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HELENA:Hierarchical Sparse Coordination over a Union of Complementary Topologies for MAS
Zhifang Mao, Linyao Zheng, Xuhang Shi, Xiuquan Hou
Intelligence
Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 90%
Last extracted: 8/9/2026, 2:07:42 AM
Summary
The paper introduces HELENA, a multi-agent system framework that constructs a union graph from complementary topologies selected via Monte Carlo Tree Search and Determinantal Point Process. It employs Hierarchical Sparse Coordination to activate sparse subgraphs and exchange compressed latent briefs, suppressing noise propagation. A Local Self-Refinement stage validates solutions using discrepancy evidence and adversarial challengers. HELENA achieves state-of-the-art results across eight benchmarks, including MMLU-Pro and GSM8K.
Entities (13)
Relation Signals (10)
HELENA → contains → Hierarchical Sparse Coordination
confidence 95% · HELENA addresses this challenge... by employing a Hierarchical Sparse Coordination module
HELENA → contains → Local Self-Refinement
confidence 95% · Finally, a Local Self-Refinement stage identifies decision units
HELENA → uses → Determinantal Point Process
confidence 95% · selects a complementary subset with a Determinantal Point Process (DPP)
HELENA → uses → Monte Carlo Tree Search
confidence 95% · HELENA employs Monte Carlo Tree Search to explore the topology space
HELENA → evaluatedon → MMLU-Pro
confidence 90% · Experiments across eight benchmarks... up to 10.34% on MMLU-Pro
HELENA → evaluatedon → GSM8K
confidence 90% · For mathematical reasoning, we use GSM8K
HELENA → usesmodel → Qwen3-Embedding-8B
confidence 90% · topology and memory embeddings use Qwen3-Embedding-8B
HELENA → usesmodel → GPT-4o-mini
confidence 90% · All agents use gpt-4o-mini
HELENA → outperforms →
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Abstract
Abstract:LLM-based multi-agent systems (MAS) typically optimize a single topology, restricting reasoning to a narrow trajectory and limiting comprehensive analytical capacity. Naively merging multiple topologies into a composite graph introduces redundant noise propagation across irrelevant connections, degrading solution quality. To address this dilemma, we propose \textbf{Hierarchical Sparse Coordination over a Union of Complementary Topologies for MAS (HELENA)}, a multi-agent framework that balances diverse reasoning paths with sparse task-dependent execution. \helena{} constructs a union MAS graph from complementary candidate topologies selected via Monte Carlo Tree Search and Determinantal Point Process, broadening the reasoning trajectory for comprehensive analysis of complex problems. A Hierarchical Sparse Coordination module then activates only a sparse subgraph at each step while agents exchange compressed latent briefs to suppress redundant noise propagation. Finally, a Local Self-Refinement stage identifies decision units with discrepancy evidence and rewrites them only when contrastive evidence simultaneously confirms a reliable solution-side failure and a challenger-side improvement. Experiments across eight benchmarks show that \helena{} achieves state-of-the-art results on all benchmarks, with an average gain of \pctup{3.47} over the strongest baseline and up to \pctup{10.34} on MMLU-Pro, achieving larger improvements on harder benchmarks at a reasonable additional cost.
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- Source: https://arxiv.org/abs/2608.04634v1
- Canonical: https://arxiv.org/abs/2608.04634v1
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HELENA: Hierarchical Sparse Coordination over a Union of Complementary Topologies for MAS Zhifang Mao1, 3, Linyao Zheng1,2, *, Xuhang Shi1, 3, Xiuquan Hou2 Abstract LLM-based multi-agent systems (MAS) typically optimize a single topology, restricting reasoning to a narrow trajectory and limiting comprehensive analytical capacity. Naively merging multiple topologies into a composite graph introduces redundant noise propagation across irrelevant connections, degrading solution quality. To address this dilemma, we propose Hierarchical Sparse Coordination over a Union of Complementary Topologies for MAS (HELENA), a multi-agent framework that balances diverse reasoning paths with sparse task-dependent execution. HELENA constructs a union MAS graph from complementary candidate topologies selected via Monte Carlo Tree Search and Determinantal Point Process, broadening the reasoning trajectory for comprehensive analysis of complex problems. A Hierarchical Sparse Coordination module then activates only a sparse subgraph at each step while agents exchange compressed latent briefs to suppress redundant noise propagation. Finally, a Local Self-Refinement stage identifies decision units with discrepancy evidence and rewrites them only when contrastive evidence simultaneously confirms a reliable solution-side failure and a challenger-side improvement. Experiments across eight benchmarks show that HELENA achieves state-of-the-art results on all benchmarks, with an average gain of 3.47% ↑ over the strongest baseline and up to 10.34% ↑ on MMLU-Pro, achieving larger improvements on harder benchmarks at a reasonable additional cost. 1 Introduction LLM-based Multi Agents System(MAS), which integrate language generation with decision-making and action-execution, have shown impressive performance across tasks ranging from reasoning and code generation to video gaming (Park et al. 2026) and autonomous driving (Liu et al. 2025). More importantly, combining multiple agents into a collaborative team consistently outperforms individuals on complex tasks (Wan et al. 2026; Yang et al. 2026). MAS can thus exhibit collective intelligence shaped by collaboration topology and information sharing. Figure 1: Problem Illustration: I) A single topology restricts reasoning to a narrow trajectory. I) Composite graph propagates irrelevant noise across redundant connections. Despite these advances, existing methods for automated multi-agent workflow construction typically optimize a single topology, as in MetaGPT (Hong et al. 2024), GPTSwarm (Zhuge et al. 2024), and G-designer (Zhang et al. 2025c). This paradigm faces a fundamental bottleneck due to its reliance on a single topology, leading to a narrow reasoning trajectory and preventing a comprehensive analysis of complex problems. A heuristic idea is to merge multiple topologies into a composite graph. Rank-and-fuse (Jiang et al. 2023), consensus (Chen et al. 2024a), and mixture-style systems confirm that aggregating multiple candidates improves robustness (Wang et al. 2025). However, a composite topology can propagate irrelevant reasoning paths and feedback signals across redundant connections, which introduces noise into the final solution (Huang et al. 2025). As shown in Figure 1, this raises a critical yet under-explored dilemma: How to maintain composite reasoning perspectives in MAS while mitigating the propagation of irrelevant noise? A key insight is that fusing complementary topologies captures diverse reasoning perspectives, while restricting irrelevant node communication prevents noise propagation. To address this dilemma, we propose HELENA. Targeting the first challenge, ① how to maintain composite reasoning perspectives in MAS, HELENA employs Monte Carlo Tree Search to explore the topology space and selects a complementary subset with a Determinantal Point Process (DPP), merging them into a union graph that structurally preserves diverse reasoning paths. To further tackle ② how to mitigate noise propagation induced by composite-graph communication, HELENA addresses this challenge at two levels: by employing a Hierarchical Sparse Coordination module that activates a sparse subgraph at each step while exchanging compressed latent briefs, and by introducing a Local Self-Refinement stage that selects reliable solutions, identifies high-risk units, and revises them based on clear evidence of flaws and alternative improvements. Our contributions are summarized as follows: ❶ Dilemma Identification. We identify a topological diversity and noise propagation dilemma in LLM-based MAS. A single topology restricts reasoning coverage, while composite topologies lead to the propagation of irrelevant noise. ❷ Practical Solution. We propose HELENA, which builds a union graph from reliable and complementary topologies, then controls noise propagation during inference through Hierarchical Sparse Coordination and Local Self-Refinement. ❸ Experimental Evaluation. Experiments on eight benchmarks show that HELENA achieves state-of-the-art performance, with gains of up to 10.34% ↑ . It also maintains favorable cost efficiency across model backbones. 2 Related Work Pre-defined MAS Protocols. Early LLM-based multi-agent systems rely on pre-defined collaboration or aggregation protocols (Wu et al. 2024), including role-based systems that assign predefined responsibilities (Li et al. 2023; Chen et al. 2024c; Hong et al. 2024), debate-style methods that encourage agents to challenge intermediate reasoning (Du et al. 2024; Liang et al. 2024), and rank-and-fuse approaches such as LLM-Blender (Jiang et al. 2023). While these methods improve over individual LLMs, their manually specified and task-agnostic protocols limit adaptability across domains and resource budgets. Automated Agentic System. To reduce manual engineering, recent work formulates agentic construction as automated optimization. AFlow and A2Flow search over code-represented workflows (Zhang et al. 2025d; Zhao et al. 2026), ADAS and AgentSquare expand the search space to modular designs (Hu et al. 2025; Shang et al. 2025), and MaAS optimizes a supernet for query-dependent architecture sampling (Zhang et al. 2025b). These methods show that topology can be searched rather than manually specified. Multi-agent Systems as Graphs. GPTSwarm represents agents as optimizable graphs (Zhuge et al. 2024), G-Designer learns task-aware topologies via graph neural networks (Zhang et al. 2025c), MacNet studies scalable graph-organized collaboration (Qian et al. 2025), and G-Memory organizes memory traces with graph structures (Zhang et al. 2025a). However, both automated optimization and graph-based methods predominantly instantiate a single topology at a time, leaving open how to preserve complementary topological perspectives without suffering from noise propagation induced by composite-graph communication. 3 Preliminaries We model a multi-agent system as a directed collaboration graph G=(V,E)G=(V,E), where V=v1,…,vNV=\v_1,…,v_N\ is the set of agent nodes, N=|V|N=|V|, and E is the set of directed edges. Each node vi∈Vv_i∈ V corresponds to an LLM-based agent and is formalized as vi=(Basei,Rolei,Memi),v_i= (Base_i,\ Role_i,\ Mem_i ), (1) where BaseiBase_i denotes the underlying large language model instance, RoleiRole_i specifies the agent’s designated role or persona, and MemiMem_i denotes its memory state. Given a user query Q, the system evolves through T communication epochs. At epoch t, each agent receives the query, its own memory state, and the information made available by its in-neighbors: oit o_i^t =vi(Q,Memit,it) =v_i (Q,Mem_i^t,C_i^t ) (2) Here, it=mj→it:vj∈−(vi)C_i^t=\m_j→ i^t:v_j ^-(v_i)\ denotes the information made available from in-neighbor agents to viv_i at epoch t, oito_i^t denotes the output generated by viv_i, which may include reasoning steps, intermediate analyses, or final proposals, and mj→itm_j→ i^t denotes the message transmitted from agent vjv_j to agent viv_i at epoch t. 4 Method Figure 2: The designing workflow of our proposed HELENA. The overall framework of HELENA is illustrated in Figure 2. 4.1 Union MAS Graph Construction HELENAgenerates candidate topologies for the query and then selects a reliable and complementary subset to construct a query-dependent union graph. Candidate Topology Generation. Given a query Q and an initial workflow W0W_0, candidate topologies are generated through legal graph edits. At search state s, an edit u changes an agent role, a prompt setting, or a directed edge while preserving workflow validity. Tree search has recently been used to navigate discrete agent workflow spaces (Zhang et al. 2025d; Li et al. 2026). In HELENA, topology search is instantiated as a query-conditioned variant of policy-value MCTS (Silver et al. 2017; Zhao et al. 2022). The edit prior Pϕ(u∣s,Q)P_φ(u s,Q) guides action selection, while the utility predictor gψg_ψ evaluates terminal topologies. The valid terminal topologies found within search budget B constitute the query-specific candidate pool (Q)C(Q). During inference, the search is reinitialized for each query. Both PϕP_φ and gψg_ψ remain frozen, and candidate workflows are not executed. Agent LLM calls begin only after the selected topologies have been merged into GUG_U. The offline learning procedure is provided in Appendix A.2. Union Graph Construction. Candidate selection is based on query-dependent quality and information-flow similarity. For each candidate topology W, the quality predictor gψg_ψ takes Q and W as input and returns the expected utility μQ,W _Q,W and predictive uncertainty σQ,W _Q,W. The lower quantile is denoted by α<0.5α<0.5, and Φ−1 ^-1 is the inverse cumulative distribution function of the standard normal distribution. The parameter η>0η>0 controls the scale of the quality weight. The query-dependent quality weight is defined as qQ,W=exp(μQ,W+Φ−1(α)σQ,W2η).q_Q,W= ( _Q,W+ ^-1(α) _Q,W2η ). (3) The weight increases with predicted utility and decreases with predictive uncertainty. Its dependence on both Q and W allows the same topology to receive different weights for different queries. Quality alone cannot determine whether two candidates provide distinct coordination patterns. Their complementarity is therefore measured from the information flows supported within T communication turns. After role-compatible nodes are mapped to a common order, AWA_W denotes the normalized adjacency matrix of topology W. Its finite-turn information flow is ΨT(W)=∑t=1TAWt. _T(W)= _t=1^TA_W^t. (4) The matrix power AWtA_W^t records the directed routes that can transmit information in t steps. Their finite-turn coverage is represented by ΨT(W) _T(W). Matrix vectorization is denoted by vec(⋅)vec(·). For two candidate topologies WaW_a and WbW_b, their information-flow similarity is Σab=⟨vec(ΨT(Wa)),vec(ΨT(Wb))⟩‖vec(ΨT(Wa))‖2‖vec(ΨT(Wb))‖2. _ab= ( _T(W_a) ),vec ( _T(W_b) ) \|vec ( _T(W_a) ) \|_2 \|vec ( _T(W_b) ) \|_2. (5) A larger Σab _ab indicates greater overlap between the directed information flows of the two topologies. Query-dependent quality and information-flow similarity define the DPP kernel Lab=qQ,WaΣabqQ,Wb.L_ab=q_Q,W_a _abq_Q,W_b. (6) The selected topology set is defined as S⋆=argmaxS⊆(Q),|S|=Klogdet(LS).S = *arg\,max_S (Q),\,|S|=K (L_S). (7) Here K is the number of retained topologies, and LSL_S is the principal submatrix indexed by S. The log-determinant criterion assigns high values to sets that combine predicted quality with nonredundant information flows. Greedy MAP inference is used to obtain S⋆S . Before merging, nodes with the same role and agent identity are aligned. Different prompt settings are retained as variants of the aligned node. The node and edge sets of topology W are denoted by VWV_W and EWE_W. The selected topologies form the union graph GU=(⋃W∈S⋆VW,⋃W∈S⋆EW).G_U= ( _W∈ S V_W, _W∈ S E_W ). (8) The union retains the directed edges of the selected topologies but does not represent the strength of their support. Predictive confidence is therefore assigned to each union edge u→vu→ v as puv=∑W∈S⋆qQ,W[(u,v)∈EW]∑W∈S⋆qQ,W.p_uv= _W∈ S q_Q,W1 [(u,v)∈ E_W ] _W∈ S q_Q,W. (9) Here [(u,v)∈EW]1[(u,v)∈ E_W] equals one when W contains the edge and zero otherwise. Support from a topology with a larger quality weight contributes more to puvp_uv. Each edge descriptor is written as eu→v=(puv,compatuv,ctxuv).e_u→ v= (p_uv,compat_uv,ctx_uv ). (10) The remaining components represent endpoint compatibility and source topology context. Their construction is provided in Appendix A.2. 4.2 Hierarchical Sparse Coordination Hierarchical Sparse Coordination realizes sparse collaboration at two levels. Node-Level Memory Composer removes irrelevant private records before they enter coordination. Edge-Level Sparse Activation restricts communication and execution to selected paths at each turn. Node-Level Memory Composer. Each node viv_i maintains a private memory store ℳit=miktM_i^t=\m_ik^t\. Let rir_i denote its role instruction. The query builder fqf_q forms the retrieval feature qit=fq(Q,ri)q_i^t=f_q(Q,r_i). The relevance scorer fsf_s assigns each memory record the score zikt=fs(qit,mikt)z_ik^t=f_s(q_i^t,m_ik^t). Both modules are trained from retrospective execution feedback as described in Appendix A.2. Sparsemax (Martins and Astudillo 2016) converts these scores into the retrieval weight vector it=sparsemax(zikt∣mikt∈ℳit). α_i^t=sparsemax (\z_ik^t m_ik^t _i^t\ ). (11) Its component αikt _ik^t is the weight of record miktm_ik^t. Records with positive weight form the selected set ℛitR_i^t. The composer returns the private context ℓit _i^t. The selected records are also represented by a latent brief bitb_i^t and a verbalized brief BitB_i^t. (ℓit,bit,Bit)=Compose((αikt,mikt)∣mikt∈ℛit).( _i^t,b_i^t,B_i^t)=Compose (\( _ik^t,m_ik^t) m_ik^t _i^t\ ). (12) This node-level sparsity removes irrelevant private records before coordination. The selected memory remains private in ℓit _i^t, while bitb_i^t supports graph control and BitB_i^t is transmitted across active edges. Edge-Level Sparse Activation. Let gtg^t denote the global state at turn t. Let Θe _e denote the parameters of the edge scorer. For each edge (vj,vi)∈EU(v_j,v_i)∈ E_U, the scorer combines gtg^t with an edge representation formed from the endpoint briefs and the descriptor defined in Section 4.1. ϕjit=ΨΘeedge(bjt,bit,gt,evj→vi). _ji^t= _ _e^edge (b_j^t,b_i^t,g^t,e_v_j→ v_i ). (13) For each target node viv_i, sparsemax produces the incoming edge weight vector it=sparsemax(ϕjit∣(vj,vi)∈EU). β_i^t=sparsemax (\ _ji^t (v_j,v_i)∈ E_U\ ). (14) Its component βjit _ji^t is the weight assigned to edge (vj,vi)(v_j,v_i). The active edge set is Et=(vj,vi)∈EU∣βjit>0.E^t=\(v_j,v_i)∈ E_U _ji^t>0\. (15) The endpoints of EtE^t form VtV^t, and Gt=(Vt,Et)G^t=(V^t,E^t) is executed at turn t. This edge-level sparsity restricts communication to EtE^t and agent execution to VtV^t, so coordination proceeds only along the selected paths. Let DitD_i^t denote the predecessor context formed from the verbalized briefs received through active incoming edges. Let PitP_i^t denote the role-conditioned prompt of viv_i. Each active Agent produces oit=vi(Pit,Q,ℓit,Dit,gt).o_i^t=v_i (P_i^t,Q, _i^t,D_i^t,g^t ). (16) The active briefs and outputs are pooled into ptp^t. Feedback available during inference forms ℱtF^t and does not contain benchmark labels. Let Θg _g denote the parameters of the global updater. The next state is gt+1=GlobalNodeΘg(gt,pt,ℱt).g^t+1=GlobalNode_ _g (g^t,p^t,F^t ). (17) The updated state conditions edge selection at the next turn, allowing the active graph to change with the current execution state. The edge scorer and global updater are trained offline and remain frozen during evaluation. When oito_i^t is a solution proposal, it is added to Y=(ak,ξk)k=1KyY=\(a_k, _k)\_k=1^K_y, where Ky=|Y|K_y=|Y|. Each ξk _k links its answer to the active graph trace and the evidence supporting that answer. The active graphs are recorded as Gact1:T=(G1,…,GT)G_act^1:T=(G^1,…,G^T). 4.3 Local Self-Refinement Residual errors can persist after Hierarchical Sparse Coordination. Local Self-Refinement addresses them through Decision Unit Localization and Adversarial Solution Validation. Decision Unit Localization. Hierarchical Sparse Coordination produces the solution set Y=(ak,ξk)k=1KyY=\(a_k, _k)\_k=1^K_y. Each candidate answer aka_k is paired with an evidence record ξk _k that links it to the activated graph trace and its supporting evidence. Each record ξk _k links an answer to the activated graph trace and the evidence that supports it. Following semantic answer equivalence (Kuhn et al. 2023), the candidates are collapsed into answer classes. The evidence-aware reliability estimator RψR_ψ scores the candidates within each class. The selected answer and its evidence record are (yrel,ξrel)=argmax(a,ξ)∈Rep(Collapse(Y))Rψ(a,ξ).(y_rel, _rel)= *arg\,max_(a,ξ) (Collapse(Y))R_ψ(a,ξ). (18) Here, Collapse(Y)Collapse(Y) groups semantically equivalent answers and RepRep retains the highest-scoring member of each class. The selected answer serves as the default solution and remains unchanged unless local validation supports a replacement. The selected solution is decomposed into locally verifiable decision units. For each unit d, the discrepancy encoder DiscθDisc_θ maps the evidence linked to d into the representation ϵd _d. This representation indicates whether the unit has consistent support. The risk scorer riskθrisk_θ maps ϵd _d to a scalar value. Let τ denote the risk threshold. The units selected for further validation are =d∈UnitParse(yrel,ξrel)∣riskθ(d,ϵd)>τ.U= \d (y_rel, _rel) _θ(d, _d)>τ \. (19) Units outside U are kept unchanged. The task-specific units and the local refinement operators are defined in Appendix A.2. This stage uses only evidence available during inference and does not access benchmark-only supervision. Adversarial Solution Validation. For each unit d∈d , a local correction subgraph is extracted from GUG_U by following the active evidence paths stored in ξrel _rel. The subgraph retains the nodes that contributed to or checked the disputed unit. A challenger ydchy_d^ch with evidence record ξdch _d^ch is retrieved from Y when another supported candidate disagrees at d. Otherwise, it is generated by executing the correction subgraph with a targeted repair prompt. The term adversarial means that the challenger contests one local decision. The unit-conditioned score RψlocR_ψ^loc applies the same reliability estimator to evidence linked to d. The local gain Δd _d is the score of the challenger minus the score of the selected solution. Let δ denote the required improvement margin. The certificate model compares the proposed replacement with the current unit using their linked evidence. Its output is cd=Certχ(ydch,ξdch,yrel,ξrel,d).c_d=Cert_χ (y_d^ch, _d^ch,y_rel, _rel,d ). (20) It equals one when the evidence supports the replacement while preserving the remaining solution. Let I denote the indicator function. The acceptance rule is Accept(ydch,d)=[Δd>δ∧cd=1].Accept(y_d^ch,d)=I[ _d>δ c_d=1]. (21) When a challenger is accepted, only unit d is replaced and the evidence record is updated. If no challenger satisfies the acceptance rule, the selected solution is preserved. 5 Experiments 5.1 Experiment Setup Datasets. We evaluate HELENA on eight widely used benchmarks covering three representative domains. For general knowledge and professional reasoning, we use MMLU (Hendrycks et al. 2021a) and MMLU-Pro (Wang et al. 2024). For mathematical reasoning, we use GSM8K (Cobbe et al. 2021), MATH (Hendrycks et al. 2021b), MATH-Lv5 (Hendrycks et al. 2021b), and SVAMP (Patel et al. 2021). For code generation, we use HumanEval (Chen et al. 2021) and MBPP (Austin et al. 2021). Baselines. We compare HELENA with representative single-agent, protocol-based, automated agentic, and graph-structured MAS baselines listed in Table 1. All results are independently reproduced under the same evaluation protocol using official implementations when available. Implementation details and budget controls are provided in Appendix A.1. Implementation Details. All agents use gpt-4o-mini; topology and memory embeddings use Qwen3-Embedding-8B with D=512D=512. Local Qwen models run on eight NVIDIA H100 GPUs. We use K=3K=3 fused topologies and average three independent runs. Code tasks are evaluated by pass@1; all other tasks use exact-match accuracy. Additional settings are provided in the supplementary material. Evaluation Protocol. Observed topology utilities are computed only from training examples. Validation labels are used only for model selection and calibration after each system output is fixed. During test-time MCTS and topology selection, the system receives the query and frozen model predictions only; gold answers, answer indices, exact-match feedback, and official held-out tests are inaccessible. Prompt-visible examples are treated as part of Q and are distinct from hidden evaluator tests. The benchmark scorer is invoked only after the final output has been fixed. 5.2 Main Results Table 1: Performance comparison under the unified reproduction protocol. All baselines are independently reproduced by us using the same gpt-4o-mini backbone, benchmark instances, decoding settings, and evaluation scripts. The best result is shown in bold and the runner-up is underlined. Red arrows denote the absolute improvement over the strongest reproduced baseline. Method Code generation Mathematical reasoning Knowledge / professional MBPP HumanEval GSM8K MATH SVAMP MATH-Lv5 MMLU MMLU-Pro Single-Agent Self-Refine (Madaan et al. 2023) 69.80 87.80 89.60 46.10 88.73 32.85 75.44 57.97 Pre-defined MAS Protocols LLM-Debate (Du et al. 2024) 70.29 88.68 89.47 52.96 91.76 38.55 81.04 63.28 AgentVerse (Chen et al. 2024c) 74.28 89.29 89.91 50.85 89.64 36.85 78.36 60.83 LLM-Blender (Jiang et al. 2023) 77.05 88.80 88.35 50.34 89.52 36.16 81.22 63.47 DyLAN (Liu et al. 2024) 77.30 90.42 89.98 51.12 88.48 37.38 79.96 62.14 Automated Agentic Systems H-Swarms (Feng et al. 2025) 81.65 89.86 95.00 65.89 92.74 50.27 83.68 69.00 AFlow (Zhang et al. 2025d) 82.20 90.06 92.30 73.35 91.73 58.82 83.10 64.35 RouterDC (Chen et al. 2024b) 75.20 87.75 93.68 73.46 91.86 58.93 82.01 63.27 A2Flow (Zhao et al. 2026) 85.00 92.40 93.80 58.50 92.15 44.76 83.29 63.42 DAAO (Su et al. 2025) 86.95 94.65 94.40 55.37 92.64 42.57 84.90 65.28 MasRouter (Yue et al. 2025) 84.00 90.62 95.45 75.42 92.95 61.36 84.25 64.85 MaAS (Zhang et al. 2025b) 82.17 92.85 92.30 74.45 91.55 60.23 83.01 63.68 BiRouter (Yang et al. 2026) 84.82 91.46 94.09 74.92 93.20 60.87 86.80 66.53 Multi-agent Systems as Graphs GPTSwarm (Zhuge et al. 2024) 75.40 86.28 94.66 68.85 92.18 54.64 82.80 64.19 G-Designer (Zhang et al. 2025c) 79.24 87.50 93.97 70.46 90.29 55.39 87.20 66.94 DAWN (Wan et al. 2026) 86.70 94.44 93.12 73.80 92.10 59.21 80.00 62.15 HELENA (Ours) 88.73 ↑ 1.78 95.20 ↑ 0.55 97.39 ↑ 1.94 78.29 ↑ 2.87 95.48 ↑ 2.28 68.21 ↑ 6.85 88.37 ↑ 1.17 79.34 ↑ 10.34 Notes. All scores are percentages. MATH-Lv5 denotes the Level-5 subset of MATH, and MMLU-Pro denotes the harder professional-knowledge variant of MMLU. Obs.❶ State-of-the-Art Performance with Difficulty-Scaled Gains. As shown in Table 1, HELENA achieves the best results on all evaluated benchmarks, with an average gain of 3.47% ↑ over the strongest baseline. The performance margin consistently widens with task difficulty: HELENA outperforms H-Swarms by 10.34% ↑ points on MMLU-Pro and exceeds MasRouter by 6.85% ↑ points on MATH-Lv5. These results demonstrate that topology-diverse coordination with noise-controlled execution yields substantial benefits on complex, multi-step reasoning tasks. 5.3 Cost Analysis (a) MATH and MATH-Lv5 (b) MMLU and MMLU-Pro Figure 3: Cost–accuracy comparison and difficulty-dependent scaling of HELENA. Left: MATH and MATH-Lv5. Right: MMLU and MMLU-Pro. Obs.❷ Difficulty-Adaptive Token Efficiency. Figure 3 shows that HELENA achieves a favorable cost–accuracy trade-off across benchmarks of varying difficulty. On standard datasets, HELENA remains in the low-cost region (2.44M tokens on MATH, 2.32M tokens on MMLU) while achieving the best accuracy. As reasoning complexity increases, HELENA adaptively scales computation: relative cost increases to 1.25× on MATH-Lv5 and 1.40× on MMLU-Pro. Concurrently, Figure 3 reveals that this efficiency primarily stems from the Node-Level Memory Composer and Edge-Level Sparse Activation. The memory composer selects and compresses task-relevant records into compact latent and verbalized briefs, while the controller activates only a sparse subgraph over the union graph, preventing full-graph communication and redundant context propagation. 5.4 Ablation Study and Sensitivity Analysis We conduct ablations on four key components of HELENA: local refinement, union graph construction, latent memory, and the global controller. Table 2 reports the results, where each variant disables one component while keeping the remaining system unchanged. Table 2: Ablation study. Variant Code generation Mathematical reasoning Knowledge / professional Summary MBPP HumanEval GSM8K MATH SVAMP MATH-Lv5 MMLU MMLU-Pro Avg. Avg. HELENA (Ours) 88.73 95.20 97.39 78.29 95.48 68.21 88.37 79.34 86.37 – w/o union graph 84.74 87.36 96.87 76.89 93.18 66.07 86.70 76.12 83.49 ↓ 2.88 w/o memory composer 84.59 88.09 96.58 76.15 94.20 63.49 87.24 75.42 83.22 ↓ 3.15 w/o sparse activation 86.15 90.77 97.04 77.24 94.83 64.11 87.85 76.80 84.35 ↓ 2.02 w/o local self-refinement 84.27 88.55 96.31 76.74 94.21 62.60 85.31 74.77 82.85 ↓ 3.52 Table 3: Structural diversity analysis of topology selection on MBPP. Selection Strategy K Active Edge [-0.15ex] Ratio ↓ ↑ D_E\, MBPP [-0.15ex] Acc. ↑ Single Best Topology 1 12.4(−34.0%)12.4\;(-34.0\%) – 84.74(−1.38)84.74\;(-1.38) Quality-only Top-K 3 18.8(ref.)18.8\;(ref.) 0.41(ref.)0.41\;(ref.) 86.12(ref.)86.12\;(ref.) Random-K 3 18.5(−1.6%)18.5\;(-1.6\%) 0.52(+26.8%)0.52\;(+26.8\%) 85.37(−0.75)85.37\;(-0.75) Max-Jaccard-K 3 19.1(+1.6%)19.1\;(+1.6\%) 0.78(+90.2%)0.78\;(+90.2\%) 86.44(+0.32)86.44\;(+0.32) HELENA (Ours) 3 18.7(−0.5%)18.7\;(-0.5\%) 0.69(+68.3%)0.69\;(+68.3\%) 88.73(+2.61)88.73\;(+2.61) Table 4: Controlled noise propagation with the union graph and the pre-injection state fixed across variants. Variant Noise [-0.15ex] Survival ↓ Reach@1 ↓ Reach@2+ ↓ Final Flip ↓ HELENA 34.7 14.6 5.3 4.1 w/o sparse activation 34.7 33.8 21.9 10.4 w/o memory composer 100.0 41.6 17.8 12.7 w/o HSC 100.0 92.4 68.7 29.6 Obs.❸ Union Graph Ensures Diversity. As shown in Table 2, removing the union graph causes an average drop of 2.88% ↓ . This result confirms its effectiveness but does not show whether the gain comes from complementary reasoning paths or greater communication cost. We therefore measure the structural diversity of the selected topologies by their average pairwise edge-Jaccard distance. For =G1,…,GKS=\G_1,…,G_K\ with edge sets Eii=1K\E_i\_i=1^K, it is defined as DE()=2K(K−1)∑i<j(1−|Ei∩Ej||Ei∪Ej|).D_E(S)= 2K(K-1) _i<j (1- |E_i∩ E_j||E_i∪ E_j| ). (22) A larger DED_E indicates lower structural overlap. As reported in Table 3, Quality-only Top-K retains redundant structures, while Max-Jaccard-K achieves higher diversity but lower accuracy. In contrast, HELENA combines a high DED_E with the best MBPP accuracy under a comparable active-edge ratio. The gain therefore comes from reliable and nonredundant reasoning paths rather than a larger communication budget or arbitrary diversity. Obs.❹ Two-Level Noise Control Ensures Reliable Reasoning. Noise control in HELENA requires two complementary mechanisms rather than a single step. Hierarchical Sparse Coordination suppresses noise during propagation by filtering private memories and maintaining a compact global state, preventing low-confidence traces from spreading through the union graph. Removing latent memory alone causes a 3.15% ↓ drop. Local Self-Refinement then targets high-risk decision units and revises them only when contrastive evidence confirms a correction. Its removal triggers the largest collapse of 3.52% ↓ , with 5.61% ↓ on MATH-Lv5 and 4.5% ↓ on MMLU-Pro. Together, the two mechanisms form a complete noise-suppression pipeline, with propagation control followed by targeted local revision. Controlled Noise Propagation. We adopt controlled error injection from prior MAS robustness analysis (Huang et al. 2025) across the full benchmark suite. For each example, a task-associated but incorrect intermediate record, sampled independently of the memory selector, is inserted into a nonterminal node under a fixed union graph and pre-injection state. NoiseSurvivalNoiseSurvival records post-filter retention; Reach@1Reach@1 and Reach@2+Reach@2+ measure one-hop and multi-hop exposure; FinalFlipFinalFlip counts outputs in the common clean-correct mask that become incorrect. The w/o HSC variant disables both levels. As shown in Table 4, the composer reduces survival to 34.7%34.7\%; at this fixed rate, sparse activation reduces Reach@1/Reach@2+Reach@1/Reach@2+ from 33.8/21.933.8/21.9 to 14.6/5.314.6/5.3 and yields the lowest FinalFlipFinalFlip of 4.1%4.1\%. Sensitivity Analysis on K. Figure 4 shows how the number of fused topologies K affects performance and cost. K=3K=3 achieves the best balance between topology diversity and redundant-edge control, so we use K=3K=3 as the default setting. Figure 4: Sensitivity analysis of HELENA. Left: performance and relative token cost as the number of fused topologies K varies. Right: Right: performance and relative token cost as the lower-confidence quantile α varies. Sensitivity Analysis on Lower-Confidence Quantile. Figure 4 also reports the effect of the lower-confidence quantile α. The default value α=0.38α=0.38 performs best, balancing uncertainty suppression and the retention of uncertain but useful candidate topologies. 5.5 Case Study Figure 5: Case study illustrating the intrinsic mechanisms of HELENA. Tuples denote (syntax_ok, entry_point_ok, passed_tests) in this case. We explore and visualize the intrinsic mechanisms of HELENA. Taking MBPP as an example, Figure 5 showcases the workflow of HELENA. 5.6 Transferability Analysis Table 5 presents cross-backbone results. HELENA transfers well to open-source models: Qwen3-32B remains within 1 point of GPT-4o-mini on complex reasoning benchmarks, including MATH, MATH Level 5, and MMLU-Pro, while Qwen3-8B also benefits consistently from the same coordination framework across all eight benchmarks. Table 5: Cross-backbone transferability of HELENA on Qwen3-32B and Qwen3-8B. Benchmark GPT-4o-mini Qwen3-32B Qwen3-8B MMLU 88.37 87.92↓ 0.45 84.12↓ 4.25 GSM8K 97.39 96.85↓ 0.54 94.23↓ 3.16 HumanEval 95.20 92.75↓ 2.45 86.34↓ 8.86 MBPP 88.73 86.90↓ 1.83 81.08↓ 7.65 MATH 78.29 77.86↓ 0.43 72.84↓ 5.45 MATH-L5 68.21 67.74↓ 0.47 61.92↓ 6.29 MMLU-Pro 79.34 78.62↓ 0.72 73.41↓ 5.93 SVAMP 95.48 94.91↓ 0.57 92.16↓ 3.32 Avg. 86.37 85.44↓ 0.93 80.76↓ 5.61 6 Conclusion HELENAaddresses the topology diversity and noise propagation dilemma in LLM-based MAS . Across eight benchmarks, it improves the strongest baseline by 3.47% ↑ on average and up to 10.34% ↑ . References J. Austin, A. Odena, M. Nye, M. Bosma, H. Michalewski, D. 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Each reported result is averaged over three independent runs. Prompt and context budget. For each agent call, the input prompt consists of the original task query, a role-conditioned instruction, the selected private-memory brief, active predecessor briefs, and the current global-state summary. To control context growth, we cap the maximum input length of each agent call at approximately 8,1928,192 tokens and the maximum generation length at 1,0241,024 tokens. The task query is truncated to at most 2,0482,048 tokens, the private-memory brief to 256256 tokens, the predecessor-context brief to 384384 tokens, and the global-state summary to 256256 tokens. When the composed prompt exceeds the budget, we first remove zero-weight memory records and then truncate lower-weight predecessor briefs. Graph and memory settings. During union-graph construction, we select K=3K=3 complementary topologies by greedy k-DPP MAP inference and use the lower-confidence quantile α=0.38α=0.38 for conservative topology scoring. In Hierarchical Sparse Coordination, we run up to T=5T=5 communication turns. Each active node retrieves at most 44 private-memory records and receives at most 33 predecessor briefs from active in-neighbors. Both memory retrieval and edge activation use sparsemax, so irrelevant memory records and inactive communication edges are assigned zero weight. Stage separation. All learned components are optimized before benchmark evaluation and are frozen at validation and test time. MCTS visit counts and action values are query-local statistics and are reinitialized for every query. Test-time memory writes are also query-local and are discarded after finalization. They do not update model parameters or affect later test examples. The data-access protocol is specified in Appendix A.2. Lightweight test-time topology search. Test-time MCTS performs legal graph edits and forward passes through the frozen edit prior and utility predictor. It does not invoke the agent backbone or execute candidate workflows. Agent LLM calls begin only after the selected topologies have been merged into the query-specific union graph. The search therefore adds no LLM token cost. Its graph operations and local model forward passes are included in end-to-end wall-clock latency. Item Setting Default backbone gpt-4o-mini Transfer backbones Qwen3-32B, Qwen3-8B Embedding model Qwen3-Embedding-8B Embedding dimension 512512 Selected topologies K=3K=3 Lower-confidence quantile α=0.38α=0.38 Maximum communication turns T=5T=5 Generation temperature 0 Verifier / finalizer temperature 0 Max input length per call 8,1928,192 tokens Max generation length 1,0241,024 tokens Task query cap 2,0482,048 tokens Private-memory brief cap 256256 tokens Predecessor-context brief cap 384384 tokens Global-state summary cap 256256 tokens Selected memory records per node up to 44 Active predecessor briefs per node up to 33 Agent LLM calls during MCTS 0 Evaluation runs 33 Code metric pass@1pass@1 Other metrics exact-match accuracy Table 6: Implementation and prompt-budget settings of HELENA. A.2 Algorithm Workflow Training–inference separation. We distinguish three disjoint data roles: a topology-supervision split supD_sup, a validation split valD_val, and the held-out test split testD_test. Observed topology utility is computed only on supD_sup. Validation labels are used only after a complete system output has been fixed, for checkpoint selection and calibration. At test time, topology search, topology selection, hierarchical coordination, and local refinement receive no gold answer, answer index, exact-match result, or official held-out test feedback. The benchmark scorer accesses such information only after the final output is irrevocably fixed. Prompt-visible examples or checks are treated as part of the query Q. They are therefore available to every compared method and are distinct from the official held-out evaluator. Across validation and test examples, all learned parameters are frozen; only per-query MCTS statistics, graph activations, and query-local memories may change. Signal Offline supervision Validation/test inference External scoring Query and prompt-visible checks Yes Yes Yes Gold answer or answer index supD_sup only No After output Observed Tier-1/Tier-2 utility supD_sup only No No Frozen predicted topology utility After fitting Yes No Official held-out code tests No No After output Cross-query parameter updates Yes No No Table 7: Data-access firewall used by HELENA. “After output” means that the corresponding signal is accessed only after the system output has been fixed and can no longer affect search, topology selection, or refinement. Multi-fidelity topology supervision. For (Q,Y)∈sup(Q,Y) _sup, executing topology W produces a candidate answer a^W a_W. A training-only multi-fidelity evaluator first uses a cheap proxy to screen all candidates, then applies Tier 1 to a shortlist and Tier 2 to the finalists. When a higher-fidelity result is available, it supersedes the lower-fidelity estimate rather than being linearly blended with it. The highest-fidelity observed utility is normalized to [0,1][0,1] and serves as the supervision target: uobs(Q,W;Y)=[a^W=Y],MCQ/EM,[Eq(a^W,Y)],Math,PassRatesup(Q)(a^W),Code.u^obs(Q,W;Y)= cases1[ a_W=Y],&MCQ/EM,\\ 1[Eq( a_W,Y)],&Math,\\ PassRate_T_sup(Q)( a_W),&Code. cases (23) Here EqEq denotes normalized numeric or symbolic equivalence, and sup(Q)T_sup(Q) contains only prompt-visible checks or checks belonging to the non-test supervision split. Official held-out evaluator tests are never included in sup(Q)T_sup(Q). At validation and test time, Eq. (23) is not evaluated inside the system; terminal topologies are valued only by the frozen predictor described below. Algorithm 1 Offline Topology Guidance Learning of HELENA 1. Input: topology-supervision set supD_sup; validation set valD_val; initial workflow W0W_0; search budget B; training-only multi-fidelity evaluator ℰMFsupE_MF^sup; trainable edit prior PϕP_φ; trainable utility-predictor ensemble gψmm=1M\g_ _m\_m=1^M. 2. Output: frozen edit prior Pϕ⋆P_φ and frozen utility predictors gψm⋆m=1M\g_ _m \_m=1^M. 3. Initialize the topology-supervision buffer toposup←∅D_topo^sup← . 4. for each (Q,Y)∈sup(Q,Y) _sup do 5. Initialize query-local MCTS statistics NQ(s,u)←0N_Q(s,u)← 0 and Q(s,u)←0Q_Q(s,u)← 0. 6. for simulation b=1,…,Bb=1,…,B do 7. Set s←W0s← W_0 and search path τb←∅ _b← . 8. while s is not terminal do 9. Select u⋆←argmaxu∈(s)PUCTQ(s,u)u ← _u (s)PUCT_Q(s,u), where PUCTQ(s,u)= _Q(s,u)= Q(s,u) Q_Q(s,u) +cpuctPϕ(u∣s,Q)NQ(s)1+NQ(s,u). +c_puctP_φ(u s,Q) N_Q(s)1+N_Q(s,u). 10. Append (s,u⋆)(s,u ) to τb _b and set s←(s,u⋆)s (s,u ). 11. end while 12. Compile terminal state s into topology W, execute W on Q, and obtain uWobs←ℰMFsup(Q,Y,W).u_W^obs _MF^sup(Q,Y,W). 13. Add (Q,W,uWobs)(Q,W,u_W^obs) to toposupD_topo^sup. 14. for each visited pair (s,u)∈τb(s,u)∈ _b do 15. Update the visit count NQ(s,u)←NQ(s,u)+1.N_Q(s,u)← N_Q(s,u)+1. 16. Update the running action value Q(s,u)←Q(s,u)+uWobs−Q(s,u)NQ(s,u).Q_Q(s,u)← Q_Q(s,u)+ u_W^obs-Q_Q(s,u)N_Q(s,u). 17. end for 18. end for 19. Update PϕP_φ from the utility-labelled MCTS trajectories; only examples in supD_sup contribute to this update. 20. end for /* Query-conditioned topology encoding and utility learning */ 21. for each (Q,W,uWobs)∈toposup(Q,W,u_W^obs) _topo^sup do 22. Encode the query eQ←Emb(Q)e_Q (Q). 23. Canonically serialize the topology as Ser(W)=roles,edges,prompts.Ser(W)=\roles,edges,prompts\. 24. Compute eW←Emb(Ser(W))e_W (Ser(W)) and hW←StructFeat(W)h_W (W). zW←ProjW([eW;hW]).z_W _W([e_W;h_W]). 25. Form the joint query–topology representation zQ,W←ProjQW([eQ;zW]).z_Q,W _QW([e_Q;z_W]). 26. end for 27. for predictor m=1,…,Mm=1,…,M do 28. Predict (μQ,W(m),ρQ,W(m))←gψm(zQ,W).( _Q,W^(m), _Q,W^(m))← g_ _m(z_Q,W). 29. Convert the scale parameter σQ,W(m)←softplus(ρQ,W(m))+ε. _Q,W^(m) ( _Q,W^(m))+ . 30. Train gψmg_ _m by minimizing ℒqual(m)=∑(Q,W,u)∈toposup[(u−μQ,W(m))22(σQ,W(m))2+12log(σQ,W(m))2].L_qual^(m)= _(Q,W,u) _topo^sup [ (u- _Q,W^(m))^22( _Q,W^(m))^2+ 12 ( _Q,W^(m))^2 ]. 31. end for 32. Select and calibrate checkpoints on valD_val only after each validation output has been fixed; freeze all selected parameters. 33. return Pϕ⋆P_φ and gψm⋆m=1M\g_ _m \_m=1^M. Algorithm 2 Test-Time Query-Conditioned Union Graph Construction of HELENA 1. Input: test query Q; initial workflow W0W_0; frozen edit prior Pϕ⋆P_φ ; frozen utility predictors gψm⋆m=1M\g_ _m \_m=1^M; search budget B; selected topology number K; PUCT coefficient cpuctc_puct; lower-confidence quantile α<0.5α<0.5; DPP temperature η; maximum communication turns T. 2. Output: union MAS graph GU=(VU,EU)G_U=(V_U,E_U) and edge descriptors eu→v=(puv,compatuv,ctxuv)e_u→ v=(p_uv,compat_uv,ctx_uv). 3. No gold answer or official held-out test is an input to this algorithm. 4. Initialize ←∅C← , NQ(s,u)←0N_Q(s,u)← 0, and Q(s,u)←0Q_Q(s,u)← 0. /* Candidate generation with predicted leaf values */ 5. for simulation b=1,…,Bb=1,…,B do 6. Set s←W0s← W_0 and τb←∅ _b← . 7. while s is not terminal do 8. Select u⋆←argmaxu∈(s)[Q(s,u)+cpuctPϕ⋆(u∣s,Q)NQ(s)1+NQ(s,u)].u ← *arg\,max_u (s) [Q_Q(s,u)+c_puctP_φ (u s,Q) N_Q(s)1+N_Q(s,u) ]. 9. Append (s,u⋆)(s,u ) to τb _b and set s←(s,u⋆)s (s,u ). 10. end while 11. Compile s into topology W and compute zQ,Wz_Q,W as in the offline algorithm. Candidate workflows are not executed. 12. for predictor m=1,…,Mm=1,…,M do 13. Predict (μQ,W(m),σQ,W(m))←gψm⋆(zQ,W)( _Q,W^(m), _Q,W^(m))← g_ _m (z_Q,W). 14. end for 15. Compute the predictive mean μQ,W←1M∑m=1MμQ,W(m). _Q,W← 1M _m=1^M _Q,W^(m). 16. Compute the predictive uncertainty σQ,W2←1M∑m=1M((σQ,W(m))2+(μQ,W(m))2)−μQ,W2. _Q,W^2← 1M _m=1^M (( _Q,W^(m))^2+( _Q,W^(m))^2 )- _Q,W^2. 17. Set u^ψ(Q,W)←μQ,W u_ψ(Q,W)← _Q,W. 18. for each (s,u)∈τb(s,u)∈ _b do 19. Update NQ(s,u)←NQ(s,u)+1.N_Q(s,u)← N_Q(s,u)+1. Q(s,u)←Q(s,u)+u^ψ(Q,W)−Q(s,u)NQ(s,u).Q_Q(s,u)← Q_Q(s,u)+ u_ψ(Q,W)-Q_Q(s,u)N_Q(s,u). 20. end for 21. Add the unique record (W,μQ,W,σQ,W)(W, _Q,W, _Q,W) to C. 22. end for /* Quality estimation and information-flow comparison */ 23. for each W∈W do 24. Compute qQ,W←exp(μQ,W+Φ−1(α)σQ,W2η).q_Q,W← ( _Q,W+ ^-1(α) _Q,W2η ). 25. Map role-compatible nodes to a common order and construct the normalized adjacency matrix AWA_W. 26. Compute ΨT(W)←∑t=1TAWt. _T(W)← _t=1^TA_W^t. 27. end for 28. for each pair (Wa,Wb)∈×(W_a,W_b) ×C do 29. Compute Σab←⟨vec(ΨT(Wa)),vec(ΨT(Wb))⟩‖vec(ΨT(Wa))‖2‖vec(ΨT(Wb))‖2. _ab← ( _T(W_a)),vec( _T(W_b)) \|vec( _T(W_a)) \|_2 \|vec( _T(W_b)) \|_2. 30. Set Lab←qQ,WaΣabqQ,WbL_ab← q_Q,W_a _abq_Q,W_b. 31. end for 32. Greedily select K candidates by the objective in Eq. (7) and denote the result by S⋆S . /* Union graph construction */ 33. Align nodes by role-agent identity and construct GU←(⋃W∈S⋆VW,⋃W∈S⋆EW).G_U← ( _W∈ S V_W, _W∈ S E_W ). 34. for each edge (u,v)∈EU(u,v)∈ E_U do 35. Compute puv←∑W∈S⋆qQ,W[(u,v)∈EW]∑W∈S⋆qQ,W.p_uv← _W∈ S q_Q,W1[(u,v)∈ E_W] _W∈ S q_Q,W. 36. Compute compatuvcompat_uv from the role and prompt representations of the two endpoints. 37. Compute ctxuvctx_uv from the finite-turn flow and the position of the edge in the selected topologies. 38. Set eu→v←(puv,compatuv,ctxuv)e_u→ v←(p_uv,compat_uv,ctx_uv). 39. end for 40. return GUG_U and eu→v∣(u,v)∈EU\e_u→ v (u,v)∈ E_U\. Offline training of hierarchical coordination. The memory query builder fqf_q, memory selector fsf_s, edge scorer ΨΘeedge _ _e^edge, and global updater GlobalNodeΘgGlobalNode_ _g are optimized only on supD_sup and frozen before validation/test inference. Memory retrieval uses retrospective contrastive credit assignment, ℒmem=−logexpzposexpzpos+∑zneg∈iexpzneg.L_mem=- z_pos z_pos+ _z_neg _i z_neg. (24) For an execution trace τ, define sΘe(τ)=∑t∑(vj,vi)∈Etlogβjits_ _e(τ)= _t _(v_j,v_i)∈ E^t _ji^t. The controller is trained by pairwise trace ranking with explicit communication-cost penalties, ℒctrl= _ctrl= −logexpsΘe(τ+)expsΘe(τ+)+∑τ−expsΘe(τ−) - s_ _e(τ^+) s_ _e(τ^+)+ _τ^- s_ _e(τ^-) (25) +λE∑t|Et|+λC∑tTok(Dt). + _E _t|E^t|+ _C _tTok(D^t). These are stage-specific objectives; gradients do not pass through discrete MCTS expansion or k-DPP subset selection. Algorithm 3 Test-Time Hierarchical Sparse Coordination of HELENA 1. Input: query Q; union graph GU=(VU,EU)G_U=(V_U,E_U); edge descriptors eu→v\e_u→ v\; private memories ℳi0\M_i^0\; role instructions ri\r_i\; role-conditioned prompts Pit\P_i^t\; turns T; frozen query builder fq⋆f_q ; frozen memory selector fs⋆f_s ; memory composer ComposeCompose; frozen edge scorer ΨΘe⋆edge _ _e ^edge; frozen global updater GlobalNodeΘg⋆GlobalNode_ _g ; feedback operator FeedbackavailFeedback_avail. 2. Output: solution set Y=(ak,ξk)k=1KyY=\(a_k, _k)\_k=1^K_y and activated graph trace Gact1:TG_act^1:T. 3. FeedbackavailFeedback_avail uses only signals available during inference and excludes benchmark labels. 4. Initialize g0←InitGlobal(Q)g^0 (Q), Y←∅Y← , and Gact1:T←∅G_act^1:T← . 5. for turn t=1,…,Tt=1,…,T do /* Node-level memory composition */ 6. for each node vi∈VUv_i∈ V_U do 7. Compute qit←fq⋆(Q,ri)q_i^t← f_q (Q,r_i). 8. Compute zikt←fs⋆(qit,mikt)z_ik^t← f_s (q_i^t,m_ik^t) for every mikt∈ℳitm_ik^t _i^t. 9. Compute it←sparsemax(zikt∣mikt∈ℳit). α_i^t (\z_ik^t m_ik^t _i^t\ ). 10. Set ℛit←mikt∈ℳit∣αikt>0.R_i^t←\m_ik^t _i^t _ik^t>0\. 11. Compute (ℓit,bit,Bit)←Compose((αikt,mikt)∣mikt∈ℛit).( _i^t,b_i^t,B_i^t) (\( _ik^t,m_ik^t) m_ik^t _i^t\ ). 12. end for /* Edge-level sparse activation and execution */ 13. for each edge (vj,vi)∈EU(v_j,v_i)∈ E_U do 14. Compute ϕjit←ΨΘe⋆edge(bjt,bit,gt,evj→vi). _ji^t← _ _e ^edge (b_j^t,b_i^t,g^t,e_v_j→ v_i ). 15. end for 16. for each target node vi∈VUv_i∈ V_U do 17. Compute it←sparsemax(ϕjit∣(vj,vi)∈EU). β_i^t (\ _ji^t (v_j,v_i)∈ E_U\ ). 18. end for 19. Define Et←(vj,vi)∈EU∣βjit>0.E^t←\(v_j,v_i)∈ E_U _ji^t>0\. 20. Let VtV^t contain the endpoints of EtE^t and set Gt←(Vt,Et)G^t←(V^t,E^t). 21. Append GtG^t to Gact1:TG_act^1:T. 22. for each active node vi∈Vtv_i∈ V^t do 23. Form DitD_i^t from the verbalized briefs received through active incoming edges. 24. Execute oit←vi(Pit,Q,ℓit,Dit,gt).o_i^t← v_i (P_i^t,Q, _i^t,D_i^t,g^t ). 25. Obtain ℱit←Feedbackavail(oit,Q).F_i^t _avail(o_i^t,Q). 26. Update the query-local memory ℳit+1←ℳit∪Write(oit,ℱit,Gt).M_i^t+1 _i^t (o_i^t,F_i^t,G^t). 27. if oito_i^t is a solution proposal then 28. Create ξit _i^t that links oito_i^t to the active graph trace and its supporting evidence. 29. Set Y←Y∪(oit,ξit)Y← Y∪\(o_i^t, _i^t)\. 30. end if 31. end for 32. Pool the active briefs and outputs into ptp^t. 33. Pool the available feedback into ℱtF^t. 34. Compute gt+1←GlobalNodeΘg⋆(gt,pt,ℱt).g^t+1 _ _g (g^t,p^t,F^t ). 35. end for 36. Set Ky←|Y|K_y←|Y| and discard the query-local memory writes. 37. return Y and Gact1:TG_act^1:T. Offline training of local refinement. The reliability estimator RψR_ψ, discrepancy encoder DiscθDisc_θ, risk scorer riskθrisk_θ, and certificate model CertχCert_χ are trained only from retrospective records derived from supD_sup. They are frozen before validation and test inference. A reliability preference pair contains two candidates produced for the same supervision query. The candidate with higher observed utility is preferred. Pairs with equal utility are omitted. Let p=((a+,ξ+),(a−,ξ−))p=((a^+,ξ^+),(a^-,ξ^-)) denote one pair in relsupP_rel^sup. The reliability estimator is trained by ℒrel=−∑p∈relsuplogσ(Rψ(a+,ξ+)−Rψ(a−,ξ−)).L_rel=- _p _rel^sup σ (R_ψ(a^+,ξ^+)-R_ψ(a^-,ξ^-) ). (26) The discrepancy encoder and risk scorer form one unit risk model. For each training unit d, DiscθDisc_θ maps the evidence linked to the unit to ϵd _d. The risk scorer maps this representation to the risk probability pdp_d. The loss below updates both parts of the model. No separate discrepancy loss is used. pd p_d =σ(riskθ(d;ϵd)), =σ(risk_θ(d; _d)), (27) ℒrisk _risk =−∑(d,ϵd,rd⋆)∈risksup[rd⋆logpd+(1−rd⋆)log(1−pd)]. =- _(d, _d,r_d ) _risk^sup [r_d p_d+(1-r_d ) (1-p_d) ]. The target rd⋆r_d indicates whether the unit lacks sufficient support under the supervision available in supD_sup. For each proposed replacement, the certificate model receives the challenger, the current solution, and the evidence linked to both versions of the unit. Its output and training loss are cd c_d =Certχ(ydch,ξdch,yrel,ξrel,d), =Cert_χ (y_d^ch, _d^ch,y_rel, _rel,d ), (28) ℒcert _cert =−∑(d,ydch,ξdch,yrel,ξrel,cd⋆)∈certsup[cd⋆logcd+(1−cd⋆)log(1−cd)]. =- _(d,y_d^ch, _d^ch,y_rel, _rel,c_d ) _cert^sup [c_d c_d+(1-c_d ) (1-c_d) ]. A positive target is assigned only when the local replacement improves the supervised outcome of unit d while preserving the remaining units. No validation or test label is used to update these models. Local refinement operators. The operator CollapseCollapse follows semantic answer equivalence (Kuhn et al. 2023). Given the resulting answer classes, RepRep retains the candidate with the highest RψR_ψ score from each class. Task type Decision unit returned by UnitParseUnitParse Multiple choice One normalized answer choice Mathematical reasoning One derivation step or the final answer field Code generation One code span identified by the available verifier evidence Table 8: Task-specific decision units used by Local Self-Refinement. For a selected solution (yrel,ξrel)(y_rel, _rel), UnitParseUnitParse returns the locally verifiable units listed in Table 8. For each unit d, DiscθDisc_θ receives the unit and the evidence linked to it. It produces the fixed-dimensional representation ϵd _d, which is passed to riskθrisk_θ and is not itself a risk score. The local score Rψloc(a,ξ,d)R_ψ^loc(a,ξ,d) uses the same reliability estimator after evidence unrelated to unit d has been masked. The global score and local score therefore share parameters. The certificate model compares the current unit with a proposed replacement using their linked evidence. Its positive output supports only the local replacement and does not certify the full solution. The operator ExtractSubgraphExtractSubgraph starts from the nodes cited by the provenance of unit d. It follows the recorded active paths to the nodes that supplied evidence for the unit. Verifier nodes that assessed the unit are retained. The union graph defines eligible edges, while the activated trace limits the correction graph to paths used in the current execution. The operator RepairExecRepairExec executes only the agents in the correction graph. Its prompt contains the original query and the selected solution. The disputed unit and its linked evidence form the repair context. The prompt asks for a replacement of that unit while preserving the remaining solution. Each challenger is returned with a new evidence record. Let Θref⋆=(Rψ⋆,Discθ⋆,riskθ⋆,Certχ⋆) _ref =(R_ψ ,Disc_θ ,risk_θ ,Cert_χ ) denote the frozen local refinement modules. Algorithm 4 Test-Time Local Self-Refinement of HELENA 1. Input: query Q; solution set Y=(ak,ξk)k=1KyY=\(a_k, _k)\_k=1^K_y; union graph GUG_U; activated trace Gact1:TG_act^1:T; frozen module bundle Θref⋆ _ref ; risk threshold τ; margin δ. 2. Output: final reliable solution y y. 3. All evidence used below is available during inference. No gold answer or official held-out test is accessible. /* Decision Unit Localization */ 4. Collapse semantically equivalent answers ←Collapse(Y)G (Y). 5. for each equivalence class G∈G do 6. Select (aG,ξG)←argmax(a,ξ)∈GRψ⋆(a,ξ).(a_G, _G)← *arg\,max_(a,ξ)∈ GR_ψ (a,ξ). 7. end for 8. Set ℛ←Rep()R (G) and select (yrel,ξrel)←argmax(a,ξ)∈ℛRψ⋆(a,ξ).(y_rel, _rel)← *arg\,max_(a,ξ) R_ψ (a,ξ). 9. Parse the selected solution unit←UnitParse(yrel,ξrel).D_unit (y_rel, _rel). 10. Initialize ←∅U← . 11. for each d∈unitd _unit do 12. Compute ϵd←Discθ⋆(d,yrel,ξrel,Y). _d _θ (d,y_rel, _rel,Y). 13. Compute rd←riskθ⋆(d;ϵd)r_d _θ (d; _d). 14. if rd>τr_d>τ then ←∪dU ∪\d\. end if 15. end for /* Adversarial Solution Validation */ 16. for each d∈d do 17. Extract Gdcorr←ExtractSubgraph(GU,Gact1:T,d,ξrel,Y).G_d^corr (G_U,G_act^1:T,d, _rel,Y). 18. Retrieve ℋdret←(a,ξ)∈Y∣Disagree(a,yrel,d)=1.H_d^ret←\(a,ξ)∈ Y (a,y_rel,d)=1\. 19. Generate ℋdgen←RepairExec(Q,Gdcorr,yrel,d,ξrel).H_d^gen (Q,G_d^corr,y_rel,d, _rel). 20. Set ℋd←ℋdret∪ℋdgenH_d _d^ret _d^gen and d←∅A_d← . 21. for each (ydch,ξdch)∈ℋd(y_d^ch, _d^ch) _d do 22. Compute Δd←Rψ⋆,loc(ydch,ξdch,d)−Rψ⋆,loc(yrel,ξrel,d). _d← R_ψ ,loc(y_d^ch, _d^ch,d)-R_ψ ,loc(y_rel, _rel,d). 23. Compute cd←Certχ⋆(ydch,ξdch,yrel,ξrel,d).c_d _χ (y_d^ch, _d^ch,y_rel, _rel,d ). 24. if Δd>δ _d>δ and cd=1c_d=1 then 25. Set d←d∪(ydch,ξdch)A_d _d∪\(y_d^ch, _d^ch)\. 26. end if 27. end for 28. if d≠∅A_d≠ then 29. Select (yd⋆,ξd⋆)←argmax(y,ξ)∈dRψ⋆,loc(y,ξ,d).(y_d , _d )← *arg\,max_(y,ξ) _dR_ψ ,loc(y,ξ,d). 30. Rewrite yrel←Rewrite(yrel,d,yd⋆).y_rel (y_rel,d,y_d ). 31. Update ξrel←UpdateEvidence(ξrel,d,ξd⋆,Gdcorr). _rel ( _rel,d, _d ,G_d^corr). 32. else preserve unit d unchanged. end if 33. end for 34. Set y^←yrel y← y_rel. 35. return y y. External benchmark scoring. After Test-Time Local Self-Refinement returns y y, the system state is frozen and no further search, topology selection, or rewriting is permitted. Only then does the external benchmark scorer compare y y with the gold answer or run the official held-out tests. Consequently, benchmark feedback cannot affect any output-producing decision. A.3 Controlled Noise Propagation Protocol The evaluation covers the full benchmark suite under a shared injection protocol. In each run, a task-associated but incorrect intermediate record is sampled independently of the memory-selection scores and inserted into a nonterminal node. Thus, the injected record is related to the task but is not constructed to rank among the node’s selected memory records. For every injection run, the query, injected record, injection source, union graph, and pre-injection state are identical across variants. Local Self-Refinement is disabled throughout this experiment. Let =HELENA,w/o sparse activation,w/o memory composer,w/o HSCA=\HELENA,w/o sparse activation,w/o memory composer,w/o HSC\ denote the evaluated variants. HELENA enables both HSC levels; w/o sparse activation retains the memory composer but executes all union edges; w/o memory composer retains sparse edge activation but bypasses memory selection; and w/o HSC disables both levels, yielding dense union-graph execution with the full private memory available to each active node. Let ℐI denote the common set of injection runs, and let m~i m_i be the record injected in run i. For variant a∈a , let ℛiaR_i^a denote the records exposed after node-level memory selection; when the memory composer is disabled, ℛiaR_i^a is the full private memory and therefore contains m~i m_i. Let i(1)V_i^(1) contain nodes whose shortest-path distance from the injection source is one, and let i(2+)V_i^(2+) contain nodes at distance 22 through the maximum communication horizon T. Because the union graph and injection source are fixed, these eligible node sets are identical across variants. The indicator xivax_iv^a equals one when the provenance of m~i m_i appears in the input context of node v under variant a. A node that is not activated, or is activated without receiving the injected record, has xiva=0x_iv^a=0. Reach is micro-averaged over all eligible downstream nodes, so every variant uses the same denominators. The reported metrics are NoiseSurvivala ^a =100|ℐ|∑i∈ℐ[m~i∈ℛia], = 100|I| _i 1[ m_i _i^a], (29) Reach@1a @1^a =100∑i∈ℐ∑v∈i(1)xiva∑i∈ℐ|i(1)|, =100 _i _v _i^(1)x_iv^a _i |V_i^(1)|, (30) Reach@2+a @2+^a =100∑i∈ℐ∑v∈i(2+)xiva∑i∈ℐ|i(2+)|. =100 _i _v _i^(2+)x_iv^a _i |V_i^(2+)|. (31) Let y^i,aclean y_i,a^clean and y^i,ainj y_i,a^inj denote the outputs of variant a before and after injection, and let yiy_i denote the reference answer. To ensure a common denominator, Final Flip is evaluated on the shared clean-correct mask =i∈ℐ∣y^i,aclean=yi for every a∈.C= \i y_i,a^clean=y_i for every a \. (32) Thus, the same examples contribute to every row of Table 4: FinalFlipa=100||∑i∈[y^i,ainj≠yi].FinalFlip^a= 100|C| _i 1[ y_i,a^inj≠ y_i]. (33) A.4 Prompt Set This appendix lists representative prompt schemas used by HELENA. We include prompts for topology-level prompt slots, role-conditioned agents, latent-guided graph execution, deductive reasoning contracts, and local verification / repair. At validation and test time, all placeholders are populated only from the input query, frozen model outputs, graph traces, query-local memories, and prompt-visible checks. They never contain the gold answer, answer index, or official held-out evaluator tests. Discrete Prompt Slot Template Representative Agent Role Prompts Latent-Guided Runtime Prompt Deductive Solver and Verifier Contracts Local Verification and Repair Prompts