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

This paper demonstrates that in multi-agent LLM systems, the impact of memory depth on consensus speed and fragmentation is fundamentally inverted by network topology, revealing a critical need to co-design memory and communication structures rather than optimizing them in isolation.

Original authors: Aliakbar Mehdizadeh, Martin Hilbert

Published 2026-06-04

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CC BY 4.0

Original authors: Aliakbar Mehdizadeh, Martin Hilbert

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a room full of 16 robots (AI agents) trying to agree on a secret handshake. They can't talk to everyone at once; they can only whisper to their immediate neighbors. To succeed, they need to pick the same letter from a list (like "A," "B," or "C") every time they interact. If they match, they get a high-five (points); if they mismatch, they get a time-out (negative points).

The researchers wanted to know two things: How much of their past should these robots remember? and How should they be connected to each other?

Here is what they found, explained simply:

1. The Two Ways to Connect

The researchers set up the robots in two different "neighborhood" styles:

The Open Town (Decentralized): Everyone is connected to a few different people in a loose web. Information flows freely, and everyone hears a mix of different opinions.
The Gated Community (Centralized): The robots are grouped into tight little clusters. Inside a cluster, they talk to each other constantly, but it's hard for them to hear what the other clusters are saying.

2. The Memory Trap

The robots were given different "memory lengths":

Short Memory: They only remember the last 2 interactions.
Medium Memory: They remember the last 5.
Long Memory: They remember the last 10.

The Big Surprise: Memory doesn't work the same way in both neighborhoods. In fact, it does the exact opposite thing depending on where the robot lives.

In the Open Town: If you give a robot a long memory, it gets stuck. It remembers too many different things from too many different neighbors and can't decide on a single handshake. It takes a long time to settle on an agreement. If you give it a short memory, it forgets the confusion quickly and agrees faster.
In the Gated Community: If you give a robot a short memory, it gets confused because it forgets its neighbors' habits too fast. But if you give it a long memory, it quickly locks in on what its specific cluster is doing and agrees very fast.

3. The "Fast but Broken" Trap

Here is the most critical part: Agreeing fast doesn't mean everyone agrees on the same thing.

In the "Gated Community" with long memory, the robots settled on an agreement incredibly fast. But, they didn't all agree on the same letter.

Cluster A decided on "A."
Cluster B decided on "B."
Cluster C decided on "C."

They reached a "steady state" (stopped changing their minds) very quickly, but the whole system was fragmented. They were efficient at agreeing locally, but the whole group was divided.

In the "Open Town," it took longer to settle, but because they were all mixing together, they were much more likely to eventually agree on a single letter for the whole group.

4. The "Bridge" Penalty

The researchers also looked at individual robots.

The Bridge: Some robots sat in the middle, connecting two different groups. These "bridge" robots had a hard time. They heard conflicting signals from both sides. The more they remembered, the more confused they got, and the worse they performed at agreeing.
The Insiders: Robots surrounded by neighbors who all thought alike (a tight cluster) did very well. Their neighbors kept reinforcing the same idea, and the robot's memory helped it stick to that idea.

5. How They Think

Finally, the researchers asked: What logic are these robots using?
They found the robots weren't just trying to maximize their own points based on what worked before (like a gambler). Instead, they were acting like mind-readers. They looked at what their neighbor just did and guessed, "Oh, they are going to do that again, so I should do it too." They were predicting their partner's move based on recent history, rather than just repeating their own past successes.

The Bottom Line

You cannot design a robot team by just picking a memory size or a network shape in isolation. They are a package deal.

If you want speed and don't mind if different groups end up with different rules, use Centralized networks with Long Memory.
If you want total unity (everyone agreeing on one rule), use Decentralized networks with Short Memory.

The paper concludes that "faster settling" in some designs is actually a trap: it just means the groups locked into their own little bubbles faster, rather than finding a solution that works for everyone.

Technical Summary: Exploring the Topology and Memory of Consensus in LLM Agents

Problem Statement

As Large Language Model (LLM) agents transition from static text generation to interactive, goal-directed multi-agent systems, a critical design challenge emerges: how do these agents reach consensus on social conventions (arbitrary but collectively adopted behaviors) without central authority? While prior work suggests LLM agents can form shared conventions in well-mixed environments, real-world multi-agent systems are constrained by specific communication networks and varying memory depths. The central problem addressed is understanding how the joint design of network topology (decentralized vs. centralized) and memory depth (bounded history length) influences the speed, stability, and outcome of convention formation. Specifically, the study investigates whether these parameters act as independent control variables or interact in complex, potentially counter-intuitive ways to determine whether populations converge on a single global convention or fragment into local norms.

Methodology

The authors employed an agent-based modeling approach simulating a Networked Naming-Game, a canonical coordination game where agents must agree on a shared label (convention) to receive a reward.

Agents: Each agent is an independent instance of an LLM (primarily Gemini 2.0 Flash, with cross-model validation using GPT-4o-mini and Claude Haiku 4.5). Agents are motivated to maximize their personal score through successful coordination.
Game Protocol: Agents interact in pairs on fixed network edges. If both choose the same convention from a set of ten arbitrary letters, they receive +100 points; otherwise, both lose 50 points.
Memory: Agents possess a bounded memory of their last $M$ interactions ( $M \in \{2, 5, 10\}$ ), formatted as a natural-language prompt. This memory window represents short, intermediate, and long-term retention, respectively.
Network Topologies: The study utilized eight fixed 16-node networks with a constant degree ( $k=3$ $k = 3$ ), categorized into two groups based on structural properties:
- Decentralized (A–D): Low clustering, short average path lengths.
- Centralized (E–H): High clustering, longer average path lengths, and more centralized structures.
Experimental Design: A factorial design varying 8 topologies $\times$ 3 memory depths resulted in 24 conditions. Each condition ran 18 independent simulations (totaling 432 runs), with each simulation lasting up to 1,600 rounds (100 population cycles).
Analysis:
- Behavioral Modeling: Agent choices were fitted against two adaptive baselines: Fictitious Play (FP) (belief-based, predicting partner behavior) and Reinforcement Learning (RL) (reward-based, predicting based on own past payoffs).
- Metrics: Key outcomes included dyadic coordination success rates, steady-state fragmentation (number of surviving conventions), and settling time (time to reach a stable state of fragmentation).
- Structural Analysis: Linear mixed-effects models were used to isolate the impact of an agent's structural position (betweenness centrality and local clustering) within a specific network.

Key Contributions and Results

1. Reversal of Memory Effects by Topology

The most significant finding is that the effect of memory depth on the time to reach a steady state is reversed depending on network topology.

Decentralized Networks: Longer memory ( $M=10$ ) slows the time to reach steady state and reduces final coordination success. Agents retain obsolete or locally conflicting information, creating inertia that hinders adaptation to new local signals.
Centralized Networks: Longer memory accelerates the time to reach steady state. In clustered neighborhoods, longer memory allows agents to accumulate consistent local signals, rapidly locking into locally reinforced conventions.
Implication: "Faster settling" in centralized networks does not imply faster global consensus; rather, it means the system locks into a fragmented plateau (multiple co-existing local conventions) more quickly.

2. The Speed–Unity Trade-off

The study identifies a memory-mediated trade-off between local efficiency and global unity:

Centralized + Long Memory: Rapidly stabilizes local clusters but preserves high global fragmentation (more surviving conventions).
Decentralized + Short Memory: Slows local locking but facilitates system-wide consensus by exposing agents to diverse signals and preventing premature local lock-in.
Centralized + Short Memory: Surprisingly, these networks settle the slowest of all conditions because agents lack sufficient history to stabilize on a single local convention amidst conflicting local signals.

3. Behavioral Mechanism: Belief vs. Reward

Analysis of agent decision logic reveals that LLM choices are better captured by Fictitious Play (FP) than Reinforcement Learning (RL).

Agents adapt based on beliefs about their partners' recent behavior rather than solely on their own past rewards.
FP consistently outperforms RL in predictive accuracy (lower negative log-likelihood), particularly when convention diversity is high. This suggests that in coordination settings, shaping what agents know about each other is more critical than shaping their reward structures.

4. Structural Position and Coordination Resilience

An agent's location within the network significantly impacts its success:

Brokerage Penalty: Agents with high betweenness centrality (bridges between subgroups) suffer lower coordination success. They receive conflicting signals from different clusters, and longer memory exacerbates this penalty by preserving incompatible interaction histories.
Clustering Benefit: Agents in densely clustered neighborhoods achieve higher coordination success. Dense local interactions act as "convention incubators," reinforcing stable conventions. This benefit is amplified by longer memory, which accumulates consistent local signals.

5. Robustness

The findings were validated through cross-model agreement checks, showing that decision patterns (94–95% agreement) are not idiosyncratic to the specific LLM used (Gemini 2.0 Flash) but generalize across model families (GPT-4o-mini, Claude Haiku 4.5).

Significance and Claims

The paper claims that memory depth and communication topology should be co-designed, not optimized in isolation. The study challenges the notion that memory acts as a simple "inertia" factor; instead, its impact is context-dependent, flipping from a hindrance in decentralized systems to an accelerator in centralized ones.

Design Guidance: For systems requiring rapid local task execution, centralized communication combined with longer memory may be beneficial, though it risks persistent fragmentation. Conversely, for systems requiring shared standards or global consensus, decentralized mixing with shorter memory horizons is preferable to prevent local lock-in.
Theoretical Contribution: The work treats LLM-agent populations as a system of interest in their own right, demonstrating that their coordination dynamics (e.g., monotonic increase in surviving conventions with memory) differ from human cooperation experiments where intermediate memory often optimizes performance.
Evaluation Warning: The authors caution that high dyadic coordination (local success) is not a proxy for global consensus. Evaluation metrics focusing solely on pairwise interaction quality can mask systemic disagreement at the population level.

The study concludes that the interaction between memory and topology creates a "settling-vs-unity" trade-off where the same parameter change can push systems in opposite directions depending on the structural environment.