Certifying Collective Reasoning in Multi-Agent Systems via Koopman Spectral Analysis

Indrakshi Dey; Nuzhat Khan

REVIEW 4 major objections 5 minor 32 references

Certifying Collective Reasoning in Multi-Agent Systems via Koopman Spectral Analysis

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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

Pith's one-line read The paper claims that the sub-dominant eigenvalue of an estimated Koopman operator certifies a reasoning collective's convergence deadline, its factions, and an auditable message code, all from interaction traces alone.

desk verdict Genuinely new and honestly evaluated, but the 'certificate' claim is undermined by an unproved spectral-dominance assumption and an unreconciled 96-vs-100% coverage inconsistency. read the letter →

arxiv 2608.05956 v1 pith:O6FJRLMV submitted 2026-08-06 cs.MA cs.SYeess.SY

Nuzhat Khan , Indrakshi Dey This is my paper · ORCID

classification cs.MAcs.SYeess.SY

keywords Koopmanoperatortheorymulti-agentLLMdebateconvergencecertificateattention-consensusdynamicseigenvalueattributionspectralmessagecompressionexplainableAI

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

This paper claims that an orchestrated collective of LLM agents debating and voting can be treated as a single nonlinear dynamical system, and that the spectrum of a linear Koopman operator estimated from recorded interaction traces turns three open questions into computable certificates: whether the collective will converge, in how many rounds, and what drove the decision. The sub-dominant eigenvalue $\lambda_2$ fixes the intrinsic timescale of reasoning and yields a convergence deadline before the debate runs, its eigenvector names the factions the collective reasons in, and the leading spectral coordinates form a compressed message basis. On an attention-consensus model, the deadline tracks observed convergence with log-log correlation 0.93 and bounds it in 96% of 24 configurations; attribution is exact whenever $|\lambda_2| > 0.9$; and a certificate learned from 15 debates holds on all 60 held-out QA debates. The authors are explicit that these validations are on a reference model, not on live language-model collectives, and that transferring the certificates to real LLM traces is the open next step.

What carries the argument

The load-bearing object is the Koopman transfer operator, an exact linear operator on a space of observable functions that represents a nonlinear map by composition; its eigenvalues encode decay timescales and its eigenfunctions encode spatial patterns. The paper approximates it from traces with extended dynamic mode decomposition (EDMD), regressing one-step dictionary values under a ridge penalty, using a dictionary of linear coordinates plus random Fourier features. From the estimated spectrum it reads the sub-dominant eigenvalue $\lambda_2$, defines the spectral gap $\gamma = 1 - |\lambda_2|$, and builds the three certificates: the deadline formula $T_{\mathrm{cert}}$, the validity flag $|\lambda_2| > 0.9$ that gates mode attribution, and the top spectral coordinates used for message compression.

What would settle it

Run the attention-consensus collective with planted factions and measure the disagreement curve; if a fresh rollout's normalised disagreement fails to cross the tolerance by the certified deadline $T_{\mathrm{cert}} = \lceil \ln(1/\epsilon)/(-\ln|\lambda_2|)\rceil$ in more than a small fraction of configurations, the certificate is unsound. A sharper test constructs an initial condition that strongly excites a faster mode than $\lambda_2$, violating the coefficient bound, and checks whether $D(t) \le C|\lambda_2|^t$ still holds.

Watch

Extended reading notes

Core claim

The central claim is that all three certification questions become spectral questions once the collective's state is lifted to a space of observable functions and the Koopman transfer operator is approximated by extended dynamic mode decomposition with a generic dictionary of linear coordinates and random Fourier features. Concretely, Proposition 1 states that if the centered deviation admits a Koopman mode expansion $\delta(t) = \sum_{j\ge 2} c_j \lambda_j^t v_j$ with $|\lambda_2| \ge |\lambda_3| \ge \cdots$ and $\sum_{j\ge 2} |c_j|\,\|v_j\| \le C\,\|\delta(0)\|$, then normalised disagreement obeys $D(t) \le C|\lambda_2|^t$, so the certified deadline is $T_{\mathrm{cert}} = \lceil \ln(1/\epsilon)/(-\ln|\lambda_2|)\rceil$ with $\epsilon$ the tolerance. The paper's validation on the attention-consensus model shows the deadline tracks observed convergence across more than a decade of timescales, the slow eigenvector recovers planted factions with perfect accuracy whenever $|\lambda_2| > 0.9$, and the top $k$ spectral coordinates preserve the final decision at 99.7% fidelity at a 4x bandwidth reduction.

Load-bearing premise

The load-bearing premise is that the centered disagreement of a debating collective has a Koopman mode expansion with one dominant slow mode, meaning the eigenvalue ordering and the coefficient bound in Proposition 1 hold; the paper states this as a working hypothesis and probes it empirically, but does not prove it for the attention-consensus dynamics, and the deadline formula collapses if it fails.

Editorial extensions

If this is right

A deployed collective can be assigned a worst-case round budget before it runs, computed from a handful of traces, rather than a fixed guess; in the paper's grid a fixed five-round budget covered only 4% of configurations.
Explanations become self-certifying: the same spectral object that names the factions also reports when no metastable structure exists, declining to attribute when the gap is wide.
Spectral compression gives an auditable channel: because the retained coordinates are the certificate basis, the compressed messages remain expressed in the coordinates of the explanation while preserving the decision.
Certification can be trained on small data and runs cheaply: a certificate learned from 15 debates generalised to all 60 held-out QA debates, and the whole pipeline runs in minutes on a CPU.

Reading between the lines

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

Beyond the paper: if the spectral dominance premise survives on real LLM traces, the deadline formula makes round budgeting an engineering input rather than a guess, but the validity of the premise depends on belief embeddings capturing argument structure; embeddings that discard disagreement content would break the certificate.
The ablation result that linear and nonlinear dictionaries perform indistinguishably suggests that near-consensus LLM debates may be well described by a linearisation, so the first deployment should test whether real debate trajectories stay in that regime; this extends the paper's own negative-result interpretation into a testable precondition.
The measured concentration rate $M^{-0.36}$ implies effective sample size is governed by trace mixing, so an experimenter can estimate how many real debates are needed before certification: roughly ten traces place $|\lambda_2|$ to within a few points in the model, and a mixing-time estimate on real transcripts would convert that into a budget.

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

Colleague,

The one thing you should know: this is a serious, honest methods paper that does something genuinely new—packaging Koopman/EDMD spectral analysis as three certification primitives for LLM debate collectives—but the headline 'certificate' language outruns the evidence. The deadline certificate rests on an unproved spectral-dominance assumption, and the paper reports two incompatible coverage numbers for the same experiment (96% in §VI-A, 100% in Table IV).

What is genuinely good: the attention-consensus reference model is well-posed and reproducible; the evaluation is careful (disjoint training and held-out runs, 20 rollouts per configuration, explicit reporting of violations); the functional-form validation (Fig. 8b) is a nice check; the dictionary ablation is an honest negative result; and code and traces are released. The paper is also explicit that Proposition 1 is a working hypothesis and that real LLM transfer is open. That honesty is real.

The soft spots, in proportion. The 96%-vs-100% discrepancy is not cosmetic: §VI-A reports one configuration where the certified deadline is 0.8× the mean observed convergence, which by definition is not an upper bound, yet Table IV lists 100% coverage. The authors need to reconcile this, or the headline claim fails. Second, the deadline 'certificate' is actually a conditional bound: the modal expansion of Prop. 1 is assumed, not proved for the attention-consensus dynamics, and EDMD gives an approximate spectrum from finite data, so the estimated λ2 does not formally bound worst-case trajectories. The paper acknowledges this, but then the word 'certificate' is misleading; 'empirically validated heuristic bound' is the honest label. Third, coverage is judged against the mean first-passage round over 20 rollouts, not per rollout, so individual debates can exceed Tcert even in 'covered' configurations. Fourth, the compression certificate uses the PCA directions of round-0 messages, not the Koopman spectrum, despite the name; that is a smaller issue but should be stated. Finally, no real LLM collectives are tested; everything is on the synthetic model, so the deployment claims are prospective.

Who this is for: anyone working on certification, monitoring, or explanation of multi-agent LLM systems. It deserves a serious referee. I would send it to review, with the expectation that the authors reconcile the coverage numbers and soften 'certificate' to 'empirically validated bound' unless the theory is supplied. As is, it's a solid conditional-accept paper, not a finished guarantee.

Referee Report

4 major / 5 minor

Summary. The paper proposes a Koopman-operator framework for certifying collective reasoning in multi-agent LLM debate systems. Treating the collective as a nonlinear dynamical system on belief embeddings, the authors estimate a Koopman transfer operator from interaction traces via extended dynamic mode decomposition (EDMD), then read off three certificates from its spectrum: a convergence deadline Tcert from the sub-dominant eigenvalue lambda_2, a faction-attribution explanation from the corresponding eigenvector with a validity flag based on |lambda_2|, and a message-compression basis from the leading spectral coordinates. The framework is validated on a reference attention-consensus model with planted factions and a QA variant, reporting that the deadline tracks observed convergence with log-log correlation 0.93 and bounds it in 96% of 24 configurations, that attribution is exact when |lambda_2|>0.9, that 8 of 32 spectral coordinates preserve decisions at 99.7% fidelity, and that a certificate learned from 15 training debates holds on 60/60 held-out QA debates. The paper explicitly labels its main theoretical assumption as a working hypothesis and discusses future work toward unconditional guarantees.

Significance. If the central claims held as stated, the paper would offer a cheap, trace-only method for predicting convergence, explaining faction structure, and compressing messages in LLM collectives, with clear relevance to trustworthy deployment. The study has real strengths: the evaluation is careful about disjoint training and held-out runs, the reference model is simple and reproducible, code and traces are released, and the paper is unusually explicit about the assumptions behind its deadline formula (Proposition 1 and Remark 1). The empirical results, especially the concentration curve and the functional-form validation, are interesting even if the word 'certificate' turns out to be too strong. However, the current manuscript contains a direct internal inconsistency in the headline coverage numbers and supports its 'certificate' language with an unproved spectral-dominance assumption, so the claims as stated require substantial revision.

major comments (4)

[§VI-A and Table IV] Section VI-A reports that Tcert upper-bounded the mean observed convergence round in 23 of 24 configurations (96% coverage), with median conservatism Tpred/Tobs = 2.0 and range 0.8–4.8. Table IV, for the same 24-configuration grid, reports 100% coverage for the Koopman certificate. These two numbers are mutually inconsistent: a ratio of 0.8 implies at least one configuration in which the mean observed convergence exceeded the predicted deadline, so coverage cannot be both 96% and 100%. Since Section VI-I's claim that the certificate is 'sound as a bound' and Table II's 'never unsound' rest on this number, the discrepancy must be resolved before the soundness claim can be accepted.
[§V-B, Proposition 1] The deadline 'certificate' is conditional on an unproved modal expansion. The proof assumes delta(t) = sum_{j>=2} c_j lambda_j^t v_j with |lambda_2| >= |lambda_3| >= ... and sum |c_j| ||v_j|| <= C||delta(0)||, but no argument is given that the attention-consensus map (5)-(6) admits such an expansion with |lambda_2| controlling the worst-case decay. Remark 1 and Section VII-C explicitly defer the finite-sample and spectral theory. Because EDMD with a finite random-feature dictionary returns only an approximate spectrum, the estimated lambda_2 is not established to be an upper bound on the true worst-case decay rate. Consequently, Tcert is currently an empirical prediction rather than a certificate, and the text's 'machine-checkable certificate' language should be qualified accordingly.
[§VI-A] Coverage is evaluated against the mean first-passage round over 20 rollouts, not against individual rollouts. The definition of Tobs as the averaged round, and of coverage as the fraction of configurations for which the issued deadline upper-bounds this mean, means that individual debates may still exceed Tcert. Because Algorithm 1 returns 'converged by round Tcert' and Table II reports per-run stability, per-rollout coverage should be reported, or the claim should be limited to mean behavior. This is not a cosmetic issue: a bound on an average does not provide the per-deployment guarantee that the word 'certificate' implies.
[§V-C and §VI-C] The validity threshold |lambda_2| > 0.9 appears to be selected on the same 60-run planted-faction benchmark used to report the 100% attribution accuracy. If the threshold was tuned on these runs, the 'self-certifying' claim is partially circular: the threshold would be calibrated to make the conditional accuracy perfect on the evaluation set. The paper should pre-specify the threshold or validate it on a separate split, and should report confidence intervals for the conditional accuracy rather than only the point value of 100%.

minor comments (5)

[Abstract and §VI-I] The abstract and Section VI-I state 96% coverage while Table IV reports 100% coverage; the numbers should be unified after the inconsistency in the major comments is resolved.
[Algorithm 1] Algorithm 1, line 5, computes Tcert as ceil(ln(1/epsilon)/(-ln|lambda_2|)), omitting the constant C from Eq. (9). Since the text sets C = 1, this is internally consistent, but the algorithm should state the assumption explicitly or include C.
[Throughout] The term 'certificate' is used for quantities that are only empirically validated under a stated working hypothesis. Consider using 'empirical certificate' or 'spectral prediction' in Section VII-A and the Conclusion to avoid overclaiming.
[Fig. 8(b) and §VII-A] There are typographical issues: 'aﬀine' in the Fig. 8(b) caption and 'suﬀices' in Section VII-A; the non-ASCII ligatures should be replaced with standard text.
[§VI-I] The graph-spectral baseline uses the expected linear update matrix P = (1-alpha)I + alpha RowNorm(A+I), which is exact only at beta=0; the text notes this, but it would be helpful to state that this baseline is the natural linearization at consensus rather than a strawman.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the deadline, attribution, and compression claims are evaluated on held-out data and do not reduce to their inputs by construction; the main weaknesses are an unproved spectral-dominance assumption and an internal coverage inconsistency, which are soundness issues rather than circular reasoning.

full rationale

The central deadline certificate is not circular: EDMD is fit on training trajectories and Tcert is evaluated on fresh rollouts (Sec. VI-A, Fig. 3), so lambda_2 is not a renamed fit to the test first-passage times. Proposition 1 derives the bound D(t) <= C|lambda_2|^t from an explicit modal-expansion assumption, and Remark 1 and Sec. VII-C(i) openly label the dominance constant and the attribution threshold as working hypotheses or empirically calibrated rather than derived; an unproved assumption is a soundness gap, not a circular reduction. The attribution validity flag is defined ex ante in Eq. (10) before the faction experiments, planted labels are used only for evaluation (Sec. IV-C), and the conditional 100% accuracy is a reported empirical property rather than a parameter fit that by construction guarantees the outcome. The compression result is measured against full-bandwidth twin runs on coupled seeds. No load-bearing self-citations appear; the Koopman/DMD citations are independent external literature. Two non-circular concerns should be weighed for correctness: (i) the coverage claim is internally inconsistent (Sec. VI-A reports 23/24 = 96% with a ratio range down to 0.8, while Table IV reports 100% for the same 24 configurations and the abstract and contributions use 96%), and (ii) the deadline's certificate status rests on the unproved spectral-dominance hypothesis of Proposition 1. These affect soundness and reproducibility, but they are not instances of a prediction reducing by construction to its input.

Assumptions & free parameters 6 free parameters · 5 assumptions · 0 invented entities

The framework rests on standard Koopman/EDMD convergence results plus two domain assumptions: the attention-consensus model stands in for real LLM collectives, and the centered dynamics admit a spectral dominance expansion. The certificate thresholds and the worst-case constant are calibrated choices rather than derived quantities.

free parameters (6)

Attribution validity threshold = 0.9
The flag VALID if |λ2|>0.9; empirically calibrated on the two-faction benchmark; acknowledged as open calibration problem in Sec. VII-C.
Near-unit exclusion threshold = 0.985
Eigenvalues above this are excluded as conserved; calibrated to the synthetic model's metastable regime; acknowledged in Sec. VI-F.
Worst-case dominance constant C = 1
Set to 1 in Prop. 1 and the deadline formula; produces the 2x conservatism; not derived from data.
Dictionary random feature scale and size = scale 0.7/sqrt(Nd), m in [40,60]
Hyperparameters of the RFF dictionary (8); chosen by hand, not optimized.
Ridge regularization = 1e-6
Ridge parameter for EDMD regression (4); chosen small, standard.
Disagreement tolerance epsilon = 0.05
Defines convergence and the deadline; chosen as the target tolerance.

assumptions (5)

standard math Koopman operator existence and EDMD convergence in the infinite-data, rich-dictionary limit
Section III uses the Koopman operator and EDMD; convergence guarantees from [20] are relied upon.
domain assumption Spectral dominance expansion for the centered dynamics (Prop. 1)
Prop. 1 assumes δ(t)=Σ c_j λ_j^t v_j with |λ2|≥|λ3|≥... and Σ|c_j|||v_j||≤C||δ(0)||; unproven for the attention-consensus dynamics.
domain assumption Attention-consensus model faithfully represents LLM collectives
Section IV states the model is a minimal analogue; transfer to real LLM embeddings is deferred to Section VII.
domain assumption Random Fourier features span the relevant eigenfunctions
The dictionary (8) is generic; the paper notes learned dictionaries [21,23] are the upgrade path.
standard math Centering removes the trivial consensus eigenspace
Subtracting the mean belief before applying the dictionary exposes the decay modes; justified in Section III-C.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Certifying Collective Reasoning in Multi-Agent Systems via Koopman Spectral Analysis." pith.science (2026). https://pith.science/paper/O6FJRLMV

@misc{pith2026260805956,
  author       = {Pith},
  title        = {Pith review of: Certifying Collective Reasoning in Multi-Agent Systems via Koopman Spectral Analysis},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/O6FJRLMV}},
  note         = {Machine review of arXiv:2608.05956}
}

abstract

Orchestrated collectives of large language model (LLM) agents that debate and vote are an emerging form of computational intelligence: the intelligent behaviour resides in the \emph{interaction}, not in any single agent. They improve task accuracy, yet remain black boxes at the system level: there is no principled test of convergence, no bound on the rounds needed, and no faithful account of what drove a decision. This paper develops a novel framework based on Koopman operator theory and validates its theoretical guarantees on multi-agent consensus dynamics. Treating the collective as one nonlinear dynamical system on a communication graph, we read its essential behaviour off the spectrum of its Koopman transfer operator, an exact linear representation of the nonlinear dynamics estimated from interaction traces. The spectrum yields three machine-checkable certificates: the sub-dominant eigenvalue $\lambda_2$ fixes the intrinsic timescale of reasoning and yields a convergence deadline computable \emph{before} the debate runs; its eigenvector names the coherent factions the collective reasons in, and $|\lambda_2|$ certifies when that explanation is valid; and the leading spectral coordinates form a compressed, auditable message basis. On an attention-consensus model, the deadline tracks observed convergence with log--log correlation $0.93$ and bounds it in 96\% of 24 configurations; attribution is exact whenever the spectrum certifies metastability; eight of 32 coordinates preserve the decision at 99.7\% fidelity; and a certificate learned from 15 debates held on 60/60 held-out debates. The study runs in minutes on a CPU, making spectral certification a practical layer for trustworthy collective reasoning.

Figures

Figures reproduced from arXiv: 2608.05956 by the authors.

**Figure 1.** What is observable differs sharply between the two settings. In (A), agents exchange numerical states over a known graph, so the update law is explicit and convergence follows from a Lyapunov argument. In (B), the agents are language models and the round-to-round map Tk+1 = f(Tk) is never available in closed form. In (C), we recover it from data: trajectories are lifted through observables ψ, the operator K is estim… view at source ↗

**Figure 2.** The four stages of the certification pipeline. Agent logs Tk are embedded into a semantic state zk = E(Tk), turning a transcript into a trajectory; snapshot pairs from that trajectory are lifted and regressed to give the operator K ≈ YX† ; and its eigenvalues supply the certificates, with ρ(K) < 1 establishing that disagreement decays and the sub-dominant eigenvalue fixing how fast. Everything after the embedding is… view at source ↗

**Figure 4.** Absolute error of the estimated sub-dominant eigenvalue against a 500-trajectory reference, versus the number of training trajectories M (12 repetitions per M, error bars giving the standard error). The fitted slope of −0.36 is shallower than the dashed M−1/2 reference, reflecting the strong dependence between snapshots drawn from a single debate. The left-hand end matters most in practice: roughly ten short traces … view at source ↗

Figures from the paper (3 more)

**Figure 5.** Figure 5: (a) The 24 agents of one debate, plotted in the top two principal coordinates of their round-1 embeddings and coloured by true faction; a black ring marks each agent the slow Koopman-mode score assigns correctly, and every agent carries one. (b) Each point is one of 60…

**Figure 7.** Figure 7: Each blue curve is the normalised disagreement of one of 12 held-out question-answering debates; the logarithmic axis renders the geometric decay as a near-straight descent. The dotted line is the tolerance ϵ = 0.05 and the red dashed line the certified deadline Tcert …

**Figure 8.** Figure 8: (a) As attention temperature β grows, agents weight like-minded peers ever more heavily, the spectral gap γ (green, right axis) collapses from about 0.10 to 0.02, and rounds-to-consensus rise by an order of magnitude, with the certificate remaining a sound envelope thr…

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

32 extracted references · 25 canonical work pages

[1]

Improving factuality and reasoning in language models through multiagent debate,

Y. Du, S. Li, A. Torralba, J. B. Tenenbaum, and I. Mordatch, “Improving factuality and reasoning in language models through multiagent debate,” arXiv:2305.14325, 2023

arXiv 2023
[2]

Encouraging divergent thinking in large lan- guage models through multi-agent debate,

T. Liang et al., “Encouraging divergent thinking in large lan- guage models through multi-agent debate,” arXiv:2305.19118, 2023

arXiv 2023
[3]

AutoGen: Enabling next-gen LLM applications via multi-agent conversation,

Q. Wu et al., “AutoGen: Enabling next-gen LLM applications via multi-agent conversation,” arXiv:2308.08155, 2023

arXiv 2023
[4]

Agentic reasoning for large language models,

T. Wei, T.-W. Li, Z. Liu, X. Ning, Z. Yang, J. Zou, Z. Zeng, R. Qiu, X. Lin, D. Fu, et al., “Agentic reasoning for large language models,” arXiv preprint arXiv:2601.12538, 2026

arXiv 2026
[5]

Prescribed- time optimal consensus for switched stochastic multiagent sys- tems: Reinforcement learning strategy,

W. Guang, X. Wang, L. Tan, J. Sun, and T. Huang, “Prescribed- time optimal consensus for switched stochastic multiagent sys- tems: Reinforcement learning strategy,” IEEE Trans. Emerg. Topics Comput. Intell., vol. 9, no. 1, 2025

work page 2025
[6]

Event-triggered random delayed impulsive consensus of multi-agent systems with time-varying delay,

W. Zhang, J. Chen, S. Wen, and T. Huang, “Event-triggered random delayed impulsive consensus of multi-agent systems with time-varying delay,” IEEE Trans. Emerg. Topics Comput. Intell., vol. 9, no. 2, 2025

work page 2025
[7]

Behavior reasoning for opponent agents in multi- agent learning systems,

Y. Hou, M. Sun, W. Zhu, Y. Zeng, H. Piao, X. Chen, and Q. Zhang, “Behavior reasoning for opponent agents in multi- agent learning systems,” IEEE Trans. Emerg. Topics Comput. Intell., vol. 6, no. 5, pp. 1125–1136, 2022

work page 2022
[8]

Adaptive trust model for multi-agent teaming based on reinforcement-learning-based fusion,

C.-T. Lin, H. Zhang, L. Ou, Y.-C. Chang, and Y.-K. Wang, “Adaptive trust model for multi-agent teaming based on reinforcement-learning-based fusion,” IEEE Trans. Emerg. Top- ics Comput. Intell., vol. 8, no. 1, pp. 229–239, 2024

work page 2024

Show all 32 references

[9]

The impact of perceived autonomous agents on dynamic team behaviors,

M. Demir, N. J. McNeese, and N. J. Cooke, “The impact of perceived autonomous agents on dynamic team behaviors,” IEEE Trans. Emerg. Topics Comput. Intell., vol. 2, no. 4, pp. 258–267, 2018

2018
[10]

Data-driven distributed output consensus control for multi-agent systems with unknown internal state,

C. Zhang, L. Ji, S. Yang, X. Guo, and H. Li, “Data-driven distributed output consensus control for multi-agent systems with unknown internal state,” Neurocomputing, vol. 615, p. 128868, 2025

2025
[11]

A survey on neural network interpretability,

Y. Zhang, P. Tiňo, A. Leonardis, and K. Tang, “A survey on neural network interpretability,” IEEE Trans. Emerg. Topics Comput. Intell., vol. 5, no. 5, pp. 726–742, 2021

2021
[12]

Towards faithfully interpretable NLP systems: How should we define and evaluate faithfulness?

A. Jacovi and Y. Goldberg, “Towards faithfully interpretable NLP systems: How should we define and evaluate faithfulness?” in Proc. 58th Annu. Meeting Assoc. Comput. Linguistics (ACL), 2020, pp. 4198–4205

2020
[13]

Lan- guage models don’t always say what they think: Unfaithful explanations in chain-of-thought prompting,

M. Turpin, J. Michael, E. Perez, and S. R. Bowman, “Lan- guage models don’t always say what they think: Unfaithful explanations in chain-of-thought prompting,” in Adv. Neural Inf. Process. Syst. (NeurIPS), 2023

2023
[14]

Hamiltonian systems and transformation in Hilbert space,

B. O. Koopman, “Hamiltonian systems and transformation in Hilbert space,” Proc. Nat. Acad. Sci. USA, vol. 17, no. 5, pp. 315–318, 1931

1931
[15]

Spectral properties of dynamical systems, model reduction and decompositions,

I. Mezić, “Spectral properties of dynamical systems, model reduction and decompositions,” Nonlinear Dyn., vol. 41, pp. 309–325, 2005

2005
[16]

Dynamic mode decomposition of numerical and experimental data,

P. J. Schmid, “Dynamic mode decomposition of numerical and experimental data,” J. Fluid Mech., vol. 656, pp. 5–28, 2010

2010
[17]

Dynamic mode decomposition and its variants,

P. J. Schmid, “Dynamic mode decomposition and its variants,” Annu. Rev. Fluid Mech., vol. 54, pp. 225–254, 2022

2022
[18]

Spectral analysis of nonlinear flows,

C. W. Rowley, I. Mezić, S. Bagheri, P. Schlatter, and D. S. Henningson, “Spectral analysis of nonlinear flows,” J. Fluid Mech., vol. 641, pp. 115–127, 2009

2009
[19]

A data-driven approximation of the Koopman operator: Extending dynamic mode decomposition,

M. O. Williams, I. G. Kevrekidis, and C. W. Rowley, “A data-driven approximation of the Koopman operator: Extending dynamic mode decomposition,” J. Nonlinear Sci., vol. 25, pp. 1307–1346, 2015

2015
[20]

On convergence of extended dynamic mode decomposition to the Koopman operator,

M. Korda and I. Mezić, “On convergence of extended dynamic mode decomposition to the Koopman operator,” J. Nonlinear Sci., vol. 28, pp. 687–710, 2018

2018
[21]

Extended dynamic mode decomposition with dictionary learning: A data- driven adaptive spectral decomposition of the Koopman opera- tor,

Q. Li, F. Dietrich, E. M. Bollt, and I. G. Kevrekidis, “Extended dynamic mode decomposition with dictionary learning: A data- driven adaptive spectral decomposition of the Koopman opera- tor,” Chaos, vol. 27, no. 10, p. 103111, 2017

2017
[22]

Multi-agent collaboration mechanisms: A survey of LLMs,

K.-T. Tran, D. Dao, M.-D. Nguyen, Q.-V. Pham, B. O’Sullivan, and H. D. Nguyen, “Multi-agent collaboration mechanisms: A survey of LLMs,” arXiv preprint arXiv:2501.06322, 2025

2025 arXiv
[23]

Data-driven approximation of the Koopman generator: Model reduction, system identification, and control,

S. Klus, F. Nüske, S. Peitz, J.-H. Niemann, C. Clementi, and C. Schütte, “Data-driven approximation of the Koopman generator: Model reduction, system identification, and control,” Physica D, vol. 406, p. 132416, 2020

2020
[24]

Modern Koopman theory for dynamical systems,

S. L. Brunton, M. Budiši’c, E. Kaiser, and J. N. Kutz, “Modern Koopman theory for dynamical systems,” arXiv:2102.12086, 2021

2021 arXiv
[25]

Langchain & langgraph in production: Architec- tures for multi-agent llm systems,

K. Pelluru, “Langchain & langgraph in production: Architec- tures for multi-agent llm systems,” Journal of Data and Digital Innovation (JDDI), vol. 2, no. 3, pp. 1–9, 2025. 13

2025
[26]

Multi-agent debate strategies: Survey, taxonomy, and challenges,

Q. Motger, M. Oriol, J. Marco, and X. Franch, “Multi-agent debate strategies: Survey, taxonomy, and challenges,” arXiv preprint arXiv:2607.26212, 2026

2026 arXiv
[27]

Consensus problems in networks of agents with switching topology and time-delays,

R. Olfati-Saber and R. M. Murray, “Consensus problems in networks of agents with switching topology and time-delays,” IEEE Trans. Autom. Control, vol. 49, no. 9, pp. 1520–1533, 2004

2004
[28]

Reaching a consensus,

M. H. DeGroot, “Reaching a consensus,” J. Amer. Statist. Assoc., vol. 69, no. 345, pp. 118–121, 1974

1974
[29]

Opinion dynamics and bounded confidence: Models, analysis and simulation,

R. Hegselmann and U. Krause, “Opinion dynamics and bounded confidence: Models, analysis and simulation,” J. Artif. Soc. Social Simul., vol. 5, no. 3, 2002

2002
[30]

Spectral lens: Activation and gradient spectra as diagnostics of llm optimization,

A. Z. Liu, E. Paquette, and J. Sous, “Spectral lens: Activation and gradient spectra as diagnostics of llm optimization,” arXiv preprint arXiv:2605.05683, 2026

2026 arXiv
[31]

Opinion dynamics with social confirmation effect: An adaptive Hegselmann-Krause model,

F. Shahbazi, H. Kebriaei, and A. Yaghmaei, “Opinion dynamics with social confirmation effect: An adaptive Hegselmann-Krause model,” IEEE Trans. Comput. Soc. Syst., 2025

2025
[32]

Diffusion maps,

R. R. Coifman and S. Lafon, “Diffusion maps,” Appl. Comput. Harmon. Anal., vol. 21, no. 1, pp. 5–30, 2006

2006

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Improving factuality and reasoning in language models through multiagent debate,

Y. Du, S. Li, A. Torralba, J. B. Tenenbaum, and I. Mordatch, “Improving factuality and reasoning in language models through multiagent debate,” arXiv:2305.14325, 2023

arXiv 2023

[2] [2]

Encouraging divergent thinking in large lan- guage models through multi-agent debate,

T. Liang et al., “Encouraging divergent thinking in large lan- guage models through multi-agent debate,” arXiv:2305.19118, 2023

arXiv 2023

[3] [3]

AutoGen: Enabling next-gen LLM applications via multi-agent conversation,

Q. Wu et al., “AutoGen: Enabling next-gen LLM applications via multi-agent conversation,” arXiv:2308.08155, 2023

arXiv 2023

[4] [4]

Agentic reasoning for large language models,

T. Wei, T.-W. Li, Z. Liu, X. Ning, Z. Yang, J. Zou, Z. Zeng, R. Qiu, X. Lin, D. Fu, et al., “Agentic reasoning for large language models,” arXiv preprint arXiv:2601.12538, 2026

arXiv 2026

[5] [5]

Prescribed- time optimal consensus for switched stochastic multiagent sys- tems: Reinforcement learning strategy,

W. Guang, X. Wang, L. Tan, J. Sun, and T. Huang, “Prescribed- time optimal consensus for switched stochastic multiagent sys- tems: Reinforcement learning strategy,” IEEE Trans. Emerg. Topics Comput. Intell., vol. 9, no. 1, 2025

work page 2025

[6] [6]

Event-triggered random delayed impulsive consensus of multi-agent systems with time-varying delay,

W. Zhang, J. Chen, S. Wen, and T. Huang, “Event-triggered random delayed impulsive consensus of multi-agent systems with time-varying delay,” IEEE Trans. Emerg. Topics Comput. Intell., vol. 9, no. 2, 2025

work page 2025

[7] [7]

Behavior reasoning for opponent agents in multi- agent learning systems,

Y. Hou, M. Sun, W. Zhu, Y. Zeng, H. Piao, X. Chen, and Q. Zhang, “Behavior reasoning for opponent agents in multi- agent learning systems,” IEEE Trans. Emerg. Topics Comput. Intell., vol. 6, no. 5, pp. 1125–1136, 2022

work page 2022

[8] [8]

Adaptive trust model for multi-agent teaming based on reinforcement-learning-based fusion,

C.-T. Lin, H. Zhang, L. Ou, Y.-C. Chang, and Y.-K. Wang, “Adaptive trust model for multi-agent teaming based on reinforcement-learning-based fusion,” IEEE Trans. Emerg. Top- ics Comput. Intell., vol. 8, no. 1, pp. 229–239, 2024

work page 2024