Sheaf Laplacian Coordination

---
name: sheaf-laplacian-coordination
description: Sheaf neural network coordination via graph Laplacians for distributed
  consensus and harmonic inference. Use when coordinating multi-agent systems, building
  sheaf-aware GNNs, or implementing distributed consensus protocols.
metadata:
  trit: 0
  color: '#26D826'
---

# Sheaf Laplacian Coordination

**Trit**: 0 (ERGODIC - coordinator)
**Color**: Green (#26D826)

## Overview

Implements sheaf neural network coordination using graph Laplacians for:
- Distributed consensus via sheaf diffusion
- Harmonic extension/restriction operators
- Spectral clustering on sheaf sections
- Multi-agent coordination with vector space representations

## Key Papers

- [Sheaf Neural Networks](https://arxiv.org/abs/2012.06333) - Hansen & Gebhart 2020
- [Neural Sheaf Diffusion](https://arxiv.org/abs/2202.04579) - Bodnar et al. 2022
- [Cooperative Sheaf Neural Networks](https://arxiv.org/abs/2507.00647) - Ribeiro et al. 2025
- [Sheaf Diffusion Goes Nonlinear](https://proceedings.mlr.press/v251/zaghen24a.html) - Zaghen et al. 2024

## Core Concepts

### Sheaf Laplacian

The sheaf Laplacian generalizes the graph Laplacian by associating vector spaces to nodes and linear maps to edges:

```latex
L_F = D^\top D

where D is the coboundary operator:
(Df)_e = F_{e,t} f_t - F_{e,s} f_s

F_{e,v} : F(v) → F(e)  (restriction maps)
```

### Diffusion Process

Sheaf diffusion for consensus:

```latex
\frac{dx}{dt} = -L_F x

At equilibrium: L_F x = 0 (harmonic sections)
```

### In/Out Degree Laplacians (Cooperative SNNs)

For directed graphs with cooperative behavior:

```latex
L_{in} = D_{in}^\top D_{in}   (gathering information)
L_{out} = D_{out}^\top D_{out} (conveying information)
```

## API

### Python Implementation

```python
import torch
import torch.nn as nn

class SheafLaplacian(nn.Module):
    """Learnable sheaf Laplacian for graph coordination."""
    
    def __init__(self, num_nodes, stalk_dim, edge_index):
        super().__init__()
        self.num_nodes = num_nodes
        self.stalk_dim = stalk_dim
        self.edge_index = edge_index
        
        # Learnable restriction maps F_{e,v}
        num_edges = edge_index.shape[1]
        self.restriction_maps = nn.Parameter(
            torch.randn(num_edges, 2, stalk_dim, stalk_dim)
        )
    
    def build_laplacian(self):
        """Construct sheaf Laplacian from restriction maps."""
        L = torch.zeros(
            self.num_nodes * self.stalk_dim,
            self.num_nodes * self.stalk_dim
        )
        
        for e, (s, t) in enumerate(self.edge_index.T):
            F_es = self.restriction_maps[e, 0]  # Source restriction
            F_et = self.restriction_maps[e, 1]  # Target restriction
            
            # Add edge contribution to Laplacian
            # L[s,s] += F_es^T F_es, L[t,t] += F_et^T F_et
            # L[s,t] -= F_es^T F_et, L[t,s] -= F_et^T F_es
            
        return L
    
    def diffuse(self, x, steps=10, dt=0.1):
        """Run sheaf diffusion for consensus."""
        L = self.build_laplacian()
        for _ in range(steps):
            x = x - dt * (L @ x)
        return x
    
    def harmonic_extension(self, boundary_values, boundary_mask):
        """Extend boundary values harmonically."""
        L = self.build_laplacian()
        # Solve L_interior x_interior = -L_boundary x_boundary
        return solve_harmonic(L, boundary_values, boundary_mask)


class CooperativeSheafNN(nn.Module):
    """Cooperative SNN with in/out degree control."""
    
    def __init__(self, in_dim, hidden_dim, out_dim, edge_index):
        super().__init__()
        self.sheaf = SheafLaplacian(
            num_nodes=edge_index.max() + 1,
            stalk_dim=hidden_dim,
            edge_index=edge_index
        )
        self.encoder = nn.Linear(in_dim, hidden_dim)
        self.decoder = nn.Linear(hidden_dim, out_dim)
        
        # Cooperative gates: control gather vs convey
        self.gather_gate = nn.Parameter(torch.ones(1))
        self.convey_gate = nn.Parameter(torch.ones(1))
    
    def forward(self, x, edge_index):
        h = self.encoder(x)
        
        # Cooperative diffusion
        h_diffused = self.sheaf.diffuse(h)
        
        # Apply cooperative gates
        h_out = self.gather_gate * h + self.convey_gate * h_diffused
        
        return self.decoder(h_out)
```

### Julia Implementation (ACSets)

```julia
using Catlab, Catlab.CategoricalAlgebra

# Define sheaf schema
@present SchSheaf(FreeSchema) begin
    V::Ob  # Vertices (nodes)
    E::Ob  # Edges
    src::Hom(E, V)
    tgt::Hom(E, V)
    
    # Stalks as vector spaces
    F_V::AttrType  # Stalk at vertex
    F_E::AttrType  # Stalk at edge
    stalk_v::Attr(V, F_V)
    stalk_e::Attr(E, F_E)
end

@acset_type SheafGraph(SchSheaf)

function build_sheaf_laplacian(sg::SheafGraph, restrictions)
    """Build sheaf Laplacian from restriction maps."""
    n = nparts(sg, :V)
    d = size(restrictions[1], 1)  # Stalk dimension
    
    L = zeros(n * d, n * d)
    
    for e in parts(sg, :E)
        s = sg[e, :src]
        t = sg[e, :tgt]
        F_s, F_t = restrictions[e]
        
        # Add contributions
        si, ti = (s-1)*d+1:s*d, (t-1)*d+1:t*d
        L[si, si] += F_s' * F_s
        L[ti, ti] += F_t' * F_t
        L[si, ti] -= F_s' * F_t
        L[ti, si] -= F_t' * F_s
    end
    
    return L
end

function sheaf_diffusion(L, x0; steps=100, dt=0.01)
    """Run sheaf diffusion to reach harmonic section."""
    x = copy(x0)
    for _ in 1:steps
        x = x - dt * (L * x)
    end
    return x
end
```

## GF(3) Triads

This skill participates in balanced triads:

```
sheaf-cohomology (-1) ⊗ sheaf-laplacian-coordination (0) ⊗ forward-forward-learning (+1) = 0 ✓
persistent-homology (-1) ⊗ sheaf-laplacian-coordination (0) ⊗ gay-mcp (+1) = 0 ✓
```

## Use Cases

### Multi-Agent Consensus

```python
# Agents reach consensus via sheaf diffusion
agents = SheafLaplacian(num_agents=5, stalk_dim=16, topology=ring_graph)
initial_beliefs = torch.randn(5, 16)
consensus = agents.diffuse(initial_beliefs, steps=50)
# All agents now have aligned beliefs in harmonic section
```

### Heterophilic GNN

```python
# Handle graphs where connected nodes have different labels
model = CooperativeSheafNN(in_dim=32, hidden_dim=64, out_dim=10, edge_index=data.edge_index)
# Sheaf structure allows nodes to maintain distinct representations
# while still communicating through restriction maps
```

### Distributed Optimization

```python
# Decentralized optimization via sheaf Laplacian flow
def distributed_optimize(local_gradients, topology):
    sheaf = SheafLaplacian(topology)
    # Average gradients via harmonic extension
    global_gradient = sheaf.harmonic_extension(local_gradients)
    return global_gradient
```

## Integration with Music-Topos

```clojure
;; In parallel_color_fork.clj
(defn sheaf-coordinate-forks
  "Coordinate parallel color forks via sheaf diffusion"
  [forks topology]
  (let [sheaf (build-sheaf-laplacian forks topology)
        consensus (sheaf-diffuse sheaf (map :color forks))]
    (mapv #(assoc %1 :coordinated-color %2) forks consensus)))
```

## See Also

- `sheaf-cohomology` - Čech cohomology for local-to-global verification
- `open-games` - Compositional game theory for agent coordination
- `acsets-algebraic-databases` - Functorial databases underlying sheaf structure
- `forward-forward-learning` - Local learning complementing sheaf diffusion

## References

```bibtex
@article{hansen2020sheaf,
  title={Sheaf Neural Networks},
  author={Hansen, Jakob and Gebhart, Thomas},
  journal={arXiv:2012.06333},
  year={2020}
}

@article{bodnar2022neural,
  title={Neural Sheaf Diffusion},
  author={Bodnar, Cristian and Di Giovanni, Francesco and others},
  journal={arXiv:2202.04579},
  year={2022}
}

@article{ribeiro2025cooperative,
  title={Cooperative Sheaf Neural Networks},
  author={Ribeiro, André and others},
  journal={arXiv:2507.00647},
  year={2025}
}
```