Weak synchronization in heterogeneous multi-agent systems

Ali Saberi; Anton A. Stoorvogel; Zhenwei Liu

REVIEW 3 major objections 5 minor 15 references

Weak synchronization in heterogeneous multi-agent systems

T0 review · 3 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper defines weak synchronization and proves that scale-free protocols for connected heterogeneous multi-agent networks automatically work on any network, preserving output synchronization within each surviving component when…

desk verdict Useful new concept (weak synchronization) with a clean equivalence theorem; the abstract overclaims and the discrete-time case is unproven, but the core math is sound. read the letter →

arxiv 2411.13806 v2 pith:ZR2ZRSXE submitted 2024-11-21 eess.SY cs.SY

Anton A. Stoorvogel , Ali Saberi , Zhenwei Liu This is my paper · ORCID

classification eess.SYcs.SY

keywords weaksynchronizationmulti-agentsystemsheterogeneousagentsscale-freeprotocolsoutputnetworkstabilitydirectedspanningtreebicomponents

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper proposes weak synchronization as the right goal for heterogeneous multi-agent systems when the communication network is unknown or can change. A network is weakly synchronized when the signals exchanged over it, the relative measurements $\zeta_i$, converge to zero, even if the agent outputs do not all converge to one common trajectory. The central result is that, for continuous-time heterogeneous agents with the paper's protocols, scale-free output synchronization and scale-free weak synchronization are equivalent: a protocol designed with no information about the network, and valid on graphs with a directed spanning tree, automatically makes the network stable on every graph. When a directed spanning tree exists, weak synchronization is exactly output synchronization; when it does not, outputs inside each basic bicomponent still synchronize and every other agent converges to a convex combination of those synchronized trajectories.

What carries the argument

The load-bearing object is the Laplacian kernel of a graph with $k$ basic bicomponents. This kernel is spanned by $k$ nonnegative vectors, each supported on one basic bicomponent together with the nodes that can reach or be reached from it, and the paper uses those vectors to split the full network dynamics into $k$ independent systems. On the support of each kernel vector, a diagonal rescaling by a matrix $\Gamma$ transforms the restricted Laplacian into a genuine Laplacian of a graph that has a directed spanning tree; the known scale-free output-synchronization protocol then applies to that graph, and summing the $k$ components recovers weak synchronization of the original network without any connectivity assumption.

What would settle it

Run the paper's 60-node continuous-time example, break the two links that remove the directed spanning tree, and record $\zeta_i(t)$: the claim is that $\zeta_i(t)\to 0$ for every agent while the disagreement inside each basic bicomponent converges to zero. A persistently nonzero $\zeta_i(t)$, or a disagreement within a basic bicomponent that does not decay, would falsify Theorem 2.

Watch

Extended reading notes

Core claim

The paper's central claim is that achieving output synchronization with a scale-free linear protocol is neither more nor less than achieving weak synchronization, as long as the protocols are of the given linear form. In practical terms, the paper proves that one never needs to know whether the network has a directed spanning tree in order to have a protocol that is safe: the same protocol that produces classical output synchronization on such networks also drives the network signals $\zeta_i$ to zero for every possible network with the same agents. If the network is disconnected into $k$ basic bicomponents, the outputs of agents within each basic bicomponent converge to a common trajectory, and the output of any agent outside these components converges to $y_j(s) = \sum_i \beta_{j,i} y_i(s)$, where the $\beta_{j,i}$ are nonnegative and sum to one and are fixed by the graph alone. For continuous-time systems this equivalence is Theorem 2, and the proof is built on decomposing the network dynamics into $k$ components aligned with the Laplacian's kernel and applying the known scale-free output-synchronization property on each component after a diagonal rescaling.

Load-bearing premise

The argument takes as given that a scale-free output-synchronization protocol already exists for the agent family, relying on earlier design results for introspective continuous-time agents and for discrete-time agents; if an agent class admits no such protocol, the weak-synchronization guarantee has nothing to attach to.

Editorial extensions

If this is right

Protocols designed without any information about the communication network, and verified only on graphs with a directed spanning tree, also make the network signals $\zeta_i$ converge to zero on every graph with the same agents.
When link failures destroy the directed spanning tree, each basic bicomponent still achieves output synchronization internally, so local consensus within surviving groups is preserved.
Every agent not in a basic bicomponent converges to a convex combination of the synchronized trajectories of the basic bicomponents, with coefficients that depend only on the graph's Laplacian and not on initial conditions.
For continuous-time heterogeneous agents the paper establishes that scale-free output synchronization and scale-free weak synchronization are equivalent, so checking one property certifies the other for the same protocol family.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A natural extension, suggested by the paper's own remark, is to test whether the reverse implication in Theorem 2 survives for nonlinear and delayed protocols; the proof's superposition step is the only place that clearly needs linearity.
Weak synchronization turns scale-free design into a fault-tolerance certificate: a protocol validated once on a spanning-tree graph carries a documented guarantee for arbitrary link failures, including the exact post-failure synchronization pattern.
Because the convex-combination coefficients $\beta_{j,i}$ are determined solely by the graph, the paper's Theorem 1 could be used in reverse, to identify the basic bicomponents of an unknown network from asymptotic output data when reference trajectories are distinguishable.
The paper's examples suggest the same guarantee holds for discrete-time heterogeneous agents; a formal discrete-time version of Theorem 2 would close the remaining gap.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Quick take: this is a real contribution. The weak synchronization notion—requiring only zeta_i -> 0 rather than output agreement—is a natural relaxation, and the iff result (weak sync iff output sync for scale-free linear protocols, Theorem 2) is new relative to the existing scale-free literature, which always assumed a spanning tree. Lemma 2 and Theorem 1 do a good job characterizing what survives when the graph loses its spanning tree: agents inside each basic bicomponent synchronize, and outside agents converge to a convex combination of those synchronized trajectories. The proofs are mostly clean. The rank argument in Theorem 2 checks out, and the decomposition into k systems with the zero-state decoupling for gamma_j^i = 0 is sound.

Soft spots, in order of importance. First, the abstract says protocols are designed that achieve weak synchronization for any network with no assumptions. That is true only conditional on the existence of a scale-free output synchronization protocol for the agent class, which comes from [4] or [13] under hypotheses (introspectiveness, right-invertibility, etc.) that are not stated here. The theorem itself is honestly conditional, but the abstract oversells it. Second, the proof of Theorem 2 has a notational imprecision: (Gamma^{-1} \otimes I) is not defined when agent state dimensions differ; it needs a block diagonal with per-agent identity matrices. That is a fix, not a flaw. Third, the discrete-time section uses protocols from [13] and simulates zeta_i -> 0, but Theorem 2 is stated only for continuous time; the discrete-time analog is not proved. Given the paper's title, an explicit discrete-time version (or a remark that the proof carries over under analogous conditions) would be appropriate.

The numerical examples are illustrative, not evidence, and the paper does not claim otherwise. The citation pattern is fine; self-citations are to the papers that actually contain the building-block protocols.

Who gets value: control theorists working on multi-agent synchronization, especially scale-free design and fault tolerance. It deserves a serious referee; the concept is worth having in the literature, and the main theorem is a genuine structural result. My recommendation: send to peer review, ask for an abstract revision and a clean-up of Theorem 2's transformation and the CT/DT gap.

Referee Report

3 major / 5 minor

Summary. The paper introduces the notion of weak synchronization for heterogeneous multi-agent systems, defined as convergence to zero of the network coupling variables zeta_i rather than of output differences. The main results are: Lemma 2, showing that weak synchronization implies ordinary output synchronization exactly when the graph has a directed spanning tree, and otherwise output synchronization is possible only in the trivial case of outputs decaying to zero; Theorem 1, characterizing the asymptotic output behavior under weak synchronization in terms of synchronization inside each basic bicomponent and convergence of non-basic agents to convex combinations of the basic synchronization trajectories; and Theorem 2, asserting that for continuous-time systems with protocols of the form (7), scale-free output synchronization and scale-free weak synchronization are equivalent. The proof of Theorem 2 uses a decomposition of the closed-loop state into components associated with the basic bicomponents and a graph-theoretic reduction to a spanning-tree network. Numerical examples for continuous- and discrete-time heterogeneous networks illustrate the behavior.

Significance. The concept of weak synchronization is natural and practically important: it separates the requirement of output synchronization from the weaker, but still useful, requirement that inter-agent communication signals vanish, which is meaningful when a network loses connectivity because of faults. Theorem 2 is a valuable robustness result: it says that any existing scale-free output-synchronization protocol automatically provides weak synchronization on every network, including those without a directed spanning tree, so the protocol degrades gracefully rather than failing catastrophically. Theorem 1's convex-combination characterization is clean and gives concrete predictions for non-basic components. The proofs are largely self-contained given the cited scale-free synchronization results, and no parameters are fitted to data. The main caveat is that the advertised design claim is conditional on the existence of scale-free output-synchronization protocols for the specific agent class, and the paper should state this condition prominently.

major comments (3)

[Abstract, Section 5 (Theorem 2)] The abstract and introduction claim that protocols are designed to achieve weak synchronization for any network without any assumptions on the communication network. However, Theorem 2 is conditional: it proves that scale-free output synchronization is equivalent to scale-free weak synchronization for protocols of the form (7), but the existence of a scale-free output-synchronization protocol for the agent family is not established in this paper. The paper cites [4] for continuous-time introspective agents and [13] for discrete-time agents but does not state their hypotheses (e.g., introspectiveness, right-invertibility, passivity where relevant). For agent classes outside those hypotheses, Theorem 2 provides no protocol and no weak-synchronization guarantee. The abstract and introduction should be rephrased to say: whenever a scale-free output-synchronization protocol exists for the agent family, that protocol automatically achieves weak synchronization on every network. The numerical examples in Section 6 also do not verify that the models satisfy the cited existence conditions, so they do not fill this gap.
[Section 5, proof of Theorem 2, around Eq. (27)] The similarity transformation xtilde = (Gamma^{-1} tensor I) x is not well-defined because the agents have heterogeneous state dimensions: the vector x stacks states of different dimensions, so a single Kronecker product with an identity matrix cannot be applied uniformly. The transformation should be defined blockwise, for example xtilde_e,l = gamma_l^{-1} x_e,l for each agent l in the support. Under that interpretation, the subsequent algebra (obtaining Ltilde_i = Gamma^{-1} L_i Gamma) is correct, but as written Eq. (27) is formally invalid and needs correction.
[Section 6.2, discrete-time case] Theorem 2 is stated only for continuous-time systems, but Section 6.2 applies the weak-synchronization conclusion to discrete-time agents using protocols from [13] and describes the result as consistent with the theory. The paper does not state or prove a discrete-time analogue of Theorem 2, although the decomposition argument appears to carry over when x^+ is interpreted as x(t+1). The discrete-time section should either be accompanied by an explicit discrete-time theorem (and a check that the cited [13] protocols satisfy its hypotheses) or be presented as a numerical illustration without claiming theorem support for the discrete-time case.

minor comments (5)

[Abstract] The sentence 'If the network happens to have a directed spanning tree, then we obtain classical synchronization' appears twice in nearly identical form in the abstract; one occurrence should be removed.
[Lemma 2 proof] The notation VL = [I -1] is confusing: [I -1] should be described explicitly as an (N-1) by N matrix whose kernel is the span of the all-ones vector, and the vector -1 should be written with a subscript (e.g., -1_{N-1}) to indicate its dimension.
[Section 6.1] The text says 'each agent is randomly assigned one of the above four models,' but only three models are listed in Section 6.1; it should say 'three models.'
[Section 5, proof of Theorem 2] In the paragraph introducing the extended null vector eta, the sentence 'where we have chosen eta_i = 0 when i not in {tau_i^1, ..., eta_{tau_i^{N_i}}}' contains a typo: it should be eta_v = 0 when v not in {tau_i^1, ..., tau_i^{N_i}}.
[References] There are several typographical errors: [3] should be 'Nojavanzadeh' not 'Nojavanzedah', and [6] should be 'Springer' not 'Spinger'; please check all author and publisher names.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Theorem 2 proves a conditional equivalence between scale-free output synchronization and scale-free weak synchronization, with a self-contained decomposition proof; protocol existence is imported from prior work but is not used as evidence for the equivalence.

full rationale

The paper's central claim, Theorem 2, is an implication: any protocol family of the form (7) that achieves scale-free output synchronization (Definition 3) also achieves scale-free weak synchronization (Definition 4), and conversely. The proof is self-contained in the manuscript. For the converse, weak synchronization implies output synchronization on spanning-tree graphs via Lemma 2; for the forward direction, the proof decomposes the initial condition into k components aligned with the basic bicomponents, builds the sub-Laplacian L_i with rank N_i-1, applies the similarity transform Gamma to obtain a classical Laplacian L-tilde_i with a spanning tree, and then invokes Definition 3 exactly as stated to get output synchronization on that subnetwork, which by Lemma 1 gives weak synchronization for that component. All structural facts (nullspace of L, nonnegativity of beta coefficients, grounded Laplacian invertibility) are proved or cited to standard external graph-theoretic sources (e.g., [2], [6], [14]). No parameter is fitted to data, no subnetwork property is redefined in terms of the conclusion, and the 'if and only if' is not tautological: the two definitions are genuinely different (output synchronization on all spanning-tree graphs versus zeta_i -> 0 on all graphs). The existence of scale-free output synchronization protocols is imported from [4] and [13], which are by overlapping authors, but those citations concern a different property (output synchronization under spanning-tree connectivity) and do not assume weak synchronization; hence they are independent support for the conditional result, not an input that is renamed as the conclusion. The abstract's phrasing that protocols are designed 'for any network' is broader than what the theorem alone establishes, since Theorem 2 applies only to agent families for which scale-free output synchronization protocols already exist, but this is a scoping/correctness concern, not circularity. No circular step satisfying the required evidence standard was found.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central results rest on standard graph theory and the cited scale-free protocol designs; no parameters are fitted to data and no new physical entities are introduced.

assumptions (4)

standard math The graph Laplacian nullspace basis in (13) exists for any graph with k basic bicomponents, with beta_{j,i} nonnegative and summing to 1 for each non-basic node.
Invoked in Theorem 1 proof, Section 4, using [6, Theorem 4.25] for nonnegativity of the grounded inverse.
domain assumption Scale-free output synchronization protocols of form (7) exist for the heterogeneous agent families considered (introspective agents in [4]; discrete-time agents in [13]).
Theorem 2 is conditional on this property; the paper does not construct new protocols.
domain assumption The closed-loop system (18) is linear, so the superposition decomposition x_e = sum x^i_e is valid.
Used in Theorem 2 proof, Section 5, equation (21).
standard math A grounded Laplacian has all eigenvalues in the open right half plane when the graph has no additional basic bicomponent.
Used in Section 2 for the structure (1).

how reviews work

0 comments

Cite this review

Pith. "Pith review of Weak synchronization in heterogeneous multi-agent systems." pith.science (2026). https://pith.science/paper/ZR2ZRSXE

@misc{pith2026241113806,
  author       = {Pith},
  title        = {Pith review of: Weak synchronization in heterogeneous multi-agent systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZR2ZRSXE}},
  note         = {Machine review of arXiv:2411.13806}
}

read the original abstract

In this paper, we propose a new framework for synchronization of heterogeneous multi agent system which we refer to as weak synchronization. This new framework of synchronization is based on achieving the network stability in the absence of any information on communication network including the connectivity. Here by network stability, we mean that in the basic setup of a multi-agent system, we require that the signals exchanged over the network converge to zero. As such if the network happens to have a directed spanning tree then we obtain classical synchronization. Moreover, we design protocols which achieve weak synchronization for any network without making any kind of assumptions on communication network. If the network happens to have a directed spanning tree, then we obtain classical synchronization. However, if this is not the case then we describe in detail in this paper what kind of synchronization properties are preserved in the system and the output of the different agents can behave.

Figures

Figures reproduced from arXiv: 2411.13806 by the authors.

**Figure 2.** A directed communication network with a spanning tree and its bicom [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** The 60-nodes communication network with spanning tree. [PITH_FULL_IMAGE:figures/full_fig_p020_3.png] view at source ↗

Figures from the paper (9 more)

**Figure 4.** Figure 4: The communication network without spanning tree. The links are broken [PITH_FULL_IMAGE:figures/full_fig_p020_4.png]

**Figure 5.** Figure 5: The trajectory of 𝜁𝑖 for continuous-time MAS. We have seen that for the 60-node network given in [PITH_FULL_IMAGE:figures/full_fig_p021_5.png]

**Figure 6.** Figure 6: Basic bicomponent 1 for continuous-time MAS: disagreement dynamic [PITH_FULL_IMAGE:figures/full_fig_p022_6.png]

**Figure 7.** Figure 7: Basic bicomponent 2 for continuous-time MAS: disagreement dynamic [PITH_FULL_IMAGE:figures/full_fig_p022_7.png]

**Figure 8.** Figure 8: Basic bicomponent 3 for continuous-time MAS: disagreement dynamic [PITH_FULL_IMAGE:figures/full_fig_p023_8.png]

**Figure 9.** Figure 9: The trajectory of 𝜁𝑖 for discrete-time MAS. 24 [PITH_FULL_IMAGE:figures/full_fig_p024_9.png]

**Figure 10.** Figure 10: Basic bicomponent 1 for discrete-time MAS: disagreement dynamic [PITH_FULL_IMAGE:figures/full_fig_p025_10.png]

**Figure 11.** Figure 11: Basic bicomponent 2 for discrete-time MAS: disagreement dynamic [PITH_FULL_IMAGE:figures/full_fig_p025_11.png]

**Figure 12.** Figure 12: Basic bicomponent 3 for discrete-time MAS: disagreement dynamic [PITH_FULL_IMAGE:figures/full_fig_p026_12.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages

[4]

Nojavanzadeh, Z

D. Nojavanzadeh, Z. Liu, A. Saberi, and A.A. Stoorvogel. Output and regulated output synchronization of heterogeneous multi-agent systems: A scale-free protocol design using no information about communication net- work and the number of agents. In American Control Conference , pages 865–870, Denver, CO, 2020

work page 2020
[13]

X. Wang, A. Saberi, and T. Yang. Synchronization in heterogeneous net- works of discrete-time introspective right-invertible agents. Int. J. Robust & Nonlinear Control, 24(18):3255–3281, 2013

work page 2013
[1]

Corfmat and A.S

J.P. Corfmat and A.S. Morse. Decentralized control of linear multivariable systems. Automatica, 12(5):479–495, 1976

work page 1976
[2]

Godsil and G

C. Godsil and G. Royle. Algebraic graph theory, volume 207 of Graduate Texts in Mathematics. Springer-Verlag, New York, 2001. 25 Figure 12: Basic bicomponent 3 for discrete-time MAS: disagreement dynamic among the agents and synchronized output trajectories

work page 2001
[3]

Z. Liu, D. Nojavanzedah, and A. Saberi. Cooperative control of multi-agent systems: A scale-free protocol design. Springer, Cham, 2022

work page 2022
[5]

Olfati-Saber and R.M

R. Olfati-Saber and R.M. Murray. Consensus problems in networks of agents with switching topology and time-delays. IEEE Trans. Aut. Contr., 49(9):1520–1533, 2004

work page 2004
[6]

Z. Qu. Cooperative control of dynamical systems: applications to au- tonomous vehicles. Spinger-Verlag, London, U.K., 2009

work page 2009
[7]

Ren and E

W. Ren and E. Atkins. Distributed multi-vehicle coordinate control via local information. Int. J. Robust & Nonlinear Control , 17(10-11):1002–1033, 2007

work page 2007

Show all 15 references

[8]

Ren and Y.C

W. Ren and Y.C. Cao. Distributed coordination of multi-agent networks . Communications and Control Engineering. Springer-Verlag, London, 2011

2011
[9]

D.D. Siljak. Decentralized control of complex systems . Academic Press, London, 1991. 26

1991
[10]

Stanoev and D

A. Stanoev and D. Smilkov. Consensus theory in networked systems. In L. Kocarev, editor, Consensus and synchronization in complex networks , pages 1–22. Spinger-Verlag, Berlin, 2013

2013
[11]

A. A. Stoorvogel, A. Saberi, Z. Liu, and D. Nojavanzadeh. 𝐻2 and 𝐻∞ almost output synchronization of heterogeneous continuous-time multi-agent systems with passive agents and partial-state coupling via static protocol.Int. J. Robust & Nonlinear Control, 29(17):6244–6255, 2019

2019
[12]

E. Tegling. Fundamental limitations of distributed feedback control in large- scale networks. PhD thesis, KTH Royal Institute of Technology, 2018

2018
[14]

C.W. Wu. Synchronization in complex networks of nonlinear dynamical systems. World Scientific Publishing Company, Singapore, 2007

2007
[15]

Wu and L.O

C.W. Wu and L.O. Chua. Application of Kronecker products to the anal- ysis of systems with uniform linear coupling. IEEE Trans. Circ. & Syst.-I Fundamental theory and applications, 42(10):775–778, 1995. 27

1995

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [4]

Nojavanzadeh, Z

D. Nojavanzadeh, Z. Liu, A. Saberi, and A.A. Stoorvogel. Output and regulated output synchronization of heterogeneous multi-agent systems: A scale-free protocol design using no information about communication net- work and the number of agents. In American Control Conference , pages 865–870, Denver, CO, 2020

work page 2020

[2] [13]

X. Wang, A. Saberi, and T. Yang. Synchronization in heterogeneous net- works of discrete-time introspective right-invertible agents. Int. J. Robust & Nonlinear Control, 24(18):3255–3281, 2013

work page 2013

[3] [1]

Corfmat and A.S

J.P. Corfmat and A.S. Morse. Decentralized control of linear multivariable systems. Automatica, 12(5):479–495, 1976

work page 1976

[4] [2]

Godsil and G

C. Godsil and G. Royle. Algebraic graph theory, volume 207 of Graduate Texts in Mathematics. Springer-Verlag, New York, 2001. 25 Figure 12: Basic bicomponent 3 for discrete-time MAS: disagreement dynamic among the agents and synchronized output trajectories

work page 2001

[5] [3]

Z. Liu, D. Nojavanzedah, and A. Saberi. Cooperative control of multi-agent systems: A scale-free protocol design. Springer, Cham, 2022

work page 2022

[6] [5]

Olfati-Saber and R.M

R. Olfati-Saber and R.M. Murray. Consensus problems in networks of agents with switching topology and time-delays. IEEE Trans. Aut. Contr., 49(9):1520–1533, 2004

work page 2004

[7] [6]

Z. Qu. Cooperative control of dynamical systems: applications to au- tonomous vehicles. Spinger-Verlag, London, U.K., 2009

work page 2009

[8] [7]

Ren and E

W. Ren and E. Atkins. Distributed multi-vehicle coordinate control via local information. Int. J. Robust & Nonlinear Control , 17(10-11):1002–1033, 2007

work page 2007