Network Topology Does Not Matter: The Breakdown of Classical Consensus in LLM Populations

Abstract

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.