# Phase Transition for Budgeted Multi-Agent Synergy

arXiv: https://arxiv.org/abs/2601.17311 · HTML: https://mlsanity.com/paper/2601.17311/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Placed in General LLM agent frameworks and benchmarks, one of 16 directions in LLM agents and tool use.
- Builds on 9 of its 29 references that are in the map.
- First posted on arXiv 24 Jan 2026.

## Placed in

- LLM agents and tool use / [General LLM agent frameworks and benchmarks](https://mlsanity.com/agents/general-llm-agent-frameworks-and-benchmarks/)

## Builds on (most cited first, 9 of 9)

- [Scaling Laws for Neural Language Models](https://mlsanity.com/paper/2001.08361/) (2020)
- [AutoGen: Enabling Next-Gen LLM Applications via Multi-Agent Conversation](https://mlsanity.com/paper/2308.08155/) (2023)
- [Scaling LLM Test-Time Compute Optimally can be More Effective than Scaling Model Parameters](https://mlsanity.com/paper/2408.03314/) (2024)
- [AI safety via debate](https://mlsanity.com/paper/1805.00899/) (2018)
- [Supervising strong learners by amplifying weak experts](https://mlsanity.com/paper/1810.08575/) (2018)
- [Towards a Science of Scaling Agent Systems](https://mlsanity.com/paper/2512.08296/) (2025)
- [G-Designer: Architecting Multi-agent Communication Topologies via Graph Neural Networks](https://mlsanity.com/paper/2410.11782/) (2024)
- [When Single-Agent with Skills Replace Multi-Agent Systems and When They Fail](https://mlsanity.com/paper/2601.04748/) (2026)
- [An Information Theoretic Perspective on Agentic System Design](https://mlsanity.com/paper/2512.21720/) (2025)

## Built on by (most cited first, 4 of 4)

- [Spend Less, Reason Better: Budget-Aware Value Tree Search for LLM Agents](https://mlsanity.com/paper/2603.12634/) (2026)
- [Partially Correlated Verifier Cascades in LLM Harnesses: Concave Log-Odds, Polynomial Reliability, and Blind-Spot Ceilings](https://mlsanity.com/paper/2607.13918/) (2026)
- [How Much Does Correctness Cost? Budgeted Placement of Strong Correctors in a Weak Multi-Agent Swarm](https://mlsanity.com/paper/2607.09765/) (2026)
- [Attention Tax, Handoff Tax: A Stylised Model of When Multi-Agent LLM Systems Help](https://mlsanity.com/paper/2610.06069/) (2026)

## Abstract (arXiv)

Multi-agent systems can improve reliability, yet under a fixed inference budget they often help, saturate, or even collapse. We develop a minimal and calibratable theory that predicts these regimes from three binding constraints of modern agent stacks: finite context windows, lossy inter-agent communication, and shared failures among similar agents. Each leaf agent is summarized by a compute-performance scaling exponent β; communication is captured by a message-length fidelity curve γ(m); dependence is captured by an effective shared-error correlation ρ; and a context window W imposes hard fan-in limits that make hierarchy necessary. For binary success/failure tasks with majority aggregation, we prove a sharp phase transition for deep b-ary trees with correlated inputs and lossy communication: a single scalar αρ (combining γ(m), ρ, and fan-in b) determines whether weak signal is amplified to a nontrivial fixed point or washed out to chance. In the amplifying regime, we derive an organization exponent s and show that budgeted synergy, i.e., outperforming the best single agent under the same total budget, occurs exactly when s>β, yielding closed-form compute allocation rules and explicit budget thresholds. We further characterize saturation via a mixing depth and provide a conservative clipped predictor that remains accurate across growth and saturation. A continuous-performance warm-up gives closed-form risks for star, chain, and tree organizations, making correlation- and communication-induced floors explicit and exposing the core design trade-offs in a smooth setting. Finally, we validate the predicted phase boundaries in controlled synthetic simulations and show how the same mechanisms explain the dominant bottlenecks reported in recent large-scale matched-budget studies of LLM agent-system scaling.