Sheaf Theoretic Coordination

Sheaf-Theoretic Coordination

Category: Phase 3 Core - Distributed Reasoning Status: Skeleton Implementation Dependencies: oriented-simplicial-networks, categorical-composition

Overview

Implements sheaf-theoretic coordination mechanisms for multi-agent systems, using sheaf Laplacians for consensus, harmonic extension for inference, and cohomology for detecting global obstructions.

Capabilities

• Sheaf Laplacian: Consensus dynamics on cellular sheaves
• Harmonic Extension: Infer missing data via global consistency
• Cohomology Detection: Identify obstructions to global agreement
• Sheaf Neural Networks: Learn sheaf structures from data

Core Components

1. Cellular Sheaf Builder (cellular_sheaf.jl)

   • Construct sheaves over cell complexes
   • Define restriction maps between stalks
   • Compute sheaf cohomology groups

2. Sheaf Laplacian (sheaf_laplacian.jl)

   • Weighted Laplacian on sheaf sections
   • Consensus dynamics and heat flow
   • Spectral analysis for convergence

3. Harmonic Extension (harmonic_extension.jl)

   • Solve for globally consistent assignments
   • Handle partial observations
   • Regularized least-squares formulation

4. Sheaf Neural Networks (sheaf_nn.jl)

   • Learn restriction maps via gradient descent
   • Sheaf diffusion layers
   • Integration with geometric deep learning

Integration Points

• Input from: oriented-simplicial-networks (base simplicial complex)
• Output to: emergent-role-assignment (coordination constraints)
• Coordinates with: categorical-composition (sheaf functoriality)

Usage

using SheafTheoreticCoordination

# Build cellular sheaf over graph
graph = SimplexGraph(adjacency_matrix)
sheaf = CellularSheaf(graph, stalk_dim=3)

# Define restriction maps (can be learned)
for edge in edges(graph)
    sheaf.restrictions[edge] = random_orthogonal_matrix(3)
end

# Solve for harmonic extension (inference)
partial_observations = Dict(1 => [1.0, 0.0, 0.0], 5 => [0.0, 1.0, 0.0])
global_assignment = harmonic_extension(sheaf, partial_observations)

# Check for cohomological obstructions
obstruction = compute_obstruction_cocycle(sheaf, global_assignment)

References

• Hansen & Ghrist "Toward a Spectral Theory of Cellular Sheaves" (2019)
• Bodnar et al. "Sheaf Neural Networks" (ICLR 2022)
• Robinson "Topological Signal Processing" (2014)

Implementation Status

• Basic sheaf data structures
• Sheaf Laplacian construction
• Full cohomology computation
• Neural sheaf learning
• Multi-agent coordination demo