Collective Regimes in Multi-Agent LLMs under Reasoning Effort and Communication Topology

Large Language Models (LLMs) are increasingly deployed in multi-agent environments that produce unanimous agreement, both in intended interactions that need coordination and in unintended emergent interactions. Consequently, understanding how LLMs with competing incentives reach consensus is crucial to better interpret and design LLM collectives. We introduce an evaluation, based on the Network Consensus game, to assess the mechanisms driving this process. Here, a network of agents must agree on one of two colours to receive a payoff, but agents are offered different payoffs for the agreed colour. We vary components such as communication access and network structure and assess six open-source models and one closed model (ChatGPT 5.6-sol). We find that open-source LLMs reach consensus less often than humans in comparable experiments and only if messages are communicated across the global network. ChatGPT 5.6-sol, however, is able to reach consensus with communication limited to its neighbours and with no communication ability. This is achieved through lower stubbornness and higher majority-following tendencies, and by providing more information about its neighbours' states in messages. However, situating agents in a highly clustered graph reduces consensus, regardless of these traits, demonstrating that both system design and individual model traits drive collective outcomes.

The shift from monolithic LLMs to distributed multi-agent architectures demands new frameworks for verifying and securing autonomous coordination. Unlike traditional multi-agent systems focused on cooperative state alignment, modern LLM patterns: multi-agent debate, constitutional oversight, helper-critic loops-rely on adversarial critique for error correction and reasoning refinement. Since LLMs are dynamical systems whose latent states are imperfectly observable from verbalized outputs, securing these networks requires understanding both macroscopic topology and microscopic agent observability. This paper establishes a rigorous graph-theoretic framework for analyzing consensus in signed, directed interaction networks, bridging graph theory and LLM reasoning by formally mapping Transformer cross-entropy log-odds to the signed Laplacian. We characterize agreement stability through structural balance theory, showing how unbalanced critique cycles produce logical frustration and persistent reasoning oscillations, and prove that unobservable latent states from hidden system prompts act as topological Trojan horses that destabilize cooperative consensus. To resolve unobservable deadlocks, we restrict interaction topologies to chordal graphs and apply matrix decomposition with Gram-Schmidt orthogonalization, proving that rank-one spectral edge perturbations deterministically break expertise symmetry by shifting eigenvalues into the stable left-half plane. Core contributions include consensus theorems, polynomial-time Perfect Elimination Ordering verification algorithms, and large-scale empirical validation on clustered ensembles of LLaMA-3, Mistral, and Gemma agents.

Practitioners deploying multi-agent large language model (LLM) systems must currently choose between communication topologies such as chain, star, mesh, and richer variants without any pre-inference diagnostic for which topology will amplify drift, converge to consensus, or remain robust under perturbation. Existing evaluation answers these questions only post hoc and only for the task measured. We introduce a structural diagnostic for multi-agent LLM communication graphs based on the successor representation $M = (I - γP)^{-1}$ of the row-stochastic communication operator, and we connect three of its spectral quantities, the spectral radius $ρ(M)$, the spectral gap $Δ(M)$, and the condition number $κ(M)$, to three distinct failure modes. We derive closed-form spectra for the chain, star, and mesh under row-stochastic normalization, and validate the predictions on a 12-step structured state-tracking task with Qwen2.5-7B-Instruct over 100 independent trials. The condition number is a perfect rank-order predictor of empirical perturbation robustness ($r_s = 1.0$); the spectral gap partially predicts consensus dynamics ($r_s = 0.5$); and the spectral radius is perfectly inverted with respect to cumulative error ($r_s = -1.0$). We trace this inversion to a regime in which linear spectra are blind to non-contracting bias drift, and we propose an affine-noise extension of the predictive map that recovers the empirical ordering. We read this as a first step toward representational, drift-aware structural diagnostics for multi-agent LLM systems, sitting alongside classical spectral and consensus theory.