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

TL;DR

HOPSE introduces a scalable, message-passing-free approach for higher-order relational data, leveraging Hasse graph decomposition to efficiently model complex interactions in graphs and topological structures.

Contribution

HOPSE provides a novel higher-order encoder that avoids message passing by decomposing domains via Hasse graphs, significantly improving scalability and efficiency.

Findings

HOPSE matches traditional TDL performance on molecular tasks.

HOPSE outperforms HOMP methods on topological benchmarks.

Achieves up to 7x speedups over message-passing models.

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.