Summary
Multi-agent systems can behave differently even with the same agents and communication limits, due to how they work together on tasks. The authors introduce a mathematical framework called Topological Sheaves of Irreducibility (TSI) to identify whether the difficulty in completing a task comes from the task itself or the system's capabilities. They represent tasks and agent capacities using mathematical objects called sheaves and complexes, then use cohomology (a topological tool) to detect structural obstacles. They tested TSI on several models and found it can precisely tell when tasks can be decided or repaired, depending on the interaction levels among agents.
What this means in practice
- •For distributed system designers: Determine whether task failures stem from intrinsic task complexity or system interaction limits to improve system design and troubleshooting.
- •For robotics control teams: Use TSI to verify which multi-agent coordination tasks require higher-order interactions versus simpler pairwise communications for effective task planning.
Abstract
Multi-agent systems can differ under identical agents and communication budgets when they support different forms of joint interaction. We introduce Topological Sheaves of Irreducibility (TSI), a sheaf-theoretic framework separating task-level compatibility obstructions from system capability. A task is represented by an interaction complex $K$ and a capacity sheaf $\mathcal{F}$; nonzero $H^n(K;\mathcal{F})$ defines a structural signature, while registered interventions test capability. For cocycle data, cochain realization is solvable exactly when the represented cohomology class vanishes. Examples show that the signature is sheaf-relative, and the degree-1 reduction yields cycle-space witnesses and observation requirements. We evaluate two finite one-round models. Across 15,625 degree-1 inputs, an all-agent additive primitive solves decision and repair exactly; the optimal pairwise system reaches 88.80% on decision, one input above the 88.79% majority-class baseline, and 26.63% on repair. On a degree-2 tetrahedral-boundary task over 531,441 inputs, the same-2-skeleton system retains all four three-agent primitives yet matches the pairwise-only optimum throughout the registered separation window, whereas adding the class-filling four-agent primitive makes both tasks exact. Status-preserving robustness checks retain exact performance but are not cost-matched to the class-restoring ablation. Conditional on the registered primitives and cost models, TSI provides task-relative, falsifiable tests of higher-order interaction.