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arxiv: 2606.18121 · v1 · pith:WLFR52F4new · submitted 2026-06-16 · 💻 cs.MA · cs.IT· math.IT

On the Reliability of Networks of AI Agents: Density Evolution, Stopping Sets, and Architecture Optimization

Ehsan Aghazadeh , Hossein Pishro-Nik This is my paper

Pith reviewed 2026-06-26 21:46 UTC · model grok-4.3

classification 💻 cs.MA cs.ITmath.IT
keywords density evolutionAI agentsfactor graphsmessage passingreliabilitystopping setsBoolean verifierserasure channels
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The pith

A density-evolution theorem predicts the asymptotic fraction of unresolved subclaims in AI agent networks modeled as role-typed factor graphs.

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

The paper treats multi-agent AI systems as message passing on sparse graphs, where agents exchange set-valued messages about binary subclaims and verifiers apply noisy Boolean functions. It adapts density-evolution analysis to track how three erasure-type failures propagate under a logical-forcing rule at each check node. This yields a recursion that forecasts the fraction of unresolved subclaims for random architectures and extends to locally tree-like deterministic graphs. The XOR specialization matches the known LDPC erasure-channel recursion, while the AND case produces an asymmetry between positive and negative certificates. The result supplies a predictive tool for when such networks succeed or fail without requiring exhaustive simulation of every configuration.

Core claim

We prove a density-evolution theorem that predicts the asymptotic fraction of unresolved subclaims on random role-typed architectures, with an extension to deterministic, locally tree-like graph sequences. The XOR case recovers the classical LDPC recursion on the binary erasure channel (BEC); the AND case exposes an asymmetry between positive and negative verifier certificates.

What carries the argument

Density evolution on sparse role-typed factor graphs whose noisy Boolean check nodes combine set-valued messages via a single logical-forcing rule.

If this is right

  • Random architectures admit a computable reliability threshold below which the fraction of unresolved subclaims vanishes.
  • The XOR verifier recovers the classical LDPC threshold on the binary erasure channel.
  • The AND verifier produces distinct thresholds for positive versus negative certificates.
  • The same recursion applies to any locally tree-like sequence of deterministic graphs.
  • Stopping-set analysis becomes available for finite-length performance bounds.

Where Pith is reading between the lines

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

  • The same message-passing model could be used to compare reliability of different role assignments before deployment.
  • Asymmetry results for AND suggest that certificate polarity should be balanced when designing verifier prompts.
  • The framework supplies a concrete way to test whether a proposed multi-agent workflow will leave a positive fraction of claims unresolved.
  • Stopping sets identified by the analysis point to minimal subgraphs whose repair would improve overall resolution.

Load-bearing premise

The three distinct failure modes can be modeled uniformly as erasures that propagate through set-valued messages combined by a single logical-forcing rule on the factor graph.

What would settle it

Monte Carlo simulation on large random role-typed graphs whose measured fraction of unresolved subclaims deviates from the density-evolution prediction by more than the predicted variance.

Figures

Figures reproduced from arXiv: 2606.18121 by Ehsan Aghazadeh, Hossein Pishro-Nik.

Figure 1
Figure 1. Figure 1: The logical-forcing rule at a single check. Check [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Bipartite factor graph for the 4-step proof scenario in Section [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Density-evolution prediction (Theorem 1, Corollary 1) vs. empirical bit-erasure rate from Monte-Carlo simulation of the role-typed configuration ensemble. Single-role (dv, dc) = (3, 6)-regular XOR ensemble with three-tier erasure (ϵ V, ϵC, η) = (swept, 0.05, 0.95) and L = 50 message-passing rounds. Empirical points are averages over 30 independent graph realizations at each of three problem sizes n ∈ {200,… view at source ↗
Figure 4
Figure 4. Figure 4: AND value-conditioned DE recursion (Proposition [PITH_FULL_IMAGE:figures/full_fig_p040_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: AND Monte-Carlo concentration on the single-role [PITH_FULL_IMAGE:figures/full_fig_p040_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Deterministic-graph validation of Theorem [PITH_FULL_IMAGE:figures/full_fig_p041_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: XOR residual P (L) DE on a constant-additive-sum surface: ϵ V = 0.40 fixed, side budget ϵ C +(1−η) = 0.10 split as ϵ C = t·0.10 and 1−η = (1−t)·0.10. (dv, dc) = (3, 6), L = 50. Residuals range over [0.146, 0.332] along the same additive sum. The two design points at the endpoints have the same additive effective-erasure score but different residual erasure: the DE map is sensitive to where erasures enter t… view at source ↗
Figure 8
Figure 8. Figure 8: The Hilbert architecture [1] mapped onto the role-typed Boolean-verifier-node framework. Top: the four cooperating LLM roles, decomposer, refactorer, type-checker, aggregator, each annotated with the erasure-tier parameter it controls. Bottom: the underlying bipartite factor graph that the framework analyzes. Variable agents are proposed proof steps; check agents are Lean kernel invocations performing AND-… view at source ↗
read the original abstract

Modern AI systems increasingly solve a task not with a single model call but with several imperfect agents working together: some propose pieces of a solution, others verify them, and the results are combined. These systems often outperform any single model, yet it is rarely clear why they succeed or when they will fail. We model such a system as message passing on a sparse graph, the structure that underlies low-density parity-check (LDPC) codes, and extend the density-evolution machinery of coding theory to this richer setting. In our model a task is a set of coupled binary subclaims, and an agent architecture is a sparse, role-typed factor graph whose check nodes are noisy Boolean verifier nodes, each computing a local Boolean function of the subclaims it touches. Three distinct failure modes, all modeled as erasures (an agent abstaining, a verifier returning no usable output, and a message lost between two agents), propagate as the agents exchange set-valued messages. The check agents combine these messages by a single logical-forcing rule that specializes to XOR, AND, OR, implication, and Horn constraints. This is more than a relabeling of LDPC theory: the verifier functions are nonlinear and value-asymmetric, and the three failure modes do not reduce to a single effective channel, so they require new threshold, finite-length, and converse results rather than a direct reuse of parity-check density evolution. We prove a density-evolution theorem that predicts the asymptotic fraction of unresolved subclaims on random role-typed architectures, with an extension to deterministic, locally tree-like graph sequences. The XOR case recovers the classical LDPC recursion on the binary erasure channel (BEC); the AND case exposes an asymmetry between positive and negative verifier certificates.

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.

Referee Report

1 major / 1 minor

Summary. The paper models networks of AI agents solving tasks via coupled binary subclaims as message passing on sparse, role-typed factor graphs whose check nodes are noisy Boolean verifiers. Three failure modes (agent abstention, verifier abstention, message loss) are modeled uniformly as erasures that propagate via set-valued messages combined at checks by a single logical-forcing rule. The authors prove a density-evolution theorem predicting the asymptotic fraction of unresolved subclaims on random role-typed architectures (with an extension to deterministic locally tree-like sequences), recovering the classical LDPC BEC recursion for XOR and exposing positive/negative asymmetry for AND.

Significance. If the density-evolution theorem is correct, the work supplies a non-trivial extension of coding-theory tools to multi-agent reliability analysis, including new threshold and finite-length results that do not reduce to standard LDPC. The explicit recovery of the XOR case and the identification of AND asymmetry are concrete strengths; the manuscript also ships a theorem rather than only simulations.

major comments (1)
  1. [Abstract / density-evolution theorem] Abstract / density-evolution theorem: the claimed single DE recursion requires that the three failure modes can be treated uniformly as erasures under one logical-forcing rule on set-valued messages for arbitrary Boolean verifier functions. The abstract itself states that the verifiers are nonlinear and value-asymmetric and that the modes "do not reduce to a single effective channel"; any mode-specific effect on admissible message sets would necessitate distinct alphabets or separate recursions, undermining the single-theorem claim. This assumption is load-bearing for the central result.
minor comments (1)
  1. Clarify whether the extension to deterministic locally tree-like sequences is proved as a separate statement or follows directly as a corollary of the random-graph theorem.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the modeling assumptions underlying the density-evolution theorem. We address the single major comment below and believe the central claim remains intact.

read point-by-point responses

  1. Referee: [Abstract / density-evolution theorem] Abstract / density-evolution theorem: the claimed single DE recursion requires that the three failure modes can be treated uniformly as erasures under one logical-forcing rule on set-valued messages for arbitrary Boolean verifier functions. The abstract itself states that the verifiers are nonlinear and value-asymmetric and that the modes "do not reduce to a single effective channel"; any mode-specific effect on admissible message sets would necessitate distinct alphabets or separate recursions, undermining the single-theorem claim. This assumption is load-bearing for the central result.

    Authors: The three erasure modes are deliberately modeled with the same message alphabet (subsets of {0,1}) and the same logical-forcing combination rule at every check node; the rule depends only on the verifier function, not on which physical mechanism produced the erasure. Consequently the message-passing dynamics remain identical across modes, permitting a single density-evolution recursion (Theorem 1) whose state is the distribution over unresolved sets. The statement that the modes “do not reduce to a single effective channel” refers to the fact that the resulting thresholds, asymmetry for AND/OR, and finite-length scaling differ from classical BEC-LDPC and therefore require fresh analysis; it does not imply that the message alphabet or recursion must be mode-specific. The derivation in Sections 3–4 makes this uniformity explicit for arbitrary Boolean verifiers. We are happy to insert one clarifying sentence in the abstract to separate the two senses of “channel.” revision: partial

Circularity Check

0 steps flagged

Density-evolution theorem extends LDPC machinery to role-typed graphs with independent content

full rationale

The paper frames its central result as a proved density-evolution theorem for the asymptotic unresolved fraction under set-valued messages and a logical-forcing rule on role-typed factor graphs. It explicitly distinguishes the setting from direct LDPC reuse by noting nonlinear asymmetric verifiers and multiple non-equivalent erasure modes, and recovers the classical BEC recursion only as the special XOR case while deriving new asymmetry for AND. No quoted step reduces the theorem to a fitted parameter, self-citation chain, or input by construction; the derivation is presented as an extension of existing coding-theory tools to a new domain with stated differences that require fresh analysis.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard density-evolution assumptions transferred to the new setting; no free parameters or invented entities are mentioned in the abstract.

axioms (1)
  • standard math Density evolution applies to large random sparse graphs that are locally tree-like
    Invoked implicitly when stating the asymptotic fraction on random role-typed architectures (abstract).

pith-pipeline@v0.9.1-grok · 5856 in / 1255 out tokens · 31446 ms · 2026-06-26T21:46:03.160047+00:00 · methodology

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Reference graph

Works this paper leans on

49 extracted references · 3 canonical work pages

  1. [1]

    Hilbert: Recursively building formal proofs with informal reasoning,

    S. Varambally, T. Voice, Y. Sun, Z. Chen, R. Yu, and K. Ye, “Hilbert: Recursively building formal proofs with informal reasoning,” inThe 5th Workshop on Mathematical Reasoning and AI at NeurIPS 2025, 2025. [Online]. Available: https://openreview.net/forum?id=ljAHonPrs1 57

    2025

  2. [2]

    Coder: Issue resolving with multi-agent and task graphs,

    D. Chen, S. Lin, M. Zeng, D. Zan, J.-G. Wang, A. Cheshkov, J. Sun, H. Yu, G. Dong, A. Aliev, J. Wang, X. Cheng, G. Liang, Y. Ma, P. Bian, T. Xie, and Q. Wang, “Coder: Issue resolving with multi-agent and task graphs,” 2024. [Online]. Available: https://arxiv.org/abs/2406.01304

    arXiv 2024

  3. [3]

    Debating with more persuasive llms leads to more truthful answers,

    A. Khan, J. Hughes, D. Valentine, L. Ruis, K. Sachan, A. Radhakrishnan, E. Grefenstette, S. R. Bowman, T. Rocktäschel, and E. Perez, “Debating with more persuasive llms leads to more truthful answers,” inProceedings of the 41st International Conference on Machine Learning, ser. ICML’24. JMLR.org, 2024

    2024

  4. [4]

    Towards a science of ai agent reliability,

    S. Rabanser, S. Kapoor, P. Kirgis, K. Liu, S. Utpala, and A. Narayanan, “Towards a science of ai agent reliability,” 2026. [Online]. Available: https://arxiv.org/abs/2602.16666

    Pith/arXiv arXiv 2026

  5. [5]

    Richardson and R

    T. Richardson and R. Urbanke,Modern Coding Theory. Cambridge University Press, 2008

    2008

  6. [6]

    Analysis of low density codes and improved designs using irregular graphs,

    M. Luby, M. Mitzenmacher, A. Shokrollah, and D. Spielman, “Analysis of low density codes and improved designs using irregular graphs,” inProceedings of the Thirtieth Annual ACM Symposium on Theory of Computing, ser. STOC ’98. New York, NY, USA: Association for Computing Machinery, 1998, p. 249–258. [Online]. Available: https://doi.org/10.1145/276698.276756

    work page doi:10.1145/276698.276756 1998

  7. [7]

    The capacity of low-density parity-check codes under message-passing decoding,

    T. Richardson and R. Urbanke, “The capacity of low-density parity-check codes under message-passing decoding,”IEEE Transactions on Information Theory, vol. 47, no. 2, pp. 599–618, 2001

    2001

  8. [8]

    Design of capacity-approaching irregular low-density parity-check codes,

    T. Richardson, M. Shokrollahi, and R. Urbanke, “Design of capacity-approaching irregular low-density parity-check codes,” IEEE Transactions on Information Theory, vol. 47, no. 2, pp. 619–637, 2001

    2001

  9. [9]

    Finite-length analysis of low-density parity-check codes on the binary erasure channel,

    C. Di, D. Proietti, I. E. Telatar, T. J. Richardson, and R. L. Urbanke, “Finite-length analysis of low-density parity-check codes on the binary erasure channel,”IEEE Trans. Inf. Theor., vol. 48, no. 6, p. 1570–1579, Sep. 2006. [Online]. Available: https://doi.org/10.1109/TIT.2002.1003839

    work page doi:10.1109/tit.2002.1003839 2006

  10. [10]

    Nonuniform error correction using low-density parity-check codes,

    H. Pishro-Nik, N. Rahnavard, and F. Fekri, “Nonuniform error correction using low-density parity-check codes,”IEEE Transactions on Information Theory, vol. 51, no. 7, pp. 2702–2714, 2005

    2005

  11. [11]

    Design of ldpc codes robust to noisy message-passing decoding,

    A. Tarighati, H. Farhadi, and F. Lahouti, “Design of ldpc codes robust to noisy message-passing decoding,” 2015. [Online]. Available: https://arxiv.org/abs/1501.02483

    Pith/arXiv arXiv 2015

  12. [12]

    Noisy density evolution with asymmetric deviation models,

    E. Dupraz and F. Leduc-Primeau, “Noisy density evolution with asymmetric deviation models,”IEEE Transactions on Communications, vol. 69, no. 3, pp. 1403–1416, 2021

    2021

  13. [13]

    SWE-bench: Can language models resolve real-world github issues?

    C. E. Jimenez, J. Yang, A. Wettig, S. Yao, K. Pei, O. Press, and K. R. Narasimhan, “SWE-bench: Can language models resolve real-world github issues?” inThe Twelfth International Conference on Learning Representations, 2024. [Online]. Available: https://openreview.net/forum?id=VTF8yNQM66

    2024

  14. [14]

    SWE-agent: Agent-computer interfaces enable automated software engineering,

    J. Yang, C. E. Jimenez, A. Wettig, K. Lieret, S. Yao, K. R. Narasimhan, and O. Press, “SWE-agent: Agent-computer interfaces enable automated software engineering,” inThe Thirty-eighth Annual Conference on Neural Information Processing Systems, 2024. [Online]. Available: https://openreview.net/forum?id=mXpq6ut8J3

    2024

  15. [15]

    CLEVER: A curated benchmark for formally verified code generation,

    A. Thakur, J. Lee, G. Tsoukalas, M. Sistla, M. Zhao, S. Zetzsche, G. Durrett, Y. Yue, and S. Chaudhuri, “CLEVER: A curated benchmark for formally verified code generation,” inThe Thirty-ninth Annual Conference on Neural Information Processing Systems Datasets and Benchmarks Track, 2026. [Online]. Available: https: //openreview.net/forum?id=IbOacMF5qd

    2026

  16. [16]

    Proving the coding interview: A benchmark for formally verified code generation,

    Q. Dougherty and R. Mehta, “Proving the coding interview: A benchmark for formally verified code generation,” in2025 IEEE/ACM International Workshop on Large Language Models for Code (LLM4Code), 2025, pp. 72–79

    2025

  17. [17]

    Solving a million-step llm task with zero errors,

    E. Meyerson, G. Paolo, R. Dailey, H. Shahrzad, O. Francon, C. F. Hayes, X. Qiu, B. Hodjat, and R. Miikkulainen, “Solving a million-step llm task with zero errors,” 2025. [Online]. Available: https://arxiv.org/abs/2511.09030

    arXiv 2025

  18. [18]

    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,” inProceedings of the 41st International Conference on Machine Learning, ser. ICML’24. JMLR.org, 2024

    2024

  19. [19]

    On scalable oversight with weak LLMs judging strong LLMs,

    Z. Kenton, N. Y. Siegel, J. Kramar, J. Brown-Cohen, S. Albanie, J. Bulian, R. Agarwal, D. Lindner, Y. Tang, N. Goodman, and R. Shah, “On scalable oversight with weak LLMs judging strong LLMs,” inThe Thirty-eighth Annual Conference on Neural Information Processing Systems, 2024. [Online]. Available: https://openreview.net/forum?id=O1fp9nVraj

    2024

  20. [20]

    Debate or vote: Which yields better decisions in multi-agent large language models?

    H. K. Choi, J. Zhu, and S. Li, “Debate or vote: Which yields better decisions in multi-agent large language models?” inThe Thirty-ninth Annual Conference on Neural Information Processing Systems, 2026. [Online]. Available: https://openreview.net/forum?id=iUjGNJzrF1

    2026

  21. [21]

    Factor graphs and the sum-product algorithm,

    F. Kschischang, B. Frey, and H.-A. Loeliger, “Factor graphs and the sum-product algorithm,”IEEE Transactions on Information Theory, vol. 47, no. 2, pp. 498–519, 2001

    2001

  22. [22]

    Low-density parity-check codes,

    R. Gallager, “Low-density parity-check codes,”IRE Transactions on Information Theory, vol. 8, no. 1, pp. 21–28, 1962

    1962

  23. [23]

    On decoding of low-density parity-check codes over the binary erasure channel,

    H. Pishro-Nik and F. Fekri, “On decoding of low-density parity-check codes over the binary erasure channel,”IEEE Transactions on Information Theory, vol. 50, no. 3, pp. 439–454, 2004

    2004

  24. [24]

    Results on punctured low-density parity-check codes and improved iterative decoding techniques,

    ——, “Results on punctured low-density parity-check codes and improved iterative decoding techniques,”IEEE Transactions on Information Theory, vol. 53, no. 2, pp. 599–614, 2007

    2007

  25. [25]

    Generalized and doubly generalized ldpc codes with random component codes for the binary erasure channel,

    E. Paolini, M. P. C. Fossorier, and M. Chiani, “Generalized and doubly generalized ldpc codes with random component codes for the binary erasure channel,”IEEE Transactions on Information Theory, vol. 56, no. 4, pp. 1651–1672, 2010

    2010

  26. [26]

    Analysis of absorbing sets and fully absorbing sets of array-based ldpc codes,

    L. Dolecek, Z. Zhang, V. Anantharam, M. J. Wainwright, and B. Nikolic, “Analysis of absorbing sets and fully absorbing sets of array-based ldpc codes,”IEEE Transactions on Information Theory, vol. 56, no. 1, pp. 181–201, 2010

    2010

  27. [27]

    Density evolution for asymmetric memoryless channels,

    C.-C. Wang, S. Kulkarni, and H. Poor, “Density evolution for asymmetric memoryless channels,”IEEE Transactions on Information Theory, vol. 51, no. 12, pp. 4216–4236, 2005

    2005

  28. [28]

    Reflexion: language agents with verbal reinforcement learning,

    N. Shinn, F. Cassano, A. Gopinath, K. R. Narasimhan, and S. Yao, “Reflexion: language agents with verbal reinforcement learning,” inThirty-seventh Conference on Neural Information Processing Systems, 2023. [Online]. Available: https://openreview.net/forum?id=vAElhFcKW6 58

    2023

  29. [29]

    React: Synergizing reasoning and acting in language models,

    S. Yao, J. Zhao, D. Yu, N. Du, I. Shafran, K. R. Narasimhan, and Y. Cao, “React: Synergizing reasoning and acting in language models,” inThe Eleventh International Conference on Learning Representations, 2023. [Online]. Available: https://openreview.net/forum?id=WE_vluYUL-X

    2023

  30. [30]

    Tree of thoughts: Deliberate problem solving with large language models,

    S. Yao, D. Yu, J. Zhao, I. Shafran, T. L. Griffiths, Y. Cao, and K. R. Narasimhan, “Tree of thoughts: Deliberate problem solving with large language models,” inThirty-seventh Conference on Neural Information Processing Systems,

  31. [31]

    Available: https://openreview.net/forum?id=5Xc1ecxO1h

    [Online]. Available: https://openreview.net/forum?id=5Xc1ecxO1h

  32. [32]

    Self-consistency improves chain of thought reasoning in language models,

    X. Wang, J. Wei, D. Schuurmans, Q. V. Le, E. H. Chi, S. Narang, A. Chowdhery, and D. Zhou, “Self-consistency improves chain of thought reasoning in language models,” inThe Eleventh International Conference on Learning Representations,

  33. [33]

    Available: https://openreview.net/forum?id=1PL1NIMMrw

    [Online]. Available: https://openreview.net/forum?id=1PL1NIMMrw

  34. [34]

    Training verifiers to solve math word problems,

    K. Cobbe, V. Kosaraju, M. Bavarian, M. Chen, H. Jun, L. Kaiser, M. Plappert, J. Tworek, J. Hilton, R. Nakano, C. Hesse, and J. Schulman, “Training verifiers to solve math word problems,” 2021. [Online]. Available: https://arxiv.org/abs/2110.14168

    Pith/arXiv arXiv 2021

  35. [35]

    Let’s verify step by step,

    H. Lightman, V. Kosaraju, Y. Burda, H. Edwards, B. Baker, T. Lee, J. Leike, J. Schulman, I. Sutskever, and K. Cobbe, “Let’s verify step by step,” inThe Twelfth International Conference on Learning Representations, 2024. [Online]. Available: https://openreview.net/forum?id=v8L0pN6EOi

    2024

  36. [36]

    minif2f: a cross-system benchmark for formal olympiad-level mathematics,

    K. Zheng, J. M. Han, and S. Polu, “minif2f: a cross-system benchmark for formal olympiad-level mathematics,” inInternational Conference on Learning Representations, 2022. [Online]. Available: https://openreview.net/forum?id= 9ZPegFuFTFv

    2022

  37. [37]

    Proofnet: Autoformalizing and formally proving undergraduate-level mathematics,

    Z. Azerbayev, B. Piotrowski, H. Schoelkopf, E. W. Ayers, D. Radev, and J. Avigad, “Proofnet: Autoformalizing and formally proving undergraduate-level mathematics,” 2023. [Online]. Available: https://arxiv.org/abs/2302.12433

    arXiv 2023

  38. [38]

    Lean-STar: Learning to interleave thinking and proving,

    H. Lin, Z. Sun, S. Welleck, and Y. Yang, “Lean-STar: Learning to interleave thinking and proving,” inThe Thirteenth International Conference on Learning Representations, 2025. [Online]. Available: https://openreview.net/forum?id= SOWZ59UyNc

    2025

  39. [39]

    On the reliability limits of llm-based multi-agent planning,

    R. Ao, S. Gao, and D. Simchi-Levi, “On the reliability limits of llm-based multi-agent planning,” 2026. [Online]. Available: https://arxiv.org/abs/2603.26993

    arXiv 2026

  40. [40]

    Representation in

    M. Mézard and A. Montanari,Information, Physics, and Computation. Oxford University Press, 01 2009. [Online]. Available: https://doi.org/10.1093/acprof:oso/9780198570837.001.0001

    work page doi:10.1093/acprof:oso/9780198570837.001.0001 2009

  41. [41]

    On the method of bounded differences,

    C. McDiarmid, “On the method of bounded differences,” inSurveys in combinatorics, 1989 (Norwich, 1989), ser. London Math. Soc. Lecture Note Ser. Cambridge Univ. Press, Cambridge, 1989, vol. 141, pp. 148–188

    1989

  42. [42]

    Mas-fire: Fault injection and reliability evaluation for llm-based multi-agent systems,

    J. Jia, Z. Deng, Z. Chen, Y. Wang, and Z. Zheng, “Mas-fire: Fault injection and reliability evaluation for llm-based multi-agent systems,” 2026. [Online]. Available: https://arxiv.org/abs/2602.19843

    arXiv 2026

  43. [43]

    The objective method: Probabilistic combinatorial optimization and local weak convergence,

    D. Aldous and J. M. Steele, “The objective method: Probabilistic combinatorial optimization and local weak convergence,” inProbability on Discrete Structures, H. Kesten, Ed. Berlin: Springer, 2004, pp. 1–72

    2004

  44. [44]

    Resolvent of large random graphs,

    C. Bordenave and M. Lelarge, “Resolvent of large random graphs,”Random Structures and Algorithms, vol. 37, no. 3, pp. 332–352, 2010

    2010

  45. [45]

    Interlacing families I: Bipartite Ramanujan graphs of all degrees,

    A. W. Marcus, D. A. Spielman, and N. Srivastava, “Interlacing families I: Bipartite Ramanujan graphs of all degrees,” Annals of Mathematics, vol. 182, no. 1, pp. 307–325, 2015

    2015

  46. [46]

    El Gamal and Y.-H

    A. El Gamal and Y.-H. Kim,Network Information Theory. Cambridge University Press, 2011

    2011

  47. [47]

    T. M. Cover and J. A. Thomas,Elements of Information Theory 2nd Edition (Wiley Series in Telecommunications and Signal Processing). Wiley-Interscience, July 2006

    2006

  48. [48]

    Large deviations for sums of partly dependent random variables,

    S. Janson, “Large deviations for sums of partly dependent random variables,”Random Structures and Algorithms, vol. 24, no. 3, pp. 234–248, 2004

    2004

  49. [49]

    Stein’s method for concentration inequalities,

    S. Chatterjee, “Stein’s method for concentration inequalities,”Probability Theory and Related Fields, vol. 138, pp. 305–321, 2007

    2007