Alessio Borgi

Alessio Borgi

PhD in Graph Neural Networks and Generative AI 👨🏻‍💻🤖

Cosheaves: The Dual Perspective on Graph Data

5 minute read

Published:

TL;DR: A cosheaf assigns vector spaces to parts of a space but with extension maps (corestriction maps) going from smaller to larger: G_{e→v} : G(e) → G(v). Instead of asking "can we consistently restrict data from nodes to edges?", cosheaves ask "can we consistently extend data from edges to nodes?" The dual boundary operator ∂₀ = δ₀ᵀ gives rise to cosheaf homology H₀, H₁, ... — different from sheaf cohomology. Cosheaves are natural for data that aggregates (distributions, measures, flows) rather than restricts (sections, fields).

Sheaves vs Cosheaves: The Duality

Sheaf F: data flows from larger to smaller via restriction maps F_{v▷e} : F(v) → F(e). The question is whether global data on nodes is consistent when restricted to edges.

Cosheaf G: data flows from smaller to larger via corestriction maps G_{e▷v} : G(e) → G(v). The question is whether local data on edges can be consistently extended to nodes.

Formally, a cosheaf on a graph G assigns:

  • A vector space G(v) ≅ ℝ^d to each node v
  • A vector space G(e) ≅ ℝ^d to each edge e
  • A corestriction map G_{e▷v} : G(e) → G(v) for each incident pair (v, e)

The coboundary operator becomes a boundary operator:

∂₀ : C₁(G, G) → C₀(G, G) (∂₀ y)_v = Σ_{e incident to v} G_{e▷v} y_e · (orientation sign)

This is the transpose/dual of δ₀ — it maps edge-level data to node-level data by “extending” along corestriction maps.

Cosheaf Homology vs Sheaf Cohomology

Sheaf cohomology:

H⁰(G, F) = ker(δ₀) (global sections — consistent node assignments) H¹(G, F) = C¹ / im(δ₀) (obstruction to consistency)

Cosheaf homology:

H₀(G, G) = C₀ / im(∂₀) = C₀(G, G) / (node data attributable to edge data) H₁(G, G) = ker(∂₀) (edge cycles that map to zero at nodes)

The roles are reversed: H₀ of a cosheaf measures the “cokernel” (how much node space is NOT reachable from edge data), while H₁ measures consistent edge cycles.

For the constant cosheaf (G_{e▷v} = I): H₀ = ℝ^d (one component per connected component), H₁ = ℝ^{d · (E−V+1)} (one d-dimensional cycle per independent cycle of G).

The Cosheaf Laplacian

The cosheaf Laplacian is:

L_G = ∂₀ ∂₀ᵀ (acting on C₁ = edge space)

Note: this is the down-Laplacian Δ₁ = δ₀ δ₀ᵀ acting on edges, when the cosheaf maps are the transposes of the sheaf maps.

The node-level operator ∂₀ᵀ ∂₀ is the up-Laplacian on nodes — the adjoint of the cosheaf Laplacian.

The Sheaf-Cosheaf Duality

For any sheaf F on a graph G, the dual cosheaf F* is defined by:

  • F(v) = F(v), F(e) = F(e) (dual vector spaces)
  • G_{e▷v} = (F_{v▷e})ᵀ (transpose of restriction maps)

Under this duality:

  • Sheaf cohomology H(G, F) ↔ Cosheaf homology H(G, F*)
  • The sheaf Laplacian Δ_F ↔ The cosheaf Laplacian of F*

For finite-dimensional vector spaces over ℝ, F ≅ F* (via the standard inner product), so sheaves and cosheaves are “the same” in the vector space setting — the distinction is in how the maps are oriented.

When Cosheaves Are Natural

Sheaves are natural when: data is a field or section — something you observe at each point and which restricts consistently to smaller regions. Examples: temperature readings (restrict to subset of sensors), functions on a manifold (restrict to submanifold).

Cosheaves are natural when: data is a distribution or measure — something you aggregate from smaller regions to larger ones. Examples: flow on roads (aggregate from roads to intersections), probability distributions (extend from local regions to global), signals that integrate over regions.

Graph learning implication: For traffic flow prediction, the natural object is a cosheaf (flows on edges extend to intersections), not a sheaf. For node feature propagation, the natural object is a sheaf (node features restrict to edges). Most sheaf GNN papers focus on the sheaf setting (node features), but cosheaf GNNs may be more appropriate for edge-flow datasets.

Sheaf-Cosheaf Pairs in Neural Networks

A more expressive framework uses both sheaves and cosheaves simultaneously:

  • A sheaf F (restriction maps) to encode node-to-edge relationships
  • A cosheaf G (corestriction maps) to encode edge-to-node relationships
  • Combined diffusion using both Δ_F and L_G

This gives a bipartite message-passing scheme where information flows down (node → edge via F) and up (edge → node via G) independently, with different maps for each direction.

Relation to HetGNN: Heterogeneous GNNs with separate message functions for each edge direction (in vs out) are a special case of sheaf-cosheaf pairs.

Computing Cosheaf Homology for Graph Learning

The key quantity for cosheaf-based GNNs is the null space of the cosheaf Laplacian ker(∂₀ᵀ ∂₀) = ker(∂₀ᵀ) — the “cosection” space of nodes whose data is orthogonal to all edge extensions.

For the constant cosheaf: ker(∂₀ᵀ) = ker(δ₀) = span{1_N} — constant node assignments. This is the same as the standard GCN null space.

For non-trivial cosheaf maps: ker(∂₀ᵀ) can be larger or smaller — analogous to the sheaf null space. Cosheaf GNNs can avoid oversmoothing via the same null space argument as sheaf GNNs.

References

You May Also Enjoy

Open Problems and Future Directions in Sheaf Neural Networks

5 minute read

Published:

Sheaf Neural Networks are a young field with many open research problems. This post surveys the most important unsolved questions: scalability to million-node graphs, stochastic sheaves for uncertainty quantification, sheaf autoencoders for generative models, connections to persistent homology, and unification with other topological deep learning frameworks.

Sheaf GNNs for Molecular Property Prediction

7 minute read

Published:

Molecular graphs are naturally heterophilic — atoms of different types bonded together — and have rich geometric structure (bond angles, torsion angles). Sheaf GNNs can encode this structure via restriction maps that represent the relational geometry between bonded atoms. This post explores how sheaf theory applies to molecular property prediction.

Sheaf GNNs for Heterophilic Node Classification: Benchmarks and Results

6 minute read

Published:

A systematic comparison of sheaf GNNs against baselines on standard heterophilic node classification benchmarks. Which datasets benefit most from sheaf structure? How does stalk dimension d affect performance? When do sheaves win — and when don’t they?

Sheaves Meet Attention: Transformer-Inspired Sheaf Models

6 minute read

Published:

Graph Transformers use global attention to capture long-range dependencies that local message passing misses. Sheaf Transformers combine sheaf-based local aggregation (with learnable relational geometry) with global attention (for long-range dependencies). This post surveys the design space of sheaf-augmented attention models.