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

2025-05-21

AI Summary

Existing graph neural networks (GNNs) struggle to model higher-order, multi-directional relationships prevalent in real-world systems. While topological deep learning (TDL) leverages structures such as simplicial complexes to support higher-order interactions, mainstream higher-order message passing (HOMP) methods suffer from combinatorial path explosion and computational redundancy, severely limiting scalability. This paper introduces the first **message-passing-free higher-order encoding framework**, achieving linear time complexity via Hasse graph decomposition while preserving both expressive power and permutation equivariance. Key innovations include: (i) hierarchical decoupling of the Hasse graph, (ii) encoding of higher-order compositional structures, and (iii) joint positional-structural embedding. Empirically, our method matches or surpasses state-of-the-art models on molecular property prediction and topological benchmarks, with up to 7× faster inference speed.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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 extit{(i)} a combinatorial explosion of message-passing routes, and extit{(ii)} significant complexity overhead from the propagation mechanism. To overcome these limitations, we propose HOPSE (Higher-Order Positional and Structural Encoder) -- a emph{message passing-free} framework that uses Hasse graph decompositions to derive efficient and expressive encodings over emph{arbitrary higher-order domains}. Notably, HOPSE scales linearly with dataset size while preserving expressive power and permutation equivariance. Experiments on molecular, expressivity and topological benchmarks show that HOPSE matches or surpasses state-of-the-art performance while achieving up to 7 $times$ speedups over HOMP-based models, opening a new path for scalable TDL.

Problem

• Addressing scalability in higher-order graph representations
• Eliminating combinatorial explosion in message-passing routes
• Reducing complexity in topological deep learning models

Innovation

• Uses Hasse graph decompositions for encoding
• Eliminates message-passing for scalability
• Scales linearly with dataset size