PaperlayerView exact revision on arXiv

Source-linked AI summary

Consensus and Factual Dynamics in Large Populations of Interacting Language Models

Emanuele Ricco, Elia Onofri, Vincenzo Sammartino, Roberto Di Pietro

arXiv:2609.39211v1cs.MA

TL;DR

The paper asks how consensus and factual outcomes in interacting LLM populations depend on their interaction structure, including whether factual fate is determined by the initial vote. It introduces Rheon, an asynchronous spin-system framework swept across 432 configurations, and finds that consensus is common while factual outcomes are not fixed by initial majorities.

  • Problem

    The paper asks whether a population’s factual fate is fixed by its initial accuracy and whether consensus is reached slowly or can be read from early dynamics.

    Source: §4 · p. 6

  • Method

    Rheon represents responses with semantic and factual spins and sweeps a Glauber-inspired asynchronous population dynamics across interaction configurations.

    Source: §1 · p. 2, §3.1 · p. 4

  • Results

    Consensus is a near-universal outcome that becomes faster and more complete with higher connectivity, while factual fate is not fixed by the initial vote.

    Source: §5 · p. 10

  • Takeaways & Limitations

    A correct minority can pull a population back from a hallucinated majority, so initial agreement does not settle its factual outcome.

    Source: §5 · p. 10

    The study uses one frozen backbone and one prompt per run on regular geometries, and its factual axis relies on automatic tags that are noisiest for false-premise questions.

    Source: §5 · pp. 10–11

Abstract

from arXiv · show

Large Language Model (LLM) agents are increasingly deployed as populations of interacting entities, in which consensus --agreement on a shared answer-- emerges as a collective, unengineered behaviour. Prior work on LLM consensus shows that agents can cross-verify their answers and converge towards more factual responses, treating agreement as a proxy for correctness. However, these studies usually fix a single interaction structure, leaving open how consensus depends on how agents interact. We address this gap by introducing RHEON, a physics-inspired framework that recasts a population drawn from a single frozen model as an evolving $O(n)$ spin system on a ladder of interaction geometries of increasing effective dimension --from a 1D ring to a full-coupling mean-field graph-- with the sampling temperature $T$ as the tunable source of thermal disorder, evolved through a Glauber-like asynchronous dynamics. Sweeping RHEON across $432$ configurations of prompt, population size, communication topology, and sampling temperature yields Eraclitus-4.7M, a tagged evolutionary corpus of $4.7$ million responses. We find that agents reach their strongest consensus gain within the first few update sweeps and that increasing the number of neighbours per agent accelerates convergence on average. We further show that whether a configuration settles on factually correct or hallucinated consensus is not predictable from its initial state alone, and that the hallucination-minimising temperature depends on how the agents are coupled, so the common near-greedy default is not automatically the safest. Finally, semantic agreement correlates positively with factual convergence, and interaction strengthens the association, yet never enough for unanimity to certify correctness.

1. Introduction

Interacting LLM populations can produce collective answers that are difficult to trust, and prior consensus work leaves open how interaction structure shapes agreement. This paper frames consensus as an emergent property and introduces Rheon to study semantic agreement and factuality together.

1. Introduction

  • LLM agents are increasingly deployed in systems that debate, negotiate, review work, and vote, making population-level interaction central to their behaviour [3].Each agent carries its own persona, role, or behavioural disposition.Source: §1 · p. 2
  • Unsupervised interaction can yield confident, unanimous wrong answers, while prior work often treats agreement as a proxy for correctness [7].The open question is how consensus depends on how agents are wired together.Source: §1 · p. 2
  • Rheon models each response with a semantic O(n) spin and a binary factual spin, enabling consensus and factual dynamics to be studied together.The framework treats the semantic spin as the response embedding and the factual spin as whether the response is correct or hallucinated.Source: §1 · p. 2
  • The study introduces a Glauber-inspired asynchronous dynamics, sweeps interaction geometries, population sizes, and sampling temperatures, and releases the tagged Eraclitus-4.7M corpus.The framework also uses physics-inspired macroscopic order parameters to analyse population behaviour and convergence.Source: §1 · p. 2

2. Related Work

Related work spans statistical-physics analyses of language generation and models of consensus on networks. Rheon applies this descriptive perspective to interacting LLM populations while varying their interaction geometry and releasing the resulting evolutionary corpus.

Consensus

  • The geometry ladder increases effective dimension from a 1D ring (k = 2) through 2D and 3D tori (k = 4 and k = 6) to mean-field all-to-all coupling.The figure highlights a reference site and its neighbours for each graph.Source: Figure 1 · p. 3

Statistical physics of LLMs

  • Prior studies of LLM-generated text examine sampling distributions and report thermodynamic phenomena, whereas Rheon uses order parameters and temperature-like disorder to describe collective population dynamics.The related work distinguishes single-model sampling analyses from the present focus on interacting agents.Source: §2 · p. 3

Consensus

  • Opinion-dynamics and control-theoretic traditions study how network geometry, noise, and connectivity affect consensus.LLM work has also examined debate-driven agreement, topology-modulated strategies [23], and conformity to majority or confident peers [7].Source: §2 · p. 3
  • Rheon differs from averaging-based accounts by modelling the population as an evolving O(n) spin system and jointly varying geometry, population size, and temperature.Its geometry ladder runs from 1D through 2D and 3D to mean-field coupling.Source: §2 · p. 3, Figure 1 · p. 3

3. Methodology

Rheon represents interacting agents on regular graphs as coupled semantic and factual spins, then evolves their responses through asynchronous rewrites. The methodology sweeps configurations to build Eraclitus-4.7M and measures semantic ordering and factual error separately.

3. Methodology

  • The study places independent instances of one frozen model on a geometry ladder of regular graphs, from lattices to a mean-field graph connecting every agent to all others.The lattice coordination numbers are k = 2ℓ in dimension ℓ.Source: §3.1 · p. 3

3.1. The Rheon model.

  • Each response becomes a semantic spin from its embedding and a binary factual spin marking correctness or hallucination.The semantic spin dimension is fixed by the encoder; the factual label is adjudicated by a judge.Source: §3.1 · pp. 3–4
  • Peer interaction aligns semantic spins while sampling temperature supplies thermal disorder, and a Glauber-like asynchronous process updates agents using live neighbour responses.The temperature sweep ranges from T = 0.1 to 1.6; update order is random-sequential rather than lockstep.Source: §3.1 · p. 4

3.2. The Eraclitus-4.7M dataset.

  • Crossing six prompts, four geometries, twelve population sizes, and six temperatures yields 432 configurations and, over five trials each, 2,160 runs.Three prompts contain false premises and three are standard factual questions.Source: §3.2 · p. 4
  • Eraclitus tags responses as correct, hallucinated, not-known, or judge-failure, with factual analysis retaining correct and hallucinated answers.The verdicts are produced by an LLM judge and a stratified sample is checked against human annotators.Source: §3.2 · pp. 4–5

3.3. Order parameters.

  • Responses are embedded and whitened using independent initial answers pooled across temperatures, sizes, and geometries, so dot products of normalized spins measure semantic alignment.Whitening is performed once per prompt to reduce the common positive similarity component of raw embeddings.Source: §3.3 · p. 5
  • Semantic magnetisation and cohesion track alignment, while hallucination density and its change track factual error separately from consensus.The change in hallucination density is negative when interaction corrects errors and positive when it manufactures them.Source: §3.3 · pp. 5–6

4. Results

Across prompts and geometries, interaction raises cohesion rapidly, but consensus does not guarantee factual recovery. Factual outcomes and the temperature that best reduces error depend on geometry and can vary across trials.

4.1. Single-prompt dynamics.

  • Every geometry orders rapidly, flattening by about t ≈4, but the eventual alignment level rises from ring and 2D to near-perfect mean-field alignment.Source: §4.1 · pp. 6–7, Figure 2 · p. 7
  • At T = 0.1, denser-than-ring geometries can lock into shared hallucinations, including total mean-field error; the ring avoids lock-in but retains a residual hallucination floor.Source: §4.1 · p. 7
  • For Prompt 6, the best hallucination reduction shifts with coupling: larger 2D/3D lattices peak near cold T, whereas mean-field peaks at T⋆≈1.0.Source: §4.1 · pp. 7–8, Figure 4 · p. 8

4.2. Aggregate behaviour across prompts.

  • Across prompts 2–6, cohesion rises in essentially every run, while factual correctness is not universal.Source: Figure 5 · p. 9, §4.2 · p. 9
  • About 5% of pooled runs (85 of 1,800) amplify hallucination, half on the ring, although no prompt’s average accuracy declines.Source: §4.2 · pp. 8–9, Figure 5 · p. 9

5. Discussion

Across the tested geometries, consensus is near-universal and becomes faster and more complete with greater connectivity, but agreement does not determine factual correctness. Temperature’s effect depends on coupling, and early trajectory patterns permit shorter runs while leaving important monitoring and scope caveats.

Insight 1

  • Consensus emerges on every geometry, with greater connectivity bringing faster and more complete ordering.Mean-field reaches near-perfect alignment fastest at every temperature, but costs more per agent.Source: §5 · p. 10

Insight 2

  • A correct minority can redirect a hallucinated majority, although neighbours sharing the same wrong answer can amplify it.The authors describe repair as the general behavior and shared hallucination as a narrower failure mode.Source: §5 · p. 10

Insight 3

  • Temperature’s effect on factual outcomes is non-monotone, and its optimum depends on coupling.Cold sampling is safest only for sufficiently large populations with local coupling; under global coupling the optimum is approximately T≈1.0.Source: §5 · p. 10

Insight 4

  • Both consensus and factuality settle within the first few sweeps, supporting truncation at t=4 with no measurable cost in either observable.This removes 6 of 11 calls per agent (54.5%); monitoring should use more than agreement because cohesion and accuracy correlate imperfectly.Source: §5 · p. 10

Scope and limitations

  • The findings concern homogeneous populations on regular geometries, and factual labels are noisiest for false-premise questions.The study uses one frozen backbone and one prompt per run; its front-loading and apparent absorbing-state claims are limited to a ten-sweep horizon.Source: §5 · pp. 10–11

6. Conclusion

The conclusion presents Rheon and Eraclitus-4.7M, then draws together findings on consensus, factual repair, and temperature-dependent outcomes. It also identifies annealing and heterogeneous populations as directions for further study.

Contribution

  • Rheon models interacting LLM agents as an evolving O(n) spin system and produced Eraclitus-4.7M, a corpus of 4.7 million factually tagged responses across 432 configurations.The sweep varied prompt, geometry, population size, and sampling temperature.Source: §6 · p. 11

Findings

  • Consensus generally settles within the first few sweeps, but the final factual state is not fixed by the initial majority and the temperature effect depends on coupling.Repair remains possible when hallucinations initially dominate, while near-greedy sampling is not automatically prudent.Source: §6 · p. 11

Next steps and resources

  • The authors propose annealing temperature during a run and relaxing population homogeneity as directions for future testing; code, figures, and the corpus are slated for public release.The planned release includes judge tags and is described as being under a permissive research licence.Source: §6 · p. 11

Appendix A. The Eraclitus-4.7M dataset

Appendix A describes Eraclitus-4.7M’s questions, response-generation setup, factuality tagging, and asynchronous update procedure. The dataset uses six FreshQA questions and fixed generation settings, with a documented judge failure for one question.

Appendix A. The Eraclitus-4.7M dataset

  • Eraclitus-4.7M uses six fixed FreshQA questions, split between false-premise questions and standard multi-hop factual questions.The split allows structural agreement to diverge from semantic correctness.Source: Appendix A · p. 14

A.1. The prompt list.

  • Three prompts contain false premises that must be rejected, while three are answerable as asked.The appendix lists the prompts, including the false-premise cube-sum question and the standard question about Queen Elizabeth II’s age at death.Source: §A.1 · p. 14

A.2. The agent prompt template.

  • The chat template varies time, temperature, and neighbour count; initial answers have no peer context, while later updates include the agent’s previous response and k neighbours.The three axes are t, T, and k, with k determined by the geometry’s degree.Source: §A.2 · p. 15

A.3. The generation parameters.

  • Generation uses Qwen3-14B in 4-bit quantisation with thinking disabled, fixes every response at 100 tokens, and makes temperature the only relevant sampling parameter.The setup sets top-p to 1 and top-k to 0.Source: §A.3 · p. 15

A.4. The factuality tagging protocol.

  • An independent Qwen2.5-72B-Instruct judge assigns factuality labels, but its labels for Prompt 1 are systematically inverted, so that prompt is excluded from pooled main-text analyses.The main factual analyses retain correct and hallucinated responses and discard not-known and judge-failure rows.Source: §A.4 · pp. 15–16

A.5. The Glauber-inspired dynamics.

  • Rheon initializes agents independently, then runs ten sweeps of asynchronous updates in which a randomly selected agent sees live neighbour responses and overwrites its answer.Each sweep performs one elementary update per agent.Source: §A.5 · p. 16, §A.4 · p. 16

A.6. The released artefact.

Eraclitus-4.7M stores generated responses and their judge tags and embeddings in configuration- and trial-specific files. Its appendices document how to reconstruct population states and how campaign calls and judge categories are organized.

A.6. The released artefact.

  • 2,160 Parquet files totaling approximately 39 GB cover the 432 configurations and five trials, with each file containing one run’s generated responses.The initial response is included; filenames identify configuration and trial.Source: §A.6 · p. 16
  • A stored row is an update event, not a snapshot; reconstruct each sweep by carrying forward each agent’s latest response.Sampling with replacement means some agents are updated repeatedly while others retain their prior answer during a sweep.Source: §A.6 · pp. 16–17
  • Each released row contains one generated response, its judge tag, and embeddings from all four encoders.Source: Table 3 · p. 17

A.7. The LLM call budget.

Appendix B. Analysis of the LLM-as-a-judge tagging

The appendix shows that judge-tagged hallucination rates vary across geometry, temperature, prompt, and sweep. Prompt 1 behaves anomalously, while several other prompts show early reductions in hallucinations that differ by prompt.

Appendix B. Analysis of the LLM-as-a-judge tagging

  • Pooled hallucination share rises with temperature, from approximately 11% at the coldest settings to 22.0% at T = 1.6.The pooled trend masks substantially different prompt-level behaviour.Source: Appendix B · p. 18, Table 7 · p. 19
  • Prompt 1’s tagged hallucination share rises from 2.9% at T = 0.1 to 81.3% at T = 1.6, but the authors suspect label inversion and draw no dynamical conclusion from it.Source: Appendix B · p. 18, Table 7 · p. 19
  • Excluding Prompt 1, temperature responses remain heterogeneous: Prompts 3 and 6 dip at intermediate temperatures, Prompt 2 plateaus, and Prompts 4 and 5 stay near zero.Across Prompts 2–6, the temperature effect is dominated by repair, though Prompt 2’s mid-range plateau and some individual runs show manufacture.Source: Appendix B · p. 19
  • Prompt-specific hallucination trajectories differ: Prompts 3 and 6 fall sharply in the first two sweeps, while Prompt 1 rises from 10.8% at t = 0 to 68.2% at t = 10.Prompts 4 and 5 reach 0% within two sweeps, while Prompt 2 remains within a 5–9% band.Source: Appendix B · p. 19, Figure 6 · p. 19
  • The pooled sweep curves are stable across five trials, but averaging across configurations obscures the bimodal outcomes found in individual runs.The pooled standard deviation is at most 4.3 percentage points, while Figure 10 shows dispersion within each configuration.Source: Appendix B · p. 20

Appendix C. Human validation protocol and results

Human validation finds strong judge agreement on Prompts 2–6 but a severe, systematic disagreement on Prompt 1. The order-parameter schematic distinguishes disorder, partial alignment, and consensus, but semantic alignment alone does not indicate factual correctness.

Appendix C. Human validation protocol and results

  • The 1,063-response audit samples every configuration and deliberately oversamples the rare not-known category, so its frequencies do not mirror the corpus.Two humans independently annotated each response while blind to the judge’s verdict.Source: Appendix C · p. 20
  • Pooled judge–human agreement is 87.6%, with κ = 0.760 and 0.791 against the annotators, above their κ = 0.707 agreement with each other.Source: Appendix C · p. 20, Table 8 · p. 20
  • On Prompt 1, the judge misclassifies correct refutations of the false premise as hallucinations, while its agreement with humans is near chance.The authors exclude Prompt 1 from pooled analyses while retaining Prompts 2–6, whose automatic tags are validated as human-level.Source: Appendix C · p. 21

Appendix D. Order parameters

  • The schematic’s order parameters depict disorder near the finite-size magnetisation floor, partial alignment, and consensus as spins align.At large population size, cohesion is approximately the square of magnetisation; neither semantic measure encodes factual accuracy.Source: Appendix D · p. 21, Figure 7 · p. 21, Appendix C · p. 21

D.1. A geometric interpretation of the order parameters.

Semantic cohesion and factual accuracy are distinct: populations can reach geometrically similar consensus states with opposite factual outcomes. The appendix extends trajectory and temperature analyses across prompts and geometries to show how ordering and hallucination dynamics vary.

D.1. A geometric interpretation of the order parameters.

  • At high cohesion, collective error correction and shared hallucination have the same semantic order parameters, so hallucination density must be tracked separately.The two outcomes correspond to ρ → 0 and ρ → 1, respectively.Source: §D.1 · p. 22, Figure 8 · p. 22

D.2. Evolution of the order parameters across the prompts.

  • Across prompts and geometries, magnetisation rises toward a plateau within the first few sweeps, with higher connectivity ordering populations faster and further.Cold curves start higher because near-greedy sampling makes initial responses nearly identical; their later level does not mean interaction did more work.Source: §D.2 · p. 22
  • Comparing the magnetisation and hallucination grids shows that strong ordering can accompany rising hallucination, indicating convergence onto a shared wrong answer.The grids disaggregate prompt-level trajectories by geometry and temperature.Source: §D.2 · pp. 22–23
  • The phase-space grid separates cohesion from hallucination: trajectories moving right and down indicate increasing cohesion as hallucination falls, while right and up indicate shared error.Cold, densely coupled runs can end at high cohesion and high hallucination.Source: §D.2 · p. 23
  • The temperature grid instead uses temperature on the horizontal axis and system size for color, with a separate size ladder for each geometry.Source: §D.2 · p. 23

Appendix E. Embedding-robustness results

The geometry ladder and configuration-level cohesion rankings remain essentially unchanged across four sentence encoders, despite slight shifts in absolute order-parameter values.

Appendix E. Embedding-robustness results

  • Across the four encoders, the geometry ladder and per-configuration cohesion rankings remain robust despite slight rescaling of order parameters.The pipeline re-embeds approximately 4.7 million responses and reruns whitening, unit-spin projection, and order-parameter estimation unchanged for each encoder.Source: Appendix E · p. 23
  • The robustness check re-embeds responses with three additional encoders and reruns the analysis pipeline unchanged for each encoder.The main-text observables use all-MiniLM-L6-v2; the alternatives are all-MiniLM-L12-v2, all-mpnet-base-v2, and paraphrase-mpnet-base-v2.Source: Appendix E · p. 23

E.1. The geometry ladder is preserved.

  • Final semantic cohesion increases monotonically from 1D through mean-field under all four encoders, with mean-field reaching approximately 0.92 in every case.The absolute levels shift slightly between encoders, but the geometry ordering remains identical.Source: §E.1 · p. 23

E.2. The per-configuration ordering is preserved.

Configuration-level cohesion rankings are highly consistent across encoders, and the geometry-level cohesion ordering also persists across encoder choices.

E.2. The per-configuration ordering is preserved.

  • Spearman rank correlations of final cohesion between encoder pairs range from 0.899 to 0.962 across all configuration-trial pairs.The authors report that encoders agree on which individual configurations cohere most strongly.Source: §E.2 · p. 23, Table 10 · p. 24
  • Table 9 averages final cohesion across prompts 2–6, sizes, temperatures, and trials, and reports the 1D < 2D < 3D < MF ladder under every encoder.The caption identifies the aggregation scope and the ordering preserved across encoders.Source: Table 9 · p. 24
  • Supplementary panels plot semantic magnetisation, hallucination density, phase-space trajectories, and interaction-induced hallucination change across prompts and geometries.The interaction-induced change panel varies temperature and system size and shows standard errors over five trials.Source: Figure 9 · p. 25, Figure 10 · p. 26, Figure 11 · p. 27
Loading 2609.39211v1…