Consensus and Factual Dynamics in Large Populations of Interacting Language Models

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

cs.MASep 30, 2026

Consensus and Factual Dynamics in Large Populations of Interacting Language Models

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

Organizations: Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division, King Abdullah University of Science and Technology (KAUST) Thuwal 23955, Saudi Arabia · Dipartimento di Informatica, Universit`a di Pisa Pisa, Italy

Abstract

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.

Figures & tables

Panel of Figure 1 — Figure 1 . The geometry ladder. Rheon couples the $N_{a}$ agents through one of four graphs of increasing effective dimension: a 1D ring ( $k=2$ ), a 2D torus ( $k=4$ ), a 3D torus ( $k=6$ ), and a mean-field all-to-all graph ( $k=N_{a}-1$ ). In each panel one reference site (red) and its $k$ neighbours (blue) are highlighted, illustrating the coordination number; the population sizes $N_{a}$ used in the sweeps are reported above each panel.

		Cost per agent (m $) through sweep$ t$
Geometry	$N_{a}$	$t=0$	$t=4$	$t=7$	$t=10$	Total
1D	8	5.148	50.686	84.827	118.969	$0.95
	16	5.148	50.689	84.821	118.935	$1.90
	32	5.148	50.681	84.816	118.940	$3.81
2D	64	5.148	66.308	112.153	157.992	$10.11
	144	5.148	66.300	112.156	158.006	$22.75

Table 1 . Estimated cost of Eraclitus at Qwen3-14B per-token cloud pricing (

0.10 / 1M input,

0.24 / 1M output; https://openrouter.ai/qwen/qwen3-14b ), broken down by configuration.

Figure 2 . Temporal evolution of the semantic magnetisation

m(t)

(top) and of the hallucination density

\rho(t)

(bottom) across network geometries for Prompt 3, shown for the 1D ring at

N_{a}=32

and for 2D, 3D, and MF at

N_{a}=64

, over ten sweeps. Trajectories are colour-coded by sampling temperature

T

Figure 5 . Global transformation from

t=0

(

x

-axis) to

t=10

(

y

-axis) of factual accuracy (left,

1-\rho

) and semantic cohesion (right,

\bar{c}

) divided by prompt (2–6, colour) and network geometry (shape). The invariant threshold (

y=x

line) is shown as a dashed line, with the upper and lower triangles indicating improved and worsened performance over time, respectively.

	$N_{a}$	$t=0$	$t=10$
1D	8	+.127	+.183 ⋆
	16	+.050	+.024
	32	+.068	+.237 ⋆
2D	64	+.066	+.162 ⋆
	144	+.095	+.327 ⋆
	256	+.097	+.361 ⋆

Table 2. Pearson correlation over 150 runs between semantic cohesion

\bar{c}

and factual accuracy (

1-\rho

) evaluated before (

t=0

) and after (

t=10

) the agents’ evolution.

{}^{\star}=p<0.05

Appendix figures & tables15 assets

Supplementary material from the paper’s appendix.

Appendix

Column	Contents
p	prompt identifier, $1$ – $6$
geometry	1D , 2D , 3D , or MF
L	lattice side; null for MF
N_a	population size
T	sampling temperature
t_1	sweep index, $0$ – $10$

Appendix

Table 3 . Row schema of a released Eraclitus-4.7M Parquet file. One row is one generated response, i.e. one elementary update of the Rheon dynamics (Algorithm 1 ), together with its judge tag and its embeddings under all four encoders.

Geometry	$N_{a}$	Calls	$\times 6$ prompts
1D	$\{8,16,32\}$	$3{,}696$	$22{,}176$
2D	$\{64,144,256\}$	$30{,}624$	$183{,}744$
3D	$\{64,216,512,1000\}$	$118{,}272$	$709{,}632$
Mean-field	$\{32,64\}$	$6{,}336$	$38{,}016$
Total (one trial)			$953{,}568$
Total ( $R=5$ trials)			$\mathbf{4{,}767{,}840}$

Appendix

Table 4 . LLM call budget by geometry. Per each geometry it is specified the set of sizes

N_{a}

used in the campaign, the number of calls for a single prompt summed over all sizes and temperatures, and the total multiplied by the six prompts. Totals are reported for one trial and for the full

R=5

campaign.

Geometry	Correct	Halluc.	Not-known	Judge-fail	Total
1D	88,703	20,081	1,958	138	110,880
2D	756,877	150,318	10,739	786	918,720
3D	2,925,544	580,723	38,427	3,466	3,548,160
MF	156,790	29,718	3,350	222	190,080
Total	3,927,914	780,840	54,474	4,612	4,767,840

Appendix

Table 5 . LLM-as-a-judge category counts by geometry, summed over the

432

configurations (

6

prompts,

12

sizes,

6

temperatures), all sweeps, and all

R=5

trials of the released dataset, with per-geometry and per-category totals. Hallucination shares quoted for this table in the text are taken over all tagged rows, as in Table 6 , and so are not directly comparable with

\rho

(see Table 7 ).

$T$	Correct	Halluc.	Not-known	Judge-fail	Halluc. %
0.1	691,107	93,661	9,403	469	11.8
0.4	703,233	83,733	7,442	232	10.5
0.7	643,959	142,643	7,637	401	18.0
1	645,110	140,496	8,323	711	17.7
1.3	635,996	145,443	11,535	1,666	18.3
1.6	608,509	174,864	10,134	1,133	22.0

Appendix

Table 6 . Judge-category counts by temperature, pooled over all geometries, prompts, sizes, sweeps, and all

R=5

trials , with per-temperature and per-category totals. Halluc. % is the share of all tagged rows in the row, not of the adjudicated ones, and so is not directly comparable with

\rho

(see Table 7 ).

Prompt	T=0.1	T=0.4	T=0.7	T=1	T=1.3	T=1.6
1	2.9	15.2	66.6	73.3	75.7	81.3
2	3.7	4.2	10.0	10.4	10.3	9.0
3	26.9	13.4	8.8	9.6	9.8	10.1
4	0.0	0.0	0.0	0.0	0.5	1.9
5	0.0	0.0	0.0	0.0	0.1	0.5
6	39.2	31.2	23.7	15.1	17.8	33.2

Appendix

Table 7 . Hallucination percentage by prompt and temperature, pooled over all configurations and all

R=5

trials (the same full dataset as Tables 5 and 6 ). Here the percentage is taken over the adjudicated responses only,

h/(c+h)

, matching the definition of

\rho

in the main text.

Figure 6 . Hallucination percentage for each prompt as a function of sweep

t

, pooled over all geometries, sizes, and temperatures. The marker is the mean over the

R=5

trials; the error bars are

\pm 1

standard deviation across trials.

		Cohen’s $\kappa$			Raw agreement (%)
Prompt	$N$	LLM–A 1	LLM–A 2	A 1–A 2	LLM–A 1	LLM–A 2	A 1–A 2
1	200	0.025	0.366	0.015	34.0	58.5	33.5
2	146	1.000	0.880	0.880	100.0	97.9	97.9
3	267	1.000	0.895	0.895	100.0	94.0	94.0
4	143	–	–	–	100.0	100.0	100.0
5	144	–	–	–	100.0	100.0	100.0

Appendix

Table 8. Inter-rater agreement between the LLM judge and the two independent human annotators ( A 1 , A 2 ), per prompt and pooled, as Cohen’s

\kappa

(left) and as raw percentage agreement (right). Dashes mark prompts where all three raters assigned the same label to every response, leaving

\kappa

undefined but raw agreement well defined at

100\%

Panel of Figure 7 — Figure 7 . Schematic of the semantic order parameters on the unit sphere, shown in 2D projection. Thin arrows are the agent spins $\mathbf{s}_{i}$ ; the thick arrow is the semantic magnetisation $\mathbf{m}$ , whose length is $m$ . Left: a disordered population, spins spread over the sphere, $m$ near its $1/\sqrt{N_{a}}$ floor and cohesion $\bar{c}\approx 0$ . Centre: partial order, $m\approx 0.5$ , a majority direction with a dissenting minority. Right: consensus, spins aligned, $m\approx 1$ and $\bar{c}\approx 1$ .

Panel of Figure 8 — Figure 8 . The two consensus outcomes. A population that starts mixed and disordered (left, $\rho\approx 0.5$ , $\bar{c}\approx 0$ ) can settle, at high cohesion, into either collective error correction (top, $\rho\to 0$ ) or a shared hallucination (bottom, $\rho\to 1$ ). Arrows are agent spins, coloured by factual state (green: correct; red: hallucinated); which branch a configuration takes is not fixed by its initial accuracy (see the main-text results).

Figure 9 . Semantic magnetisation

m(t)

for all five analysed prompts (Prompts 2–6, rows) across all four geometries (columns), one curve per temperature.

Figure 10 . Hallucination density

\rho(t)

for all five analysed prompts (Prompts 2–6, rows) across all four geometries (columns), one curve per temperature.

Figure 11 . Phase-space trajectories

(\bar{c}(t),\rho(t))

for all five analysed prompts (Prompts 2–6, rows) across all four geometries (columns), one curve per temperature.

Figure 12 . Interaction-induced change in hallucination,

\Delta\rho=\rho(t{=}10)-\rho(t{=}0)

, against sampling temperature

T

for all five analysed prompts (Prompts 2–6, rows) across all four geometries (columns). One curve per system size, on that geometry’s colour ramp from light (small) to dark (large); error bars are the standard error over the

R=5

trials, and the dashed line marks the

\Delta\rho=0

expected of a symmetric voter model. The vertical scale is shared within a row but not across rows, since

\Delta\rho

spans a very different range from prompt to prompt.

Geometry	MiniLM-6	MiniLM-12	MPNet-b	MPNet-p
1D	0.621	0.613	0.592	0.667
2D	0.634	0.635	0.609	0.710
3D	0.731	0.723	0.704	0.782
MF	0.926	0.920	0.916	0.919

Appendix

Table 9. Final semantic cohesion

\bar{c}_{\mathrm{final}}

by geometry under each of the four sentence encoders, averaged over prompts (2–6), sizes, temperatures and trials. The geometry ladder 1D

<

<

<

MF holds under every encoder.

	MiniLM-6	MiniLM-12	MPNet-b	MPNet-p
MiniLM-6	1.000	0.962	0.934	0.916
MiniLM-12	0.962	1.000	0.947	0.899
MPNet-b	0.934	0.947	1.000	0.923
MPNet-p	0.916	0.899	0.923	1.000

Appendix

Table 10. Spearman rank correlation of the per-configuration final cohesion

\bar{c}_{\mathrm{final}}

between the four encoders, over all

(\text{config},\text{trial})

pairs. Values

\geq 0.89

throughout: the ordering of configurations by cohesion is essentially encoder-invariant.

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Jun 20, 2026, cs.CLJ/K move · Enter open · S save

When Is Emergent Consensus Real? A Measured Coupling Gain and a Validity Diagnostic for LLM Agent Societies

Dongxu Yang

DeepLethe

Social Dynamics Conformity Frontier Models Validation

Open paper AskPDF arXiv

LLM "agent societies" are studied via demonstrations of emergent consensus or polarization -- with no measurable control parameter, no theory of when each regime appears, and no test of whether an outcome is a genuine social dynamic or a model artifact. We introduce the coupling gain gamma, measured per-agent by counterfactually perturbing a neighbour's stated opinion. (i) gamma is stable and model-distinguishing -- across five frontier models it spans 0.15-0.43 (n=20, 95% CIs <= 0.025), paraphrase-invariant; social-neighbour gamma roughly equals numeric-anchor gamma, so gamma is evidence-coupling, not uniquely social. (ii) Classical dynamics with measured (not assumed) coefficients organise the regime: Friedkin-Johnsen for consensus/pluralism, signed-Laplacian/structural-balance for polarization. (iii) Frontier LLMs do not spontaneously backfire (beta <= 0), so default societies do not self-polarize -- polarization is always induced; the beta>0 branch arises only in the FJ surrogate, never in the agents. (iv) A randomized-initial-condition diagnostic -- the (slope, bias) of final vs. initial opinion -- separates genuine averaging from model-prior artifacts (boundary-censoring ruled out by construction via interior-valued facts); applied to a published "emergent consensus" result (Chuang et al. 2023) it reveals a model-specific conflation: averaging on debatable claims, prior-artifact on settled facts. (v) Coupling is context-dependent: pairwise gamma does not predict multi-neighbour outcomes -- it can order them backwards -- whereas a modality-matched group coupling does (sixteen closed+open models, Pearson r=-0.70, permutation p=0.008). The regime laws take this matched coupling, not the single-neighbour gamma: emergent consensus must be read from coupling in the target interaction. We contribute a measurement protocol and a validity instrument, not new theory.

Abstract

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