FlowMAS: Learning Multi-Agent Workflow Topology via Information-guided Generative Flow Network

1 Introduction

• Agentic workflow topology describes how agents or operators are connected to coordinate reasoning, communication, and decision making, shaping collaboration quality and execution efficiency.Source: §1 · p. 1
• Existing approaches include manually crafted, search-based, textual-gradient-based, and generation-based methods, each with limitations in effort, cost, feedback granularity, or workflow suitability.Search- and generation-based methods may also yield insufficient workflow diversity.Source: §1 · pp. 1–2
• Directly applying GFlowNets leaves information exchange among operators unmodeled and may inefficiently explore diverse workflow topologies.The paper identifies information exchange as key to task success and insufficient structural exploration as a potential source of lower task performance.Source: §1 · p. 2
• FlowMAS combines a GFlowNet backbone, curiosity-driven structural exploration, and information-guided evaluation of operator contributions and communication efficiency.The backbone models generation as reward-guided flow over the topology space; the other modules favor structurally novel and more informative collaboration patterns.Source: §1 · p. 2
• Across multiple benchmarks and LLM backbones, FlowMAS achieves the best performance among 15 baselines.The paper attributes this result to its ability to generate diverse, query-specific workflows.Source: §1 · p. 2

2.1 Generative Flow Networks

• GFlowNets model generation over a directed acyclic graph, learning a forward policy whose terminal-state probabilities are proportional to reward.The passage defines states, valid transitions, and a backward policy for the reverse process.Source: §2.1 · pp. 2–3
• GFlowNets can generate discrete structures from reward functions while representing construction decisions, dependencies, and logical constraints.The paper describes these properties as making GFlowNets suitable for automatic exploration and generation of multi-agent workflows.Source: §2.1 · p. 3

2.2 Partial Information Decomposition

• Partial Information Decomposition separates sources’ target-relevant information into unique, redundant, and synergistic contributions.For multiple sources, the framework also considers higher-order redundancy and synergy among source subsets.Source: §2.2 · p. 3

3.1 Problem Formulation

• FlowMAS searches query-specific DAG workflows for high-quality solutions using an evaluation function that measures effectiveness and efficiency.Each node is an operator, and edges represent information flow; chains, fan-in, and fan-out form basic topology primitives.Source: §3.1 · p. 4

3.2 FlowMAS

• Intermediate rewards guide topology construction by assessing structural and information quality before terminal evaluation, directing more flow toward promising partial workflows.This addresses the limited guidance terminal rewards provide about whether an intermediate structure is worth expanding.Source: §3.2 · p. 4

3.3 Information-guided Topology Optimization

• The information-guided module rewards local collaborations whose combined messages provide target-relevant synergy beyond unique contributions and shared redundancy.It also uses an effective information channel reward to favor more evenly distributed communication and discourage redundant agents or edges.Source: §3.3 · p. 5
• The curiosity-driven module rewards structurally novel states to encourage exploration of under-visited topologies during training.Its structural reward is based on the difference between a fixed target network and a trainable prediction network.Source: §3.3 · pp. 5–6
• FlowMAS combines curiosity-driven exploration and information-guided optimization into a unified transition reward for evaluating intermediate topologies.The reward balances the structure-aware exploration and information-guided optimization components.Source: §3.3 · p. 6

4.1 Experimental Setup

• Experiments cover six public benchmarks, three LLM backbones, and five independent trials per experiment type, with mean and standard deviation reported.The benchmarks span mathematical reasoning, code generation, and reading comprehension; token usage is tracked alongside task-specific accuracy metrics.Source: §4.1 · p. 6

4.2 Performance Analysis

• FlowMAS surpasses hand-crafted multi-agent methods by 8.17 points and automated workflow-design methods by 6.12 points on average, and maintains advantages over MaAS and AFlow across all three backbones.On Llama-3.1-8B-Instruct, it exceeds AgentNet and SELFORG by 11.07 points on average.Source: §4.2 · p. 6, Table 1 · p. 7, Table 2 · p. 7

4.3 Cost Analysis

• On HumanEval, AFlow uses more than 15× FlowMAS’s inference tokens without matching its accuracy; MaAS uses fewer training tokens but scores 6.9 accuracy points lower.FlowMAS is also described as achieving the highest accuracy among optimization-oriented workflow methods with relatively low training-token consumption.Source: §4.3 · p. 7, Table 3 · p. 8

4.4 Ablation Study

• GFlowNet achieves the best result on all three benchmarks among five generator-backbone variants tested under aligned settings.The comparison removes CDM and IOM and includes Graph Diffusion, VAE, Transformer, and MLP controllers.Source: Table 4 · p. 8, §4.4 · p. 8
• Removing either CDM or IOM degrades performance, and removing both lowers HumanEval from 69.6 to 62.8; alternative CDM and IOM designs also underperform the full model.The full model discovers more executable topologies than component-ablated and alternative-reward variants.Source: §4.4 · pp. 8–9, Table 5 · p. 9

Static and search-based methods

• Static workflows rely on manual design, while search-based methods incur evaluation costs and depend on their workflow representation and search strategy.Examples include AgentVerse, LLM-Debate, and MetaGPT for static methods, and AFlow, ADAS, and EvoAgent for search.Source: §5 · p. 9

Pruning and textual-gradient methods

• Topology pruning starts from a fixed super-graph, while textual-gradient methods use coarse feedback that makes structure-level credit assignment difficult.The paper notes that pruning performance is bounded by the initial super-graph's quality.Source: §5 · p. 9

Generation-based methods

• VAE-, MLP-, and diffusion-based generators are not designed specifically for discrete workflows with complex dependencies and may lack autonomous exploration.The paper contrasts these methods with FlowMAS's fine-grained training-time generation and active exploration over the combinatorial workflow space.Source: §5 · p. 9

Workflow representations

• FlowMAS focuses on DAG-structured workflow topologies, while other representations include Python code, domain-specific languages, and symbolic primitives.The cited examples are MAS-GPT, AutoFlow, and AutoRAS, respectively.Source: §5 · p. 9

6 Conclusion

• FlowMAS generates more diverse, task-adaptive workflows and consistently outperforms strong baselines across different LLM backbones.The conclusion attributes this to components addressing structural exploration and effective collaboration patterns.Source: §6 · pp. 9–10

A FlowMAS Algorithm

• The algorithm samples graph edits to build workflow trajectories, computes intermediate exploration, synergistic-information, and effective-channel rewards, then verifies completed topologies for terminal reward.The algorithm also checks for fan-out or fan-in structures when computing the synergistic-information reward.Source: A FlowMAS Algorithm · p. 13
• After each batch, FlowMAS optimizes the forward and backward policies and state-flow parameters with the trajectory objective, then updates the curiosity-driven predictor from visited states.The algorithm returns the trained graph-edit GFlowNet.Source: A FlowMAS Algorithm · p. 13

B Derivation of Trajectory Balance Constraint

• Intermediate transition rewards augment flow consistency, whose standard form aligns trajectory probability from the initial to terminal state with terminal reward.The derivation defines forward and backward policies and expands the constraint across trajectory transitions.Source: B Derivation of Trajectory Balance… · pp. 13–14
• The resulting trajectory-balance formulation uses edge-based intermediate reward augmentation, with terminal flow set to the task reward.The text refers to [50] for details of state-based intermediate reward augmentation.Source: B Derivation of Trajectory Balance… · p. 14

C Detailed Calculation of the PID-inspired Reward

• The PID-inspired reward is computed per training instance and local DAG primitive, comparing incoming-message representations with the target to characterize shared, unique, and synergistic information.It uses target alignment of individual and aggregated incoming messages; the target representation and reward are used only during training.Source: C Detailed Calculation of the… · pp. 14–15
• The reward calculation also defines each message’s unique contribution and combines it into a PID-inspired synergistic-information reward.Neither the ground-truth answer nor the IOM reward is required during inference.Source: C Detailed Calculation of the… · p. 15

D Derivation of Spectral Entropy

• The spectral-entropy derivation embeds normalized outputs from upstream operators, computes their cosine-similarity Gram matrix, and trace-normalizes it.The eigenvalues of the normalized matrix define its spectral entropy.Source: D Derivation of Spectral Entropy · p. 15
• The effective-channel reward is computed from the spectral entropy for a local DAG primitive.The displayed reward expression is given in the derivation.Source: D Derivation of Spectral Entropy · p. 15

E Case Study

• On DROP, FlowMAS produces a more task-adaptive workflow that better separates evidence extraction, reasoning, and answer aggregation than AFlow and MaAS.Figure 3 presents the three methods’ workflow topologies.Source: E Case Study · p. 15, Figure 3 · p. 16

F Baseline Methods and FlowMAS Variants

• The comparison spans single-agent test-time scaling, hand-crafted multi-agent systems, and automated multi-agent systems, including ADAS, AgentNet, SELFORG, AFlow, and MaAS.The listed baselines also include Vanilla LLM, CoT, CoT-SC, ReAct, Reflexion, LLM-Majority, LLM-Debate, LLM-Blender, DyLAN, and AgentVerse.Source: F Baseline Methods and FlowMAS Variants · pp. 16–17
• FlowMAS variants remove IOM or CDM, replace information rewards with alternatives, or substitute graph-edit-distance and intrinsic-curiosity exploration rewards.The listed information-reward alternatives include O-Information, embedding similarity, and an LLM-as-Judge reward.Source: F Baseline Methods and FlowMAS Variants · p. 17
• Generation-backbone comparisons use graph diffusion, VAE, MLP, and Transformer models with method-specific optimizations removed.The descriptions follow GTD, G-Designer, and MaAS for their respective backbones.Source: F Baseline Methods and FlowMAS Variants · p. 18

G.1 Benchmark Statistics

• Each benchmark is split into training/validation and test portions at a 1:4 ratio; MATH uses 605 difficulty-level-5 problems, while HotpotQA and DROP each use 1,000 sampled examples.Table 6 is identified as summarizing the resulting dataset statistics.Source: §G.1 · p. 18, Table 6 · p. 18

• Detailed results for GPT-4o-mini and Gemma-4-26B-A4B-it are reported in Tables 7 and 8, respectively.Source: §G.2.1 · p. 18

G.2.2 Scalability with Larger Operator Spaces

• The larger operator-space experiments add less-relevant candidates, including RhetoricalCritic and CreativeStoryteller, to increase search-space size and introduce distracting choices.Source: §G.2.2 · p. 18
• 65.6 accuracy with 20 candidate operators compares with 58.8 for AFlow and 59.5 for MaAS.The scalability comparison covers operator spaces of 10, 15, and 20 candidates; all methods lose performance as the space grows, while optimization cost generally increases.Source: §G.2.2 · pp. 18–19, Table 9 · p. 20

G.2.3 Generalization on BIG-Bench Hard

• FlowMAS outperforms AFlow and MaAS on BBH under both Llama-3.1-8B-Instruct and GPT-4o-mini, including evaluation across seen and unseen task types.The evaluation uses 5,000 test examples; the unseen types include Multistep Arithmetic, Temporal Sequences, and Logical Deduction.Source: §G.2.3 · pp. 19–20, Table 10 · p. 20

G.2.4 Hyperparameter Sensitivity

• FlowMAS remains relatively stable across the evaluated reward-coefficient ranges, with α = 0.2 and β = 0.8 used as the default configuration.The sensitivity experiments vary α while fixing β = 0.8, and vary β while fixing α = 0.2, on HumanEval with Llama-3.1-8B-Instruct.Source: §G.2.4 · p. 20, Table 11 · p. 20

G.2.5 Training and Inference Wall-clock Time

• FlowMAS requires substantially less training time than AFlow and ADAS and has the shortest inference time among compared methods.MaAS requires slightly less training time, but FlowMAS provides a better trade-off between optimization cost and task performance; measurements use Llama-3.1-8B-Instruct on one NVIDIA GeForce RTX 5090.Source: §G.2.5 · pp. 20–21, Table 12 · p. 21

G.2.6 Detailed Computational Overhead

• IOM and CDM require 1.81 and 0.74 minutes, respectively, of the 39-minute total training time, together accounting for approximately 6.5%.The IOM timing includes encoding operator outputs and computing the information-guided reward; embedding time tracks embedding-model invocations during training.Source: §G.2.6 · p. 21, Table 13 · p. 21

OPERATOR REGISTRY

• The operator registry includes operators for decomposing tasks, aggregating inputs, writing and testing code, iterative refinement, and early stopping.Source: OPERATOR REGISTRY · p. 22

OPERATORS

• The base operator binds an LLM and uses structured ActionNode filling, falling back to a direct text query when the structured result is empty.The fallback text is inserted into a result field when available, including reflection_and_solution or response.Source: OPERATORS · p. 22
• Generation operators create or revise code, while MultiGenerateCoT deduplicates variant-based candidates and falls back to a single generation if none are produced.VoteEnsemble combines parent solutions for a generated implementation.Source: OPERATORS · p. 23
• The Decomposer parses the raw response into subtasks, strips list prefixes, deduplicates tasks using lowercase keys, and returns at most four tasks.Source: OPERATORS · p. 24
• Test runs HumanEval public tests in a separate process with a timeout, then can request reflection and revised code based on execution and test feedback.The operator reports success when the resulting execution has no error.Source: OPERATORS · pp. 24–25

1. Claims

• The checklist answers [Yes] that the abstract and introduction accurately reflect the paper’s claims and scope.Source: §1 · p. 26

2. Limitations

• The checklist answers [Yes] that the paper discusses its limitations, identifying the experimental section and supplementary appendix as locations for discussion.Source: §2 · p. 26

3. Theory assumptions and proofs

• The checklist marks theoretical assumptions and proofs [N/A], stating that the paper is not a theory paper.Source: §3 · p. 27

4. Experimental result reproducibility

• The authors say the source code is included in the appendix and experimental settings are described in the experimental section and appendix, supporting reproduction of the main results.Source: §4 · p. 27

5. Open access to data and code

• The source code is included in the supplementary materials, and the authors say all data and code will be open-access upon publication.Source: §5 · p. 28

6. Experimental setting/details

• The authors report that training and test details needed to understand the results are provided in the experimental section and appendix.Source: §6 · p. 29

7. Experiment statistical significance

• The authors report variance alongside mean values for the experiments.Source: §7 · p. 29

8. Experiments compute resources

• The experiments were conducted on computer clusters with multiple Nvidia H100 (80G) GPUs.Source: §8 · p. 29

9. Code of ethics

• The authors say they follow the NeurIPS Code of Ethics and use publicly available datasets, with no human subjects or private datasets involved.Source: §9 · p. 30
• The authors state that LLMs polish the paper and serve as necessary base models for the agentic workflows they study.Source: §9 · p. 32