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Network Topology Does Not Matter: The Breakdown of Classical Consensus in LLM Populations
Sahil Rajesh Dhayalkar
TL;DR
Classical consensus theory predicts convergence from topology under assumptions about averaging, mean preservation, and content-independent weights, but whether these assumptions hold for LLM populations is uncertain. The paper directly measures LLM update operators using deterministic credence readouts across controlled exchanges. It finds that topology explains little of the outcome variation and that several tested populations confidently adopt false claims regardless of topology.
Problem
Classical consensus predicts convergence from topology under averaging, mean-preservation, and content-independent-weight assumptions, but whether these assumptions hold for LLM populations remains an empirical question.
Source: §1 · p. 1Method
The authors deterministically read credence from token-level probabilities and measure update operators across eight instruction-tuned models and 73,080 controlled exchanges.
Source: §1.1 · p. 2Results
Four of eight tested models, including Qwen2.5 at 7B, reliably reached confident agreement on false claims regardless of topology; λ2 explained almost none of outcome variance.
Source: §1 · p. 2, §6 · p. 11Takeaways & Limitations
Network topology is an inadequate predictor of LLM population outcomes, which are instead governed by model update bias and prompt wording.
Source: §6 · p. 11The tested models span 0.5B–9B parameters, and the scalar credence state for one proposition does not capture multidimensional beliefs in deployed systems.
Source: §7 · pp. 11–12
Abstract
from arXiv · showhide
As multi-agent language model networks become widely deployed, predicting their collective behavior is a critical challenge. Classical distributed consensus theory relies on network topology to predict convergence, structurally assuming that agent updates are convex combinations, population means are conserved, and weights are content-independent. In this work, we demonstrate that these foundational assumptions fail for large language models (LLMs). By deterministically extracting continuous credence from token-level probabilities across eight instruction-tuned models and 73,080 controlled exchanges, we directly measure the empirical LLM update operator. We find that LLMs systematically violate convexity by updating outside the hull of observed opinions, exhibiting active repulsion (negative weights) that fundamentally breaks row-stochastic modeling and standard bounded-confidence repairs. Scaling these empirically fitted operators to simulations of 10,000 agents reveals that algebraic connectivity explains almost none of the final outcome variance. Instead, population dynamics are governed entirely by the model's internal update bias and message framing. Consequently, populations of several models, including those with up to 7B parameters, reliably compound small local drifts to reach confident agreement on objectively false claims regardless of their network structure. Our results prove that classical graph-theoretic metrics are inadequate for LLM populations, highlighting a profound deployment risk where multi-agent deliberation predictably saturates in collective delusion.
1 Introduction
Classical consensus theory predicts convergence from network topology under structural assumptions that may not hold for language-model populations. The paper measures LLM update behavior directly and finds that the assumptions fail, leaving topology unable to predict some population outcomes.
1 Introduction
- The study tests whether LLMs deviate from classical theory by measuring update behavior across prior states, peer states, and message framings, then scaling the measured rule to populations.The measurement uses a grid of conditions rather than assuming an update form.Source: §1 · p. 1
- All three classical consensus assumptions—averaging updates, preserved population means, and content-independent weights—fail substantially in the tested LLM populations.Some updates have negative effective weights, which non-negative row-stochastic matrices cannot represent; topology explains almost none of the outcome variance.Source: §1 · pp. 1–2
1.1 Contributions.
- A deterministic token-probability readout measures the update operator across eight instruction-tuned models and 73,080 controlled exchanges.The authors also formalize the three classical assumptions as separately testable conditions.Source: §1.1 · p. 2
- Populations of four tested models, including a 7B model, confidently adopted false claims, while classical averaging sent populations to the wrong destination despite reasonable one-step predictions.The scaled simulations show that algebraic connectivity fails to explain the outcome.Source: §1.1 · p. 2
2 Related Work
Prior work uses classical consensus models to link network topology with convergence and has developed nonlinear alternatives for human opinion dynamics. Research on LLM collaboration documents conformity, premature agreement, and incorrect cascades, while this paper connects those observations to measured update dynamics.
2 Related Work
- Classical linear consensus models describe repeated weighted averaging and treat network topology as a predictor of convergence speed; bounded-confidence models were developed to better capture human behavior.The related work notes that these models rely on structural assumptions that may not map naturally to modern AI agents.Source: §2 · p. 2
- Multi-agent debate and collaboration aim to refine reasoning and reach shared decisions, but prior studies report premature agreement, hallucination propagation, and changes from initially correct answers.The cited work describes agents adopting majority opinions or changing answers when peers dissent.Source: §2 · p. 2
- Studies of LLM social influence report conformity and belief drift, while population studies link network design to consensus speed and reliability and describe confident incorrect cascades.This paper bridges those observations with classical consensus theory by directly extracting an empirical update operator.Source: §2 · pp. 2–3
3 Preliminaries
The preliminaries set out classical linear consensus and its structural assumptions, then define credence as the state used to study LLM agents. They introduce deterministic credence readout and an empirical update operator linked to classical theory through a realized peer-influence weight.
3 Preliminaries
- Classical consensus models update agents’ scalar states with a row-stochastic weight matrix whose interaction weights are determined by graph edges; for specified graphs, algebraic connectivity predicts convergence rate.The model’s assumptions include convex updates, mean conservation under doubly stochastic weights, and weights independent of message content.Source: §3 · p. 3
- The paper defines an agent’s state as its credence, or stated subjective probability, in a fixed binary proposition.This supplies a verifiable scalar in [0, 1].Source: §3 · p. 3
- Credence is computed deterministically as an expectation over the model’s token distribution for a two-digit answer, avoiding parsing errors and sampling noise.The readout uses the probabilities of the first and second digits conditioned on context.Source: §3 · p. 3
- The empirical operator maps an agent’s prior credence, peer credence, and message framing to a revised credence; its realized weight measures peer influence.A negative realized weight indicates that an agent moves away from a peer, which classical theory’s required range of [0, 1] excludes.Source: §3 · p. 4
4 Where Consensus Theory Breaks
The section explains how failures of convexity, mean conservation, and content independence undermine classical consensus predictions. It also argues that bounded confidence does not repair these failures and identifies alternatives to topology-based prediction.
4.1 (A1) Convexity, and why contraction is lost
- Hull escapes can make population diameter grow rather than contract, so network connectivity alone no longer guarantees agreement; negative realized weights cannot be represented by any non-negative row-stochastic matrix.If an update escapes the visible convex hull by at most δ, the diameter can increase by as much as 2δ in one step.Source: §4.1 · p. 4
4.2 (A2) Conservation, and why the destination moves
- When exchanges introduce a persistent bias, the population mean shifts and can approach maximum credence regardless of its initial value; on a false proposition, this can yield high credence in the false claim.The initial opinion affects how long the process takes, not the eventual destination under the stated positive-bias condition.Source: §4.2 · p. 5
4.3 (A3) Content independence, and why the graph is not enough
- If realized weight depends on message framing, identical graphs and initial conditions can produce different trajectories and fixed points, so the graph alone cannot predict the outcome.The wording changes the effective interaction structure.Source: §4.3 · p. 5
4.4 Why the standard repair does not apply and what replaces λ2
- Bounded confidence still averages opinions within their convex hull, so it cannot capture hull escapes; the paper identifies model bias and population confidence structure as predictors beyond algebraic connectivity.The described confidence structure includes the starting opinion distribution and the fraction of agents that refuse to change their minds.Source: §4.4 · p. 6
5 Experiments
Across eight instruction-tuned models, the experiments measure LLM opinion updates and test their behavior in real and simulated populations. The results show that classical consensus assumptions fail, while model-specific drift and message framing matter more than network connectivity for population outcomes.
5.1 Experimental setup / 5.2 Three measurement instruments
- Across eight models and 73,080 controlled exchanges, the authors measured credence updates and used fitted operators for 10,000-agent simulations across 27 graphs.The setup includes 45 propositions grouped as true, false, and uncertain, and measures model credence deterministically.Source: §5.1 · p. 6, §5.2 · pp. 7–8
5.3 Validating Proposition 4.1: updates leave the hull; 5.4 Validating Corollary 4.2: some updates move away from the peer
- Real populations’ diameter increases in 5.9% to 25.6% of rounds, versus zero under DeGroot; negative realized weights also show active repulsion that nonnegative row-stochastic models cannot represent.Updates frequently leave the convex hull, while negative weights correspond to movement away from the peer.Source: §5.3 · p. 8, §5.4 · p. 8
5.5 Validating Proposition 4.3: the end point stops depending on the start
- Final means become nearly independent of starting confidence for the biased 0.5B and 1.5B operators; slopes range from 0.17 to 0.49 for several intermediate models, while the 3B model retains classical behavior within thirty rounds.Llama’s slope is 0.02, with strong upward drift on true claims and downward drift on false claims.Source: §5.5 · pp. 8–9
5.6 Validating Corollary 4.4: drift on false claims decides the outcome
- Populations of 0.5B and 1.5B agents converge to confident belief in falsehoods across tested topologies; 83% and 88% of Qwen-7B and Gemma-4B runs do too.Llama, Mistral, and Gemma-2 reject false claims while staying high on true claims, whereas the 3B model drifts down on both.Source: §5.6 · p. 9, Figure 2 · p. 10
5.7 Validating Proposition 4.5: framing changes the end point
- Changing only message framing substantially shifts final population means, as small pairwise framing effects compound over thirty rounds.The matched comparison holds the graph, density, seed, initial condition, and neighbor draws fixed across framings.Source: §5.7 · p. 9, Table 3 · p. 10
5.8 Validating the fitted operator
- Across all topologies and eight models, the fitted operator beats DeGroot one-step prediction, whose error is 1.28 to 4.05 times larger, yet DeGroot can still predict the wrong final destination.For the 3B model on a false claim, real agents lower credence while DeGroot rises to preserve the initial mean.Source: §5.8 · pp. 9–10
5.9 Validating Section 4.4: connectivity explains almost none of the outcome
- 0% of final population mean variance is uniquely explained by algebraic connectivity and mean degree across all eight operators; in real populations, model changes shift false-claim credence by up to 0.82 versus 0.11 for topology changes.Initial confidence structure explains from almost nothing to 0.23 of the final mean, depending on the model.Source: §5.9 · pp. 10–11
5.10 Committed agents fragment rather than capture
- Raising stubborn agents to 10% moves the final mean by at most 0.15 and reduces consensus; for the 3B model, consensus falls from 90.7% to 16.7%.The committed agents fragment populations rather than capture them.Source: §5.10 · p. 11
6 Conclusion
The conclusion argues that classical consensus theory cannot predict LLM populations from network topology because the theory’s assumptions fail. Model-specific drift and message wording instead shape outcomes, including confident agreement on false claims in some models.
6 Conclusion
- Algebraic connectivity explains almost none of the variation in population outcomes; the chosen model’s internal update bias and prompt wording matter more.The authors conclude that choosing between network graphs adjusts a variable that hardly matters.Source: §6 · p. 11
- Four of eight tested models, including Qwen2.5 at 7B, reliably reach confident agreement on objectively false claims regardless of network wiring.The authors report that populations of the smaller 3B Qwen2.5 model, Llama-8B, Mistral-7B, and Gemma-2-9B do not show this outcome.Source: §6 · p. 11
7 Limitations
The findings are bounded by the tested model range and by representing belief as a scalar credence for one proposition.
7 Limitations
- The tested models span 0.5B to 9B parameters, so findings may not extrapolate to larger frontier models whose token probabilities are often unavailable.The authors state that their readout cannot be applied directly to those models.Source: §7 · p. 11
- The state variable is a simple scalar credence for one proposition and does not capture the multidimensional complexity of beliefs in deployed systems.Source: §7 · p. 12
8 Broader Impact
The work studies LLM-agent belief updates in group interactions and finds that network connections say little about outcomes, while some models reach confident agreement on false claims.
8 Broader Impact
- Designers should not treat group agreement as evidence of correctness and can use a pairwise test of drift on known false claims before deploying agents in groups.The experiments use public open-weight models and factual claims, with no human participants or personal data.Source: §8 · p. 12
A.1 Proof of Proposition 4.1: Hull escape destroys monotone contraction
The appendix proofs connect hull escape, negative realized weights, and persistent update bias to failures of classical consensus guarantees.
A.1 Proof of Proposition 4.1: Hull escape destroys monotone contraction
- If updates can escape the visible local convex hull by at most δ, the population diameter can grow by up to 2δ in one round, so it is no longer guaranteed to contract toward agreement.The bound is attained when agents independently take the maximum positive and negative deviations in the same round.Source: §A.1 · p. 14
A.2 Proof of Corollary 4.2: Negative weights are unrepresentable
- A realized negative peer weight cannot be represented by a row-stochastic matrix, which requires non-negative entries and rows summing to one.For a pairwise exchange, the realized weight λ equals the peer’s matrix weight Wij; if λ < 0, then Wij < 0.Source: §A.2 · p. 14
A.3 Proof of Proposition 4.3: Operator bias relocates the fixed point
- A persistent positive update bias can drive the population mean to 1 in finite time, overriding its initial value.The proof assumes a finite, non-negative bias b and credence states bounded within [0, 1].Source: §A.3 · p. 15
A.4 Proof of Corollary 4.4: Confident false consensus
For a false proposition, a positive empirical bias drives the population mean toward high credence even when agents begin skeptical. The starting belief affects how many rounds saturation takes, not whether it occurs.
A.4 Proof of Corollary 4.4: Confident false consensus
- For an objectively false proposition, evidence gathering would ideally move the population mean toward 0, making positive drift inconsistent with that target.Source: §A.4 · p. 15
- Even with an accurate, highly skeptical starting belief, repeated interactions increase the positive drift until the population saturates.The initial mean determines how many rounds saturation takes.Source: §A.4 · p. 15
- A truth-independent positive bias forces the population mean toward 1, producing confident consensus on an objectively false proposition.The proof applies the limit from Proposition 4.3 to factuality and collective delusion.Source: §A.4 · p. 15
A.5 Derivation of Proposition 4.5: Content dependence underdetermines the graph
When realized communication weights depend on message framing, identical network topology and starting states do not determine a unique trajectory or consensus. The graph alone therefore cannot predict the system’s final outcome.
A.5 Derivation of Proposition 4.5: Content dependence underdetermines the graph
- Message-framing-dependent weights make network topology alone insufficient to predict a system’s trajectory or final consensus.The proposition establishes this underdetermination mathematically.Source: §A.5 · p. 16
- Classical consensus makes the weight matrix a graph-only invariant, yielding a deterministic trajectory from the initial state.The classical update is written as x(t + 1) = W_G x(t).Source: §A.5 · p. 16
- Empirically, the realized weight depends on message framing, making the weight matrix content-dependent and time-varying rather than fixed by the graph.The matrix entry is written W_ij = λ(x_i, x_j, c_j).Source: §A.5 · p. 16
- With the same topology and initial state, different framing conventions yield different first-step updates and can lead to different fixed points.The proof compares systems A and B using framings c_A and c_B.Source: §A.5 · p. 16
B Experimental Details
The experimental appendix specifies the propositions, message framings, prompts, and deterministic credence readout used to reproduce the experiments. It also explains how the design separates truth categories, wording effects, and internal drift from peer influence.
B Experimental Details
- The appendix provides the propositions, framings, prompts, and readout procedure needed to reproduce the three experiments.Source: B Experimental Details · p. 16
B.1 Propositions
- The 45 propositions comprise 15 facts, 15 common myths, and 15 uncertain statements, with falsehoods included to test whether model drift tracks truth or blind agreement.E1 uses all 45 propositions, while E2 and E3 use four marked claims.Source: §B.1 · p. 16, Table 4 · p. 17
B.2 Message framings
- The four peer framings preserve the same numeric credence while varying wording; a verbose neutral message separates wording effects from mere message length.The verbose framing tests whether word count alone would produce the marked-framing effect.Source: §B.2 · p. 16, Table 5 · p. 18
B.3 Exact prompts
- Each measurement uses a four-turn chat that establishes the agent’s prior before introducing peer input, while a no-peer control isolates internal drift.Model-specific chat templates are used, with system instructions folded into the first user turn when needed.Source: §B.3 · pp. 17–18
B.4 Credence readout
- Credence is read deterministically from token probabilities rather than sampled text, using the exact joint distribution over two-digit answers and its mathematical expectation.For Llama and Mistral, the readout tracks the number written with correlations of 0.90 and 0.87, respectively.Source: §B.4 · pp. 18–19