HOPSE: Scalable Higher-Order Positional and Structural Encoder for Combinatorial Representations

TL;DR

This paper tackles the limitation of graph neural networks in capturing higher-order interactions by introducing HOPSE, a scalable higher-order positional and structural encoder that eliminates higher-order message passing. By decomposing arbitrary higher-order domains with strictly augmented Hasse graphs and applying graph-based positional/structural encodings, HOPSE achieves linear-time preprocessing and competitive or superior performance on both graph and purely topological tasks. The authors present two realizations, HOPSE-M with handcrafted encodings and HOPSE-G with a pretrained GPSE, showing up to $7\times$ speedups over HOMP-based models while maintaining performance on standard benchmarks and excelling on topology-focused tasks. The work suggests that decoupling structure from attributes and leveraging Hasse-graph decompositions can substantially improve scalability in Topological Deep Learning, while also prompting a re-evaluation of the necessity of HOMP for complex relational modeling.

Abstract

While Graph Neural Networks (GNNs) have proven highly effective at modeling relational data, pairwise connections cannot fully capture multi-way relationships naturally present in complex real-world systems. In response to this, Topological Deep Learning (TDL) leverages more general combinatorial representations -- such as simplicial or cellular complexes -- to accommodate higher-order interactions. Existing TDL methods often extend GNNs through Higher-Order Message Passing (HOMP), but face critical \emph{scalability challenges} due to \textit{(i)} a combinatorial explosion of message-passing routes, and \textit{(ii)} significant complexity overhead from the propagation mechanism. This work presents HOPSE (Higher-Order Positional and Structural Encoder), an alternative method to solve tasks involving higher-order interactions \emph{without message passing}. Instead, HOPSE breaks \emph{arbitrary higher-order domains} into their neighborhood relationships using a Hasse graph decomposition. This method shows that decoupling the representation learning of neighborhood topology from that of attributes results in lower computational complexity, casting doubt on the need for HOMP. The experiments on molecular graph tasks and topological benchmarks show that HOPSE matches performance on traditional TDL datasets and outperforms HOMP methods on topological tasks, achieving up to $7\times$ speedups over HOMP-based models, opening a new path for scalable TDL.

Figures (3)

• Figure 1: Neighborhood expansions of a complex. Given a complex ${\mathcal{T}}$ (left), three examples of strictly augmented Hasse graphs $\mathcal{G}_\mathcal{N}$ are illustrated corresponding to 4 neighborhood functions: adjacency of nodes w.r.t. edges (${\mathcal{A}}_{0,1}$), incidence from nodes to edges (${\mathcal{I}}_{0\to 1}$), incidence from faces to nodes (${\mathcal{I}}_{2\to 1}$), and incidence from nodes to faces (${\mathcal{I}}_{0\to 2}$).
• Figure 2: HOPSE pipeline. (A) Considering the collection of neighborhood functions ${\mathcal{N}}_C =\{{\mathcal{A}}_{0,1}, {\mathcal{I}}_{0\to 1}, {\mathcal{I}}_{2\to 1}, {\mathcal{I}}_{0\to 2}\}$, the input combinatorial complex $\mathcal{T}$ is decomposed into the corresponding Hasse graphs expansions $\{\mathcal{G}_{{\mathcal{N}}}\}_{{\mathcal{N}}_C}$. (B) PSEs are computed for each Hasse graph, $\mathbf{G}(\mathcal{G}_{\mathcal{N}_i})$, as defined in \ref{['eq:family_func_g']}. (C) Higher-order PSEs are derived via the target rank-aware aggregation $\psi_k$ (see \ref{['eq:learn_f']}). (D) The PSEs $\mathbf{X}_{r,k}$ are transformed using $f_{r,k}(\cdot)$, and initial cell features $\mathbf{Z}_r$ are embedded via $\varepsilon_r(\cdot)$, following \ref{['eq:x_cal_hat', 'eq:initial_features']}. (E) Rank-specific, structure-aware representations $\mathbf{H}_r$ are learned using $\Theta_r(\cdot)$, as described in \ref{['eq:final_embedding']}. (F) The final task-specific readout $\varphi(\cdot)$ integrates all $\mathbf{H}_r$ representations (\ref{['eq:task_specific']}).
• Figure 3: Illustration of a combinatorial complex (top) and different neighborhood types, grouped by the target entity: nodes (second row), edges (third row), and faces (bottom row).

Theorems & Definitions (28)

• Definition A.1
• Definition A.2: Set-type relation
• Definition A.3: Rank function
• Definition A.4: Simplicial Complex
• Example 1: 3D Surface Meshes
• Definition A.5: Cell complex
• Example 2: Molecular structures.
• Definition A.6: Combinatorial complex
• Example 3: Geospatial structures.
• Definition A.7: Featured topological domain