cs.MAOct 13, 2026

Degrees of Freedom in Multi-Agent Coordination: Between Centralized and Decentralized Regimes

Authors: Yiqing Peng, Hehai Lin, Zixuan Wang, Dongyi Zhang, Zhengjie Zhu, Yi Han, Yangyang Zheng, Hao Wang, 

Organizations: The Hong Kong University of Science and Technology (Guangzhou) · Wake Forest University · University of Science and Technology of China · Shandong University

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Abstract

Large language model (LLM)-based multi-agent systems (MAS) are commonly organized around two coordination regimes. Centralized coordination provides explicit control but can create a bottleneck at a single coordination locus, whereas decentralized coordination offers local autonomy but must achieve effective collaboration through distributed decisions and experience. These trade-offs motivate a systematic study of the space between the two regimes. We introduce Degrees of Freedom (DoF), a continuous configuration framework that quantifies the size of the decision space delegated to downstream agents. DoF controls the granularity of delegated work and the target probability that each task unit is assigned to the branches. Its endpoints recover the two coordination regimes. Across multiple task domains, DoF reveals a hierarchy of organizational effects. Delegated granularity and within-regime routing produce structured but bounded profile reallocations, whereas task domain and retained experience are associated with stronger differences between the centralized- and decentralized-branch profiles. The resulting performance landscape is task-dependent: different domains favor different DoF configurations, while atomic-target conditions are more consistent across runs. In-domain retained experience brings the two branch profiles closer together. The same controlled sweep reveals a task-dependent coordination landscape: some domains exhibit higher-performing configurations, whereas others show flatter responses and no single configuration dominates across domains. In five of the six evaluated state, an intermediate DoF configuration achieves higher mean accuracy than both endpoints, demonstrating the value of exploring the coordination space between them. Together, these findings provide empirical guidance for studying and designing intermediate coordination regimes in LLM-based MAS.

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Figure 1: Illustration of the DoF framework. (a) Existing automatic MAS instantiate centralized or decentralized coordination through architecture-specific mechanisms, leaving intermediate configurations difficult to compare under matched conditions. (b) DoF moves from architecture selection to controlled coordination configuration by jointly specifying delegated granularity and per-unit branch-routing probability.
Figure 2: Overview of the DoF workflow. The Coordinator decomposes the input at the configured granularity, assigns each task unit to the centralized or decentralized branch through Bernoulli-based routing, and aggregates the completed unit outputs, linking the piecewise DoF schedule to the two branch implementations.
Figure 3: Delegated granularity and routing affect branch-specific profiles. Panel (a) reports the change in branch-specific profile composition from atomic-target to subtask-target delegation. Panel (b) compares matched- qq granularity differences, within-regime routing differences, and same-DoF repeated-run effects. Colors identify the effect type, while opacity distinguishes the centralized and decentralized branches.
Figure 4: Branch-specific profiles across task domains and retained agent states. Bars show branch-normalized profile-call shares pooled over intermediate DoF conditions ( 0<d<10<d<1 ). The upper row reports centralized worker-profile selections, and the lower row reports decentralized node activations. The two MMLU-Pro columns distinguish the BBH-trained and MMLU-Pro-trained retained states. The branches are therefore compared by profile composition rather than by absolute workload volume.
 BBHAIMEMMLU-Pro (BBH-trained)MMLU-Pro (MMLU-Pro- trained)R-Bench-TSMFR
d=0d=00.217_{{\color[rgb]{1,0,0}\scalebox{0.60}{-0.017}}}^{{\color[rgb]{0,0.5,0}\scalebox{0.60}{+0.013}}}0.619_{{\color[rgb]{1,0,0}\scalebox{0.60}{-0.019}}}^{{\color[rgb]{0,0.5,0}\scalebox{0.60}{+0.037}}}0.662_{{\color[rgb]{1,0,0}\scalebox{0.60}{-0.007}}}^{{\color[rgb]{0,0.5,0}\scalebox{0.60}{+0.004}}}0.705_{{\color[rgb]{1,0,0}\scalebox{0.60}{-0.003}}}^{{\color[rgb]{0,0.5,0}\scalebox{0.60}{+0.003}}}0.558_{{\color[rgb]{1,0,0}\scalebox{0.60}{-0.052}}}^{{\color[rgb]{0,0.5,0}\scalebox{0.60}{+0.026}}}0.243_{{\color[rgb]{1,0,0}\scalebox{0.60}{-0.014}}}^{{\color[rgb]{0,0.5,0}\scalebox{0.60}{+0.007}}}
d=0.0001d=0.00010.750_{{\color[rgb]{1,0,0}\scalebox{0.60}{-0.010}}}^{{\color[rgb]{0,0.5,0}\scalebox{0.60}{+0.010}}}0.600_{{\color[rgb]{1,0,0}\scalebox{0.60}{-0.022}}}^{{\color[rgb]{0,0.5,0}\scalebox{0.60}{+0.022}}}0.824_{{\color[rgb]{1,0,0}\scalebox{0.60}{-0.009}}}^{{\color[rgb]{0,0.5,0}\scalebox{0.60}{+0.009}}}0.839_{{\color[rgb]{1,0,0}\scalebox{0.60}{-0.018}}}^{{\color[rgb]{0,0.5,0}\scalebox{0.60}{+0.018}}}0.633_{{\color[rgb]{1,0,0}\scalebox{0.60}{-0.026}}}^{{\color[rgb]{0,0.5,0}\scalebox{0.60}{+0.022}}}\mathbf{0.411}_{{\color[rgb]{1,0,0}\scalebox{0.60}{-0.036}}}^{{\color[rgb]{0,0.5,0}\scalebox{0.60}{+0.036}}}
d=0.1d=0.10.807_{{\color[rgb]{1,0,0}\scalebox{0.60}{-0.007}}}^{{\color[rgb]{0,0.5,0}\scalebox{0.60}{+0.003}}}0.611_{{\color[rgb]{1,0,0}\scalebox{0.60}{-0.033}}}^{{\color[rgb]{0,0.5,0}\scalebox{0.60}{+0.022}}}0.832_{{\color[rgb]{1,0,0}\scalebox{0.60}{-0.010}}}^{{\color[rgb]{0,0.5,0}\scalebox{0.60}{+0.013}}}0.827_{{\color[rgb]{1,0,0}\scalebox{0.60}{-0.024}}}^{{\color[rgb]{0,0.5,0}\scalebox{0.60}{+0.024}}}0.690_{{\color[rgb]{1,0,0}\scalebox{0.60}{-0.012}}}^{{\color[rgb]{0,0.5,0}\scalebox{0.60}{+0.006}}}0.378_{{\color[rgb]{1,0,0}\scalebox{0.60}{-0.056}}}^{{\color[rgb]{0,0.5,0}\scalebox{0.60}{+0.069}}}
d=0.2d=0.20.813_{{\color[rgb]{1,0,0}\scalebox{0.60}{-0.003}}}^{{\color[rgb]{0,0.5,0}\scalebox{0.60}{+0.007}}}0.607_{{\color[rgb]{1,0,0}\scalebox{0.60}{-0.030}}}^{{\color[rgb]{0,0.5,0}\scalebox{0.60}{+0.015}}}0.835_{{\color[rgb]{1,0,0}\scalebox{0.60}{-0.013}}}^{{\color[rgb]{0,0.5,0}\scalebox{0.60}{+0.010}}}0.842_{{\color[rgb]{1,0,0}\scalebox{0.60}{-0.015}}}^{{\color[rgb]{0,0.5,0}\scalebox{0.60}{+0.015}}}0.704_{{\color[rgb]{1,0,0}\scalebox{0.60}{-0.014}}}^{{\color[rgb]{0,0.5,0}\scalebox{0.60}{+0.022}}}0.396_{{\color[rgb]{1,0,0}\scalebox{0.60}{-0.010}}}^{{\color[rgb]{0,0.5,0}\scalebox{0.60}{+0.010}}}
d=0.25d=0.250.815_{{\color[rgb]{1,0,0}\scalebox{0.60}{-0.005}}}^{{\color[rgb]{0,0.5,0}\scalebox{0.60}{+0.005}}}0.594_{{\color[rgb]{1,0,0}\scalebox{0.60}{-0.028}}}^{{\color[rgb]{0,0.5,0}\scalebox{0.60}{+0.028}}}0.838_{{\color[rgb]{1,0,0}\scalebox{0.60}{-0.010}}}^{{\color[rgb]{0,0.5,0}\scalebox{0.60}{+0.007}}}\mathbf{0.848}_{{\color[rgb]{1,0,0}\scalebox{0.60}{-0.009}}}^{{\color[rgb]{0,0.5,0}\scalebox{0.60}{+0.009}}}0.692_{{\color[rgb]{1,0,0}\scalebox{0.60}{-0.014}}}^{{\color[rgb]{0,0.5,0}\scalebox{0.60}{+0.010}}}0.375_{{\color[rgb]{1,0,0}\scalebox{0.60}{-0.052}}}^{{\color[rgb]{0,0.5,0}\scalebox{0.60}{+0.042}}}
d=0.4d=0.40.810_{{\color[rgb]{1,0,0}\scalebox{0.60}{-0.040}}}^{{\color[rgb]{0,0.5,0}\scalebox{0.60}{+0.060}}}0.596_{{\color[rgb]{1,0,0}\scalebox{0.60}{-0.019}}}^{{\color[rgb]{0,0.5,0}\scalebox{0.60}{+0.015}}}0.842_{{\color[rgb]{1,0,0}\scalebox{0.60}{-0.003}}}^{{\color[rgb]{0,0.5,0}\scalebox{0.60}{+0.003}}}0.839_{{\color[rgb]{1,0,0}\scalebox{0.60}{-0.006}}}^{{\color[rgb]{0,0.5,0}\scalebox{0.60}{+0.006}}}0.696_{{\color[rgb]{1,0,0}\scalebox{0.60}{-0.006}}}^{{\color[rgb]{0,0.5,0}\scalebox{0.60}{+0.006}}}0.399_{{\color[rgb]{1,0,0}\scalebox{0.60}{-0.014}}}^{{\color[rgb]{0,0.5,0}\scalebox{0.60}{+0.017}}}
Table 1: Accuracy across the canonical 15 DoF conditions. Cell deviations are observed run ranges around the mean, not confidence intervals; red is mean-to-min and green is max-to-mean.
Appendix figures & tables2 assets

Supplementary material from the paper’s appendix.

Appendix

SubjectTrainEval.SubjectTrainEval.
Math6719Physics6518
Chemistry5716Law5515
Engineering4814Other4613
Economics4212Health4111
Psychology4011Business3911
Biology3610Philosophy257
Appendix
Table 3: Subject-proportional MMLU-Pro sampling quotas.
Figure 5: Target and realized decentralized task-unit assignment fractions across repeated DoF sweeps. Each panel represents one evaluation collection and retained-state setting. Small points show individual runs, large points show condition means, and the diagonal denotes exact agreement with the target q(d)q(d) . Marker shape distinguishes atomic-target and subtask-target conditions. Endpoint conditions are excluded because they bypass probabilistic per-unit routing.
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