HodgeNet: Neural Architectures via Hodge Theory

• HodgeNet is a neural architecture suite that integrates combinatorial Hodge theory and simplicial signal processing to capture topological and geometric features from nodes, edges, and higher-order simplices.
• It leverages learned discrete Hodge Laplacians and spectral filtering via generalized Laguerre polynomials to enforce spatial locality and enhance representation learning.
• HodgeNet demonstrates superior performance across tasks such as flow data analysis, mesh processing, and brain connectomics, offering robust interpretability and scalability.

HodgeNet is a family of neural architectures that integrate combinatorial Hodge theory and graph/simplicial signal processing to enable learning from data supported on nodes, edges, or higher-order simplices. HodgeNet leverages discrete Hodge Laplacians as graph shift operators, providing inductive biases aligned with the topology and geometry of graphs, hypergraphs, and meshes. This approach is realized in multiple settings, including edge-centric GNNs for flow data (Roddenberry et al., 2019), spectral geometric pipelines for meshes (Smirnov et al., 2021), and heterogeneous GNNs for high-dimensional brain connectomics data (Huang et al., 2023). The core principle is to exploit Hodge Laplacian operators defined on $k$-simplices, enabling native handling of node, edge, or higher-order features and enforcing topologically meaningful structure in learned representations.

1. Mathematical Foundations: Hodge Laplacians and Simplicial Structures

The k-th Hodge Laplacian, $\Delta_k$, is defined for a simplicial complex $G$, with boundary operators $d_k : C_k \to C_{k-1}$ encoding incidence among $k$- and $(k-1)$-simplices. The Hodge Laplacian generalizes the standard graph Laplacian ($k=0$) and enables operators for edge and higher-order (e.g., triangle) signals: $\Delta_k = d_k^T d_k + d_{k-1} d_{k-1}^T$ Key specializations include:

• Node-Laplacian ($k=0$): $\Delta_0 = \partial_1 \partial_1^T$ (standard Laplacian).
• Edge-Laplacian ($k=1$): $$\Delta_1 = \partial_2 \partial_2^T + \partial_1^T \partial_1$$ (Huang et al., 2023, Roddenberry et al., 2019).

On graphs, $d_0$ is the node-edge incidence matrix, and higher-order $d_1$ encodes edge-triangle incidence. In discrete exterior calculus (DEC), these structures support a rich hierarchy of Laplacian shifts aligned with geometric and topological properties (Smirnov et al., 2021).

2. HodgeNet Architectures: Signal Types and Operator Parameterization

2.1 Edge-Centric GNNs for Flow Data

Classical GNNs perform convolutions on node data via powers or polynomials of the adjacency or Laplacian. HodgeNet generalizes to edge signal (flow) data, where signals $f \in \mathbb{R}^E$ exhibit antisymmetry ($f(i,j) = -f(j,i)$). This requires operators that capture both gradient (potential) and cycle (conservative) components: $L_1 = B_0^T B_0 + B_1 B_1^T$ On simple graphs (no triangles), $B_1 = 0$, yielding $L_1 = B_0^T B_0$ (Roddenberry et al., 2019).

2.2 Mesh Spectral Geometry via Learnable DEC

For 3D meshes, HodgeNet replaces fixed cotangent or barycentric weights with parameterized Hodge star matrices whose diagonal entries are learned functions (e.g., via MLPs) of local geometry: $$L = \star_0^{-1} d_0^T \star_1 d_0 \quad M = \star_0$$ Spectral features are constructed from low-order eigenpairs and used as per-vertex, per-face, or global mesh descriptors (Smirnov et al., 2021).

2.3 Heterogeneous/Multilevel Brain Graph GNNs

The HL-HGCNN architecture incorporates node-level, edge-level, and joint node/edge convolutions with spectral filters on $\Delta_0$ and $\Delta_1$, combined with temporal convolutions and pooling (TGPool). Learned Laguerre polynomial expansions of $\Delta_k$ define spectral filterbanks, with receptive field controlled by polynomial order (Huang et al., 2023).

3. Spectral Filters and Localization: Laguerre Polynomial Approximations

HodgeNet employs generalized Laguerre polynomials for scalable, spatially-local spectral filtering: $$g(\Delta_k) \approx \sum_{m=0}^M \theta_m L_m^\alpha (\Delta_k / \lambda_{\max})$$ where $L_m^\alpha(x)$ is the general Laguerre polynomial, $\theta_m$ learnable coefficients, and $\lambda_{\max}$ the maximal eigenvalue of $\Delta_k$. The order $M$ of the polynomial controls the receptive field, restricting nontrivial coefficients of $g(\Delta_k)$ to $k$-simplices within $M$ hops of each other.

This design ensures spatial locality while circumventing the need to compute full eigendecompositions, supporting large, heterogeneous domains such as brain graphs (Huang et al., 2023).

4. Pooling and Hierarchical Coarsening: TGPool and Efficient Backpropagation

HodgeNet introduces topological graph pooling (TGPool) to coarsen $k$-simplicial structures by multilevel clustering (e.g., Graclus), aggregating features and updating boundary operators in a manner topologically consistent with the original complex.

TGPool pseudocode proceeds as follows:

1. Build $A_k$ adjacency of $k$-simplices.
2. Cluster via normalized cut, pairing or singleton padding.
3. Construct binary tree structure.
4. Pool features by averaging or max.
5. Update simplicial complex and recompute boundary operators.
6. Recompute $\Delta_k$ on the pooled structure.

For spectral mesh learning, HodgeNet enables gradient backpropagation through partial eigendecompositions of sparse operators by analytic derivatives with respect to Hodge star entries, leveraging eigenpair identities to avoid full matrix computation (Smirnov et al., 2021).

5. Empirical Performance and Interpretability

5.1 Functional Brain Graphs and IQ Prediction

On ABCD resting-state fMRI (7,693 subjects, 268-ROI graphs), HL-HGCNN outperforms node-based GNNs (GAT, dGCN, BrainGNN, BrainNetCNN, Hypergraph NN) in predicting general intelligence (IQ). RMSEs: | Method | RMSE (mean±std) | |----------------|----------------| | HL-Node | 7.134 ± 0.011 | | HL-Edge | 7.009 ± 0.012 | | HL-HGCNN | 6.972 ± 0.015 | | GAT | 7.165 ± 0.020 | | BrainGNN | 7.144 ± 0.013 | | dGCN | 7.151 ± 0.012 | | BrainNetCNN | 7.118 ± 0.016 | | Hypergraph NN | 7.051 ± 0.022 |

Improvements for HL-HGCNN are statistically significant ($p < 0.03$). Grad-CAM–style saliency on learned edge features reveals that the most significant connections for intelligence prediction link occipital, prefrontal, parietal, salience, and temporal networks, consistent with P-FIT (Huang et al., 2023).

5.2 Edge Signal Learning and Flow Tasks

For edge-supported tasks on graphs (traffic flow interpolation, community source localization), HodgeNet demonstrably outperforms line-graph and node-based GNNs, both in predictive accuracy and convergence speed. In source localization, Hodge-AGG reaches ≈85% accuracy, substantially exceeding line-graph (≈65%) and nodespace (≈70%) counterparts (Roddenberry et al., 2019).

5.3 Mesh Analysis Benchmarks

On shape segmentation/classification (COSEG, SHREC ’11, MIT animation), HodgeNet achieves competitive or superior accuracy (e.g., >99% on SHREC ’11 split 16) and robust performance on high-resolution meshes, generalizing across mesh densities without decimation. For mesh regression tasks (e.g., dihedral angle recovery), mean error is as low as 0.17° (Smirnov et al., 2021).

6. Interpretations, Limitations, and Extensions

HodgeNet architectures provide topologically faithful operations for signals defined over arbitrary $k$-simplices, supporting not just standard node-based filtering, but true flow and higher-order feature learning. The framework extends to:

• Arbitrary simplicial complexes (supporting triangle/tetrahedral data),
• Learnable DEC operators (via parameterized Hodge stars),
• Hierarchical coarsening aligned to topological structure,
• Efficient partial eigendecomposition for global spectral features (Smirnov et al., 2021, Huang et al., 2023).

Limitations include computational cost for high-order Laplacians and, in edge-centric architectures, absence of direct higher-order (e.g., triangle) shift operator implementations. Extension to volumetric or higher-form operators is plausible, with applications suggested in geometry processing and generalized graph/simplicial learning (Smirnov et al., 2021). Advanced GNN variants (attention, residuals, graph transformers) remain open for integration within this topological framework (Roddenberry et al., 2019).

7. Significance and Research Directions

HodgeNet bridges deep learning for graphs and meshes with combinatorial topology and geometry, introducing operator-centric architectures that respect the orientation, conservation, and underlying structure of data. The use of Hodge Laplacians and DEC-inspired parameterizations establishes a rigorous analytical connection between topology, geometry, and representation learning. Expected future directions include:

• Higher-order flow/curvature learning on arbitrary discretizations,
• Scalable mesh and manifold learning via DEC-based operators,
• Interpretability and visualization of learned topological features in neuroimaging and physics,
• Exploration of optimization and architectural variants for further flexibility and computational efficiency (Smirnov et al., 2021, Huang et al., 2023, Roddenberry et al., 2019).