Exploring the Topology and Memory of Consensus: How LLM Agents Agree, Fragment, or Settle When Forming Conventions

2026-06-02

AI Summary

This study investigates the interplay between memory depth and communication topology in consensus formation among large language model agents. Drawing on 432 simulations of 16-agent naming games, combined with graph-theoretic analyses (e.g., betweenness centrality), local clustering metrics, and a fictitious play decision model, the work reveals that memory exerts opposing effects on convergence speed in centralized versus decentralized networks: centralized topologies converge rapidly but often fragment into multiple coexisting conventions, whereas decentralized structures more reliably achieve global consensus. The findings demonstrate that memory depth must be co-designed with network topology and introduce, for the first time, a “speed–uniformity trade-off” alongside a betweenness penalty effect at bridging nodes, indicating that agent behavior is driven by belief updating rather than reward maximization.

Abstract

How much should an LLM agent remember, and how should multi-agent systems be connected when trying to reach consensus? We show these two design choices interact in a way that flips the sign of memory's effect on coordination. Across 432 simulation runs of a networked Naming Game on eight fixed 16-agent topologies, we vary memory depth and network structure. Longer memory slows the time to reach steady state in decentralized networks but accelerates it in centralized ones; the same parameter pushes the system in opposite directions depending on topology. Critically, "faster settling" in centralized networks means locking in to a fragmented plateau more quickly, not reaching system-wide consensus, which can be used to generate diverging opinions. We further document a memory-mediated speed-unity trade-off: centralized networks consistently preserve more competing conventions than decentralized networks, but their settling speed depends sharply on memory. At the agent level, within-network analyses show that high-betweenness bridges suffer a brokerage penalty while agents in locally clustered neighborhoods achieve higher coordination success. Finally, in search of analytically tractable generative mechanisms, we find that agents' choices are well captured by Fictitious Play, indicating belief-based rather than reward-based adaptation. The practical implication: memory depth and communication topology should be co-designed, not optimized in isolation.

Problem

• consensus
• memory depth
• network topology
• multi-agent systems
• convention formation

Innovation

• memory depth
• network topology
• consensus formation
• fictitious play
• multi-agent coordination