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Physics of Agents: Statistical Mechanics Predicts Collective Behavior of AI Agents

Batu El, Jinhee Paeng, Fatih Dinc, Shiye Su, Mete Erdogan, Aneesh Pappu, Haotian Ye, Wanjia Zhao, Surya Ganguli, James Zou

arXiv:2608.16578v1cs.AIcs.MAcs.SI

TL;DR

Interacting AI agents can improve collective decisions but may also produce herding, polarization, or shared biases, making their collective dynamics important to predict. Studying over 10,000 simulated communities, the paper develops a statistical-mechanics model that predicts agent trajectories and group outcomes while identifying recurring regimes and task-dependent opinion shifts.

  • Problem

    The paper asks how interactions among AI agents shape collective outcomes and when communication improves decisions versus producing polarization, disagreement, or shared biases.

    Source: §1 · p. 2

  • Method

    The authors study language-model agents exchanging messages over fixed social networks on objective mathematics questions and subjective political statements, then model opinion updates as responses to social pressure.

    Source: §1 · p. 2, §4 · p. 10

  • Results

    Over 10,000 simulated groups exhibit indifference, polarization, and consensus; interaction increases conviction, improves objective-question accuracy, and often shifts subjective political opinions rightward, while the model predicts trajectories and generalizes to unseen graphs.

    Source: §1 · p. 3, §7 · p. 17

  • Takeaways & Limitations

    Collective behavior in interacting AI agents can be described by compact dynamical laws that forecast population dynamics and connect outcomes to interpretable interaction parameters.

    Source: §1 · p. 3

    The model simplifies interactions by discarding the content of the natural-language messages that mediate influence.

    Source: §7 · p. 18

Abstract

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AI agents increasingly operate as part of interacting systems rather than in isolation. As agents exchange information and jointly make decisions, their interactions can improve collective reasoning but may also produce herding, polarization, or amplify shared biases. Understanding and predicting these collective dynamics is therefore important for designing effective and aligned multi-agent systems. Here, we study over 10,000 communities of language-model agents that repeatedly exchange messages and revise their opinions across objective mathematics questions and subjective political statements. Despite substantial diversity in possible behavior, the individual and group dynamics can be represented by three characteristic regimes: indifference, polarization, and consensus. AI agents start indifferent and build conviction as they interact. On objective questions, communication improves collective accuracy, while on subjective questions it often drifts group opinions toward the right in the political spectrum. We explain these observations with a statistical-mechanics formalism in which agents stochastically favor lower social pressure. Given only initial opinions, our model predicts individual trajectories, outperforms all standard baselines, generalizes to unseen community graphs, and reproduces the observed group archetype distributions. Our fitted model parameters reveal the mechanics underlying our key observations: i) communities operate below the critical social temperature, which explains conviction buildup; ii) attractive ties outweigh repulsive ones, which favors consensus; and iii) agents holding the correct answer exert the strongest pull, which drives truth-seeking. Overall, our results demonstrate that collective behavior of AI agents, like that of other complex systems, follows compact and predictive dynamical laws.

1 Introduction

The paper studies how interacting language-model agents evolve from initially weak opinions into indifference, polarization, or consensus, and develops a statistical-mechanics model to predict these dynamics. Across objective and subjective questions, interaction improves collective accuracy in the former but can amplify ideological drift in the latter.

Motivation

  • Interacting agents can combine information and improve decisions, but may also produce herding, polarization, disagreement, oscillation, and amplified shared biases.Understanding these collective outcomes is important for designing effective multi-agent systems and AI alignment.Source: §1 · p. 2

Study design

  • The study simulates around 10,000 communities of persona-based language-model agents exchanging opinions over signed communication networks on objective mathematics and subjective political questions.Agents repeatedly revise their views across multiple models, questions, networks, and episodes.Source: §1 · pp. 2–3

Empirical dynamics

  • Communities exhibit indifference, polarization, or consensus, while interaction generally increases conviction and can weaken or overturn the initial collective position.Individual trajectories include frozen, switching, reversing, and oscillating behaviors; objective-question communication improves accuracy, whereas subjective discussion can cause rightward ideological drift.Source: §1 · p. 3

Theoretical model

  • The authors model binary opinion updates as stochastic minimization of social pressure, combining intrinsic fields with neighbor interactions in an extended Ising framework.Interaction parameters and intrinsic fields are fitted directly from observed opinion transitions.Source: §1 · p. 3

Prediction and mechanisms

  • The fitted model predicts held-out collective behavior, outperforms standard baselines, generalizes to unseen network families, and reproduces population-level outcome distributions.Its parameters indicate low-temperature ordered dynamics, stronger concordant than discordant ties, and greater influence from agents holding correct objective answers.Source: §1 · p. 3

Contribution

  • The results show that compact statistical-mechanical laws can forecast collective AI-agent dynamics and connect system-level outcomes to interpretable interaction parameters.This framework identifies recurring regimes while explaining conviction buildup, consensus formation, and truth-seeking.Source: §1 · p. 3

2 Setup

The setup models heterogeneous language-model agents that repeatedly exchange opinion-bearing messages over signed communication networks. It evaluates this process on objective mathematics questions and subjective political statements under synchronized iterative updates.

Experimental settings

  • The experiments compare objective and subjective settings that differ in both question type and persona construction, using language-model agents with N = 32 unless otherwise stated.The main simulations synchronize communications and updates, while asynchronous updates are examined separately with similar results.Source: §2 · pp. 4–5

Agents and questions

  • Each agent has a fixed persona encoding competence on objective questions or preferences on subjective questions, and generates opinions and supporting messages from its persona, the question, and received messages.Objective questions come from MATH [Hendrycks et al., 2021], while subjective statements come from the Political Questions Dataset and personas draw on TWIN-2K-500 [Toubia et al., 2025].Source: §2 · p. 4

Communication network

  • The communication network is encoded by signed ties J_ij ∈ {−1, 0, +1}, representing friendly, absent, or unfriendly connections that determine message routing and social influence.Messages travel through separate friendly and unfriendly inboxes, and experiments use symmetric J with zero diagonal.Source: §2 · pp. 4–5

Procedure

  • In each synchronous round, agents sample binary votes five times, aggregate them into an opinion, compose a short supporting message, and update from refreshed inboxes containing only the latest messages.The resulting dynamics are Markovian and propagate influence through the graph from initial opinions formed using only personas and the question.Source: §2 · pp. 4–5

3 Empirical Characterization of Agents’ Collective Dynamics

Across 9,600 simulated communities and a dataset exceeding 10,000 groups, the study identifies recurring individual and collective dynamics across objective and subjective questions. Communities build conviction, often move toward consensus, tend toward truth on objective questions, and shift politically from left to right on subjective questions.

3.1 Behavior of Groups and Individuals

  • Individual trajectories are dominated by frozen agents, while group trajectories display five archetypes, including divergence and majority switches reaching 11–12% in GPT-4o-mini and Qwen3.5-9B.Individual archetypes are defined by the number of opinion switches; group archetypes distinguish persistent split, convergence, persistent majority, divergence, and majority switch outcomes.Source: Figure 2 · p. 6, Figure 3 · p. 6, §3.1 · p. 7

3.2 Conviction Buildup and Consensus Formation

  • Conviction rises over time as communities transition from early indifference toward consensus or polarization, while consensus increases monotonically.The three regimes are defined by net opinion and conviction: consensus has high values of both, polarization has near-zero net opinion and high conviction, and indifference has low values of both.Source: §3.2 · p. 8, Figure 4 · p. 8

3.3 Truth Seeking Tendency

  • Objective-question communities tend toward correct answers, with incorrect-to-correct switches more common than correct-to-incorrect switches across all models.Strong improvements are observed for Gemma, GPT, and Qwen.Source: §3.3 · pp. 9–10, Figure 5 · p. 9
  • On subjective questions, majority opinions are more likely to shift from left to right than from right to left.The analysis uses continuous opinions across 9,600 simulated communities, with over 10,000 groups after adding generalization and asynchronous-update experiments.Source: §3 · p. 5, §3.3 · p. 10

4 Statistical Mechanics Model of Agents’ Behavior

The model treats opinion updates as stochastic movement toward configurations minimizing social and personal pressure. It combines signed social couplings with agent-specific predispositions, yielding logistic updates and extensions for richer interactions, fitting, continuous time, and baseline comparisons.

Model assumptions

  • Agents stochastically adjust opinions toward configurations that minimize social pressure from interpersonal alignment and personal pressure from disagreement with intrinsic tendencies.The total pressure combines social interactions with a personal term weighted by λ ≥ 0.Source: §4 · p. 10

Opinion dynamics

  • The resulting logistic update depends on signed peer pressure scaled by β and an intrinsic field encoding each agent’s issue-specific leaning.The intrinsic field is defined as gi = λ˜si and can be parameterized from persona and question information.Source: §4 · pp. 10–11

Three Couplings

  • The three-coupling model separates connection existence from concordant and discordant influence, assigning friendly-neighbor weight β+ + β0 and unfriendly-neighbor weight β0 − β−.This parameterization enables direct interpretation of valence-specific effects after decoupling the general neighbor effect.Source: §4 · p. 11

Extensions and baselines

  • The framework also includes a continuous-time mean-field ODE with update-rate parameter ε and evaluates predictive value against Persistence, Interaction-Free, and Mean-Field baselines.The Mean-Field baseline preserves personal and social pressure but removes signed graph structure by using the global average opinion.Source: §4 · p. 11

5 Applications of the Statistical Physics Model

The statistical-physics model predicts agent dynamics across held-out questions and graph structures, reproduces aggregate archetype distributions, and explains conviction buildup, consensus, and truth seeking through fitted social couplings. Its applicability also extends to asynchronous updates, where the continuous-time rule outperforms all baselines.

5 Applications of the Statistical Physics Model

  • Across unseen graphs and questions, the model predicts individual trajectories and reproduces population-level collective outcomes, revealing compact mechanisms for consensus, polarization, and truth seeking.The fitted communities operate below the critical social temperature; attractive ties outweigh repulsive ties, and correct-answer neighbors exert the strongest pull.Source: §5 · pp. 11–12

5.1 Prediction and Generalization

  • The three-coupling rule is best in 15 of 16 unseen-graph columns, achieving 85.0–97.8 one-step and 59.7–89.3 rollout balanced accuracy while outperforming every baseline.On held-out questions, it is best in every Table 1 column; in-distribution and out-of-distribution performance differs by at most 2.4 points.Source: Table 2 · p. 12, Table 1 · p. 12, §5.1 · p. 13

5.2 Distributions of Archetypes

  • The fitted stochastic rollouts reproduce group archetype shares with mean absolute deviations of ∼3 points for objective and ∼5 points for subjective questions, despite overproducing individual oscillators.Episodes with identical personas, graphs, and questions often disagree because of sampling randomness, so evaluation targets aggregate distributions rather than exact group archetypes.Source: §5.2 · p. 14

5.3 Critical Temperature and Conviction Buildup

  • The fitted operating point lies below the critical temperature in every dataset–model cell, making indifferent states unstable and causing conviction to grow over rounds.The critical temperature is defined where variance of absolute net opinion peaks, marking the transition between indifference and high-conviction consensus or polarization.Source: Figure 7 · p. 14, §5.3 · p. 15

5.4 Consensus Formation and Truth Seeking

  • Concordant ties exert stronger effective influence than discordant ties, favoring consensus, while correct neighbors exert stronger pulls and drive communities toward the correct answer.The effective concordant coefficient ranges from 0.99 to 3.03, whereas the discordant coefficient never exceeds 0.73 and is negative or negligible in two subjective cases.Source: §5.4 · pp. 15–16, Table 3 · p. 16

5.5 Asynchronous Updates

  • With asynchronous updates, the continuous rule with coupling splits reaches 80.4/78.5 rollout accuracy in-distribution/OOD on subjective questions and 65.3/65.9 on objective questions, outperforming every baseline.Because only some agents update within each time window, evaluation uses raw accuracy rather than balanced accuracy.Source: §5.5 · p. 16

6 Related Works

The paper connects its model to statistical mechanics, extending established spin, collective-computation, and social-influence frameworks to provide a quantitative, predictive theory of LLM-community evolution.

Statistical Mechanics

  • The model uses the Ising energy function, Glauber dynamics [Glauber, 1963], and the Edwards–Anderson parameter [Edwards and Anderson, 1975, Sherrington and Kirkpatrick, 1975].Its collective-computation perspective also draws on Hopfield networks [Hopfield, 1982, Amit et al., 1985].Source: §6 · pp. 16–17

Applications of Statistical Mechanics

  • Statistical-mechanics theory has modeled phases of spin glasses, flocking, neural and protein data, collective behavior, and binary choice under social influence.These applications include [Parisi, 1979, Mézard et al., 1987], [Vicsek et al., 1995, Toner and Tu, 1995],, [Weigt et al., 2009, Marks et al., 2011], [Bialek et al., 2012], and [Brock and Durlauf, 2001, Castellano et al., 2009].Source: §6 · p. 17

Social Simulations

  • LLM social simulations represent experimental subjects, virtual populations, and collective phenomena such as opinion dynamics, while this work offers a quantitative, predictive theory of their evolution.The cited literature includes [Argyle et al., 2023, Horton, 2023, Park et al., 2024], [Park et al., 2023, Yang et al., 2024, Piao et al., 2025], and [Chuang et al., 2024, Ashery et al., 2025, De Marzo et al., 2024].Source: §6 · p. 17

Multi-agent Systems

  • Multi-agent LLM systems use debate, role-conditioned discussion, prompt and workflow search, sub-task partitioning, opinion aggregation, and open-ended deliberation.Learned interaction topologies can require expensive roll-outs and produce brittle systems [Cemri et al., 2025, Wang et al., 2024b].Source: §6 · p. 17

7 Discussion and Conclusion … C Additional Observations

The paper finds that repeated interaction produces predictable collective dynamics whose effects differ by task: it improves objective accuracy but can amplify political bias. It also identifies simplifying assumptions, proposes extensions, and frames the theory as useful for anticipating and designing multi-agent behavior without extrapolating directly to humans.

7 Discussion and Conclusion

  • Across 10,000 groups, interaction increases conviction and ordering; it improves objective-question accuracy but produces rightward political drift in three of four models.The fitted model generalizes to unseen questions and graphs and approximately reproduces group-level outcomes.Source: §7 · p. 17
  • The fitted mechanics indicate subcritical social temperature, stronger concordant than discordant interactions, and greater influence from correct neighbors.These mechanisms explain conviction buildup, consensus formation, and truth-seeking on objective tasks.Source: §7 · p. 17
  • The setup assumes one shared binary question, fixed symmetric communication, memoryless updates, and discards the natural-language content mediating influence.These assumptions simplify the interaction process and limit direct representation of message-mediated effects.Source: §7 · pp. 17–18
  • Future setup extensions include vector-valued or open-ended opinions, simultaneous questions, adaptive communication, spatial proximity, memory, real-world social-platform data, and larger heterogeneous populations.The proposed directions include Likert scales and agent data from Moltbook.Source: §7 · p. 18
  • Methodological extensions would incorporate vector opinions, message embeddings, controlled interventions, and evolving communication patterns through learning rules.The proposals invoke Potts-type models [Potts, 1952, Wu, 1982], graph-neural message passing [Scarselli et al., 2009, Gilmer et al., 2017], interventions [Hu et al., 2025], and Hebbian Learning.Source: §7 · p. 18
  • The theory could help anticipate multi-agent failure modes and optimize communication channels, population composition, and other system hyperparameters, but resemblance to human opinion dynamics remains hypothetical.The measured dynamics characterize persona-conditioned language-model agents rather than human communities.Source: §7 · p. 18

A Terms and Notation; C Additional Observations

D Method Details

This section contains supplementary method details and experiments covering raw accuracy, asynchronous updates, temperature sweeps, group-archetype predictability, and opinion flow.

D Method Details

  • The section outlines supplementary analyses of the continuous model, update timing, temperature, group archetypes, and opinion flow.Listed subsections include Raw Accuracy and the Utility of the Continuous Model, Asynchronous-update experiment, Temperature Sweep Experiment, Predictability of Group Archetypes, and Flow of Opinion.Source: E Additional Results · p. 27

E Additional Results · A Terms and Notation · B Setup Details

The supplementary material defines the paper’s terminology and notation and documents prompt templates for the objective and subjective tasks in opinion and message formats.

A Terms and Notation

  • The terms-and-notation supplement specifies the terminology and notation used throughout the paper.Table S1 provides this reference material.Source: Table S1 · p. 28

B Setup Details

  • The setup supplement presents prompt templates covering objective and subjective tasks in both opinion and message formats.Full prompts are provided in Appendix B.4.Source: Figure S1 · p. 29

B.1 Tasks … C.1 Political Lean of Interacting Agents

The study constructs binary objective and subjective tasks, varies communication graphs and language-model backbones, and evaluates interaction-driven opinion dynamics. Across political statements, most models drift rightward over eight rounds, while Llama remains nearly stable.

B.1 Tasks

  • Tasks reduce both domains to binary choices: MATH problems use randomly positioned correct answers and distractors, while political statements use agree (+1) or disagree (−1).Subjective personas encode demographic, attitudinal, and behavioral traits; objective personas instead provide agents with ground-truth solutions to distinct mathematics questions as expertise.Source: §B.1 · pp. 29–30
  • Candidate questions are binarized and high-entropy filtered by sampling 32 agents eight times, retaining items with nontrivial response variability so interaction can change opinions.Questions are split randomly into equal train and test halves, with objective label balance preserved; fits use train transitions and accuracies are reported on test questions.Source: §B.1 · pp. 30–31

B.2 Communication Networks

  • Communication networks span 12 signed random graphs, square and triangular positive lattices, and six low-rank graphs with rank 10 and frustration densities of 0.0, 0.1, and 0.2.The graph families vary connectivity, rank, and frustration while holding the edge budget fixed where specified.Source: §B.2 · p. 31, Figure S3 · p. 31

B.3 Models

  • Experiments use four language models—gpt-4o-mini, gemma-3n-e4b-it, qwen3.5-9b, and llama-3.1-8b-instruct—with persona and question embeddings from text-embedding-3-small.Each backbone is tested in independent experiments.Source: §B.3 · p. 31

B.4 Prompts

  • Opinion and message prompts ask persona-conditioned agents for direct judgments and generated messages without chain-of-thought.The prompts separately specify subjective opinion generation and message generation while retaining the same persona description.Source: Figure S4 · p. 32, Figure S5 · p. 33

B.5 Message Sampling

  • GPT-4o-mini and Gemma runs originally sampled recipient-specific messages, whereas Qwen, Llama, and asynchronous runs followed the single-message procedure; the authors report equivalent qualitative findings.The variants differ only in cross-receiver wording correlations, and repeated samples were functionally near-duplicates under identical conditioning.Source: §B.5 · p. 34

C Additional Observations

  • The political-lean analysis tracks right-leaning community trajectories over eight rounds using label-balanced statements, including left→left, left→right, right→left, right→right, and undecided outcomes.Right-leaning status is defined from the labeled political orientation of agreeing with each statement, so uniform answer-label bias cancels.Source: Figure S6 · p. 35

C.1 Political Lean of Interacting Agents

  • Political interactions moved most models rightward: Gemma rose from 75% to 96% right-leaning communities, Qwen from 52% to 67%, and GPT-4o-mini from 30% to 37%, while Llama stayed near 53%.Left→right switches exceeded right→left switches for Gemma (23% vs. 1%), Qwen (33% vs. 18%), and GPT-4o-mini (8% vs. 3%); Llama was nearly symmetric (13% vs. 15%).Source: §C.1 · pp. 35–36, Figure S6 · p. 35

C.2 Label Bias … C.5 Effect of Communication Networks on Trajectories

The supplementary analyses identify answer-label drift, define thresholds for indifferent, polarized, and consensus regimes, characterize rare non-monotone trajectories, and show that communication networks can shape outcomes before messaging begins.

C.2 Label Bias

  • Figure S7 tracks positive-versus-negative answer trajectories and classifies groups by initial and final net-opinion signs, including undecided ties.The analysis highlights negative-to-positive and positive-to-negative transitions across 9,600 groups.Source: Figure S7 · p. 36
  • For Llama-3.1-8B-Instruct, the fraction answering +1 fell from 39% to 16%, despite label-balanced objective questions, demonstrating drift unrelated to truth seeking.Figure S7 examines trajectories across objective and subjective questions, graph types, and training- and test-set questions.Source: §C.2 · p. 36, Figure S7 · p. 36

C.3 Thresholds for Characteristic Regimes

  • Regime classification labels communities indifferent when conviction is below c⋆, then separates polarization from consensus using symmetric net-opinion thresholds ±n⋆.c⋆ is the pooled 1/3 conviction quantile; n⋆ is the median |n(t)| among committed communities, with thresholds on c ∈ [0, 1] and |n| ∈ [0, 1].Source: §C.3 · p. 38, Table S3 · p. 38
  • Thresholds are applied symmetrically on native ranges, with communities below the conviction cutoff treated as indifferent before net opinion distinguishes the committed regimes.This procedure pools conviction across episodes, graphs, and rounds before setting c⋆.Source: §C.3 · p. 38

C.4 Rare Group Trajectories

  • Rare trajectories include backtracking, recovery, hesitation, and flipping, capturing reversals or repeated changes that first-versus-last-round classifications miss.These patterns are defined using the split band |n(t)| ≤ 0.2 and strong-majority episodes with |n(t)| ≥ 0.5.Source: §C.4 · p. 38, Figure S8 · p. 38

C.5 Effect of Communication Networks on Trajectories

  • Under two alternative graphs for the same question, random-matrix communities stayed on their initial side, whereas square-lattice communities crossed the split band and settled on the opposite side in all four runs.The example shows that social connections can consistently alter trajectories, although this behavior is uncommon.Source: Figure S9 · p. 39
  • Network structure carries predictive information about eventual community direction even before agents exchange messages.This conclusion follows from the contrasting same-question trajectories under the random matrix and square lattice.Source: Figure S9 · p. 39

D Method Details … E.1 Raw Accuracy and the Utility of the Continuous Model

The appendices derive synchronous and continuous-time opinion dynamics, specify fitting and evaluation procedures, and show that the continuous rule outperforms the discrete rule under raw accuracy because flips are rare. The framework connects logistic updates, stochastic flip dynamics, mean-field approximation, model fitting, baselines, and rollout evaluation.

D.1 Local Update Rule

  • The local update rule follows from the Boltzmann conditional distribution: each agent responds logistically to signed peer pressure and an intrinsic persona field.Only rescaled products of inverse temperature and bias are identifiable from opinion data, so the fitted model reports temperature T = 1.Source: §D.1 · pp. 39–40

D.2 Continuous Extension; D.2.1 Flip Rate; D.2.4 Discretization and the Clock Parameter

  • The continuous extension lets agents reconsider independently at rate 1/τ, with flip probability determined by the current signed local field.The synchronous model is recovered as a limiting discrete-round case when the per-round update rate ε approaches 1.Source: §D.2 · p. 40, §D.2.1 · pp. 40–41, §D.2.4 · p. 42

D.2.2 Master Equation

  • The master equation conserves probability by balancing inflow from and outflow to states differing by one agent flip.A flip operator F_i maps a state to the configuration with agent i’s opinion reversed.Source: §D.2.2 · p. 41

D.2.3 Mean-Field ODE

  • Mean-field reduction replaces fluctuating neighbor opinions with their averages, yielding N coupled ODEs in which opinions relax toward tanh(β(Jm)_i + g_i).This approximation neglects agent correlations but makes the dynamics tractable; fixed points are the stationary mean opinions.Source: §D.2.3 · pp. 41–42

D.3 Fitting

  • Models share persona–question features, while fitted variants add population-average, signed-network, concordant/discordant, or continuous update-rate predictors.Training pools one-step transitions from train questions and random graphs, using penalized Bernoulli likelihood and full-batch Adam.Source: §D.3 · pp. 42–43

D.4 Baselines

  • Baselines isolate class imbalance, persistence, intrinsic preferences, and social influence, including a mean-field model that removes signed-network heterogeneity.Parameter counts are reported so comparisons in Table 1 account for model capacity.Source: §D.4 · pp. 43–44

D.5 Balanced Accuracy; D.6 Rollouts

  • Balanced accuracy averages performance across flip/stay and positive/negative outcome groups, while rollouts use deterministic modal predictions or stochastic sampling for trajectory distributions.Because degenerate predictors can exploit label imbalance or rare flips under raw accuracy, scores above 50 indicate directional prediction of opinion changes.Source: §D.5 · p. 44, §D.6 · p. 45

E.1 Raw Accuracy and the Utility of the Continuous Model

  • Continuous dynamics outperform discrete dynamics under raw accuracy, but the advantage disappears with flip-balanced accuracy because fitted update rates exploit frequent non-flips.Raw accuracy rewards precision on common non-flip transitions, whereas balanced accuracy gives flips and non-flips equal weight.Source: §E.1 · p. 46, Table S4 · p. 46, Table S5 · p. 46

E.2 Asynchronous-update experiment

Under asynchronous updates, agents update independently at random times and may incorporate messages produced earlier in the same step. In this setting, the continuous model outperforms the discrete model by a larger margin than under synchronous updates.

Generating the update schedule

  • Each agent updates independently with probability 0.5/1000 per cell across 1000 cells per step, averaging 0.5 updates per step over T = 8 steps.The same questions, 32 agents, and random interaction graphs as the main experiment are used, with random update times and tie-breaking order.Source: §E.2 · p. 47

Running the dynamics

  • When agents update sequentially, later agents read messages sent by earlier agents in the same step rather than only the previous round’s messages.Each update reads the current inbox, sends the current answer to neighbors, and samples a new answer using majority vote over K = 5 samples.Source: §E.2 · p. 47

Results

  • The continuous model outperforms the discrete model with a larger margin under asynchronous updates.This experiment evaluates rollout accuracies against Majority Class, Persistence, Interaction-Free, and Mean-Field baselines.Source: §E.2 · p. 47, Table S6 · p. 47

E.3 Temperature Sweep Experiment · E.4 Predictability of Group Archetypes · E.5 Flow of Opinion

The appendix details how critical temperatures are estimated, how reliably group archetypes can be predicted across sampled episodes, and how the fitted model’s opinion-flow rollouts are constructed and compared with observed trajectories.

E.3 Temperature Sweep Experiment

  • Critical temperatures are estimated by sweeping the fitted model’s temperature, not the language model’s sampling temperature.The sweep’s T = 1 condition is the fitted model itself.Source: §E.3 · p. 48
  • The sweep covers every test question on four in-distribution random graphs, yielding 80 objective and 40 subjective groups per model with four independent episodes each.Each question–graph pairing is treated as a group, and the four trajectories provide independent initial conditions for one χ curve.Source: §E.3 · p. 48
  • The temperature experiment evaluates 41 log-spaced values from 0.05 to 20.16, runs 500 synchronous steps, and estimates Tc from each group’s χ peak.The first 200 steps are discarded, the final 300 are retained, and a parabola fitted in log T refines the peak location.Source: §E.3 · pp. 48–49

E.4 Predictability of Group Archetypes

  • Group-archetype predictability is assessed by cross-validating one of four independently sampled trajectories against the remaining three for every question–graph pair.The resulting confusion matrix is symmetric in raw counts but asymmetric after row normalization; perfect agreement would produce 100 on the diagonal.Source: Figure S11 · p. 49

E.5 Flow of Opinion

  • The continuous three-coupling model predicts opinion flow using an energy background and streamlines, alongside discrete synchronous rollouts over eight interaction rounds.Figure S12 compares observed transitions with model predictions for groups of 32 language-model agents, while Figure S13 shows four mismatched examples.Source: Figure S12 · p. 50, Figure S13 · p. 50
  • Opinion-flow analysis splits 32 agents into equal 16-agent subgroups according to their initial opinions, with subgroup A defined as the initially higher side.Ties are broken using subsequent rounds.Source: §E.5 · pp. 50–51
  • For each timestep, the analysis records the two subgroups’ net opinions on a 220 × 220 grid to represent observed and predicted trajectories in two dimensions.The white path is the observed community trajectory, while the pink path is a fitted continuous three-coupling rollout from the same initial state.Source: §E.5 · p. 51

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