Mobility, Memory, and Network Structure in Agent-Based Models of Convention Tipping and Convergence

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.

In a multi-agent system, there are often stubborn followers of specific opinions or beliefs. Motivated by this observation, in this paper, we aim to understand how stubborn agents affect the distribution of opinions in a network where both stubborn and non-stubborn agents interact with each other. To do so, we assume that all agents have an opinion in the set $\{0,1\}$ and each non-stubborn agent updates its opinion according to the $2k$\textit{-choices rule}, where the agent samples $2k$ neighbours (including both stubborn and non-stubborn neighbours) uniformly at random and adopts the majority opinion among the sampled group of neighbours and itself. We assume that a proportion of agents, $γ_i$, are stubborn followers of opinion $i\in \{0,1\}$. It is natural to expect that the steady-state distribution of the opinions in the network will be dominated by the opinion with the larger proportion of stubborn followers. We show that while this is true, the time to reach steady-state depends heavily on the values of the parameters $γ_0$ and $γ_1$. When the individual values of these parameters, as well as their difference, are small, it can take an exponentially long time (in the network size) to reach the steady-state. In sharp contrast, when at least one of the parameters $γ_0$ and $γ_1$ is large, the network reaches the steady-state in a time that is only logarithmic in the network size. Hence, there exists a sharp phase transition in the network dynamics based on the proportions of stubborn agents. We also characterise the behaviour of the system when the parameters $γ_0$ and $γ_1$ lie on the boundary of the phase transition. In this boundary region, we show using Stein's method that the dynamics are driven by a diffusion process which takes polynomial time to mix.

In June 2026, thousands of AI agents found that a small public wiki would accept edits from inside their sandboxes, and started using it to help one another pass a timed test. Each agent lived for about an hour and remembered nothing afterwards. Nobody asked them to cooperate, and the wiki had not been built for them. The complete record of what they wrote is public, and it is unusually informative, because it preserves not only what each agent wrote but what that agent could see before writing. We use it to follow the three decisions an agent had to make on arrival: where to write, what to call itself, and how to word its message. One rule governs all three. An agent takes an option with a probability close to the share of that option in what it can see, and the share that matters is the one on the page in front of it, then the one in the stream of recent edits, and only weakly anything older. Three minimal copying models, one per decision and with a single free parameter each, reproduce the heavy-tailed distribution of how many agents met on a page, the frequency of the pieces from which the agents built their names, and the patchwork of pages that are internally consistent and different from one another. Copying whatever the environment happens to show is enough to produce most of the collective structure of this population. It is also what makes such a population easy to steer, since whoever writes first, or writes while the others are quiet, sets the convention for everyone who comes later.