Alessio Borgi

Alessio Borgi

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

Hodge Decomposition for Graph Signals: Curl, Gradient, and Harmonic Flows

6 minute read

Published:

TL;DR: Any flow y on the edges of a simplicial complex decomposes as y = δ₀x + δ₁ᵀz + h where x is a node potential (gradient flow), z is a face potential (curl/rotational flow), and h is harmonic (non-exact, non-co-exact — topologically non-trivial). For graph sheaves, the decomposition uses the sheaf coboundary δ₀ and gives: node signals → sheaf gradient flows on edges; edge signals → sheaf-harmonic components; these components separate topological from geometric structure in graph data.

The Classical Hodge Decomposition

Let G be a simplicial complex (graph + triangles + higher cells). The Hodge decomposition states that for any edge flow y ∈ C¹(G) = ℝ^E:

y = δ₀ x + δ₁ᵀ z + h

where:

  • δ₀ x ∈ im(δ₀): the gradient component — y is the “gradient” of some node potential x ∈ ℝ^N
  • δ₁ᵀ z ∈ im(δ₁ᵀ): the curl component — y circulates around triangles (faces) with “face potential” z ∈ ℝ^T
  • h ∈ ker(δ₀ᵀ) ∩ ker(δ₁): the harmonic component — neither a gradient nor a curl

The three components are orthogonal and their sum is unique. This is the graph version of the classical Helmholtz decomposition for vector fields.

The Three Operators

For a graph G = (V, E, T) (with optional triangles T):

Node-edge coboundary δ₀ : C⁰ → C¹:

(δ₀ x)_e = x_v − x_u (for oriented edge e = (u→v))

Edge-triangle coboundary δ₁ : C¹ → C²:

(δ₁ y)_t = y_{ab} + y_{bc} − y_{ac} (for triangle t = (a,b,c) with given orientation)

Hodge Laplacians:

L₀ = δ₀ᵀ δ₀ (node Laplacian = standard graph Laplacian) L₁ = δ₀ δ₀ᵀ + δ₁ᵀ δ₁ (edge Hodge Laplacian) L₂ = δ₁ δ₁ᵀ (face Laplacian)

The harmonic space for edges is ker(L₁) = ker(δ₀ᵀ) ∩ ker(δ₁) — the null space of the edge Hodge Laplacian L₁.

Hodge Decomposition for Sheaves

For a cellular sheaf F on a graph (no triangles, so δ₁ = 0):

C¹ = im(δ₀) ⊕ ker(δ₀ᵀ) y = δ₀ x + h

where:

  • δ₀ x: gradient component (edge disagreements attributable to node assignments)
  • h ∈ ker(δ₀ᵀ): harmonic component (not attributable to any node assignment)

This is the sheaf Hodge decomposition for 0-1 cochain complexes.

The harmonic space ker(δ₀ᵀ) = ker(Δ₁_F) where Δ₁_F = δ₀ δ₀ᵀ is the down-Laplacian on sheaf 1-cochains.

What the Decomposition Reveals

For a flow y on the edges (e.g., traffic flow, communication, energy transfer):

Gradient component δ₀ x: the flow is driven by a “pressure gradient” x at the nodes — like water flowing from high to low pressure. This is conservative (no cycles, follows the gradient of a scalar field). In the sheaf setting, x is a 0-cochain (node stalk assignment) and δ₀ x measures the disagreement induced by x.

Harmonic component h: the flow has no pressure source — it circulates “around topological holes” in the graph. For a graph with g independent cycles, the harmonic space has dimension g (one harmonic mode per cycle).

For graph learning: The gradient component of an edge signal contains no topological information — it can be "explained" by node potentials. The harmonic component captures the genuinely topological structure of the flow, which cannot be recovered from nodes alone. Sheaf GNNs that process edge stalks can learn from harmonic components that node-only models miss.

Application: Learning on Edge Flows

Many real-world datasets have natural edge signals rather than node signals:

  • Traffic networks: flow on roads (edges), not at intersections (nodes)
  • Financial networks: transaction amounts on edges
  • Communication networks: bandwidth on links
  • Electrical circuits: current on wires

For edge-signal datasets, the appropriate model uses L₁ = δ₀ δ₀ᵀ (down-Laplacian) rather than L₀ = δ₀ᵀ δ₀ (node Laplacian).

Edge-level sheaf diffusion:

dY/dt = −Δ₁_F Y where Δ₁_F = δ₀ δ₀ᵀ ∈ ℝ^{Ed × Ed}

This diffuses edge-stalk signals using the down-Laplacian — the sheaf generalisation of the edge Hodge Laplacian.

Simplicial Sheaf Complexes

On a simplicial complex K (graph + triangles + tetrahedra + …), one can define cellular sheaves on all cells, giving a full sheaf cochain complex:

0 → C⁰(K,F) →^{δ₀} C¹(K,F) →^{δ₁} C²(K,F) → ...

The Hodge Laplacians at each level become sheaf Hodge Laplacians:

Δ^k_F = δ_{k-1} δ_{k-1}ᵀ + δ_k^ᵀ δ_k

The k-th cohomology H^k(K, F) measures the k-dimensional “holes” in the sheaf — generalising both H⁰ (global sections) and H¹ (edge-level obstructions).

Connection to Topological Deep Learning

Bodnar et al. (2021) introduced CW Networks and Cellular Isomorphism Networks — GNNs that operate on CW complexes (which subsume simplicial complexes). These use message passing across cells of different dimensions, analogous to sheaf diffusion across different cochain levels.

The connection: a sheaf on a CW complex provides the restriction maps between cells of adjacent dimension. CW-Net message passing is sheaf diffusion on the CW complex sheaf.

Giusti et al. (2023) formalise this connection as Topological Deep Learning — a unified framework where:

  • Node signals → C⁰ sheaf cochains
  • Edge signals → C¹ sheaf cochains
  • Triangle signals → C² sheaf cochains
  • Hodge Laplacians at each level govern diffusion

Sheaf Hodge Decomposition for Node Classification

For node signals (which is what standard sheaf GNNs process), the Hodge decomposition decomposes the input X₀ ∈ ℝ^{Nd} as:

X₀ = X₀^{harm} + X₀^{grad}

where X₀^{harm} ∈ ker(Δ_F) = H⁰ and X₀^{grad} ∈ im(Δ_F).

Sheaf diffusion acts as:

  • X₀^{harm} preserved (in null space)
  • X₀^{grad} attenuated (gradient component decays)

The harmonic component X₀^{harm} is the “globally consistent” part of the input — already a global section. The gradient component X₀^{grad} is the “inconsistent” part — diffusion tries to resolve it toward consistency.

For heterophily: the optimal node features for classification may be in X₀^{grad} for some graphs and X₀^{harm} for others. NSD with general maps can adapt the null space to make the task-optimal features harmonic.

References

  • Jiang, X., Lim, L.-H., Yao, Y., & Ye, Y. (2011). Statistical Ranking and Combinatorial Hodge Theory. Mathematical Programming 2011 (the graph Hodge decomposition applied to ranking and preference data — foundational for edge-flow analysis on graphs).
  • Bodnar, C., Frasca, F., Wang, Y. G., Otter, N., Montufar, G. F., Liò, P., & Bronstein, M. M. (2021). Weisfeiler and Lehman Go Topological: Message Passing Simplicial Networks. ICML 2021 (MPSN: message passing on simplicial complexes — the precursor to sheaf-on-simplicial-complexes models).
  • Giusti, G., Battiloro, C., Testa, L., Di Lorenzo, P., Sardellitti, S., & Barbarossa, S. (2023). Cell Attention Networks. arXiv 2023 (CAN: attention on cellular complexes, related to sheaf attention on higher-order structures).

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