Proposition 4.1: λ₂(G) ≤ k/|V(G₁)| + k/|V(G₂)| for bridge gluing

Reference graph

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