Positional Encoding meets Persistent Homology on Graphs

2025-06-06

AI Summary

Message-passing GNNs suffer from inherent local inductive biases, limiting their ability to capture global graph structural properties (e.g., connectivity, cycles). While positional encodings (PE) and persistent homology (PH) show promise for encoding global structure, their expressive power boundaries remain uncharacterized rigorously. This work first proves that PE and PH are incomparable—neither dominates the other—and constructs explicit counterexamples where each fails while the other succeeds. We propose PiPE, a learnable, topology-aware positional encoding framework, and theoretically establish that PiPE strictly subsumes both PE and PH in expressive power. PiPE integrates multi-scale structural modeling, efficient PH computation, and differentiable topological feature embedding. Extensive experiments demonstrate significant improvements over state-of-the-art methods on molecular property prediction, graph classification, and out-of-distribution generalization. Our implementation is publicly available.

Technology Category

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

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

The local inductive bias of message-passing graph neural networks (GNNs) hampers their ability to exploit key structural information (e.g., connectivity and cycles). Positional encoding (PE) and Persistent Homology (PH) have emerged as two promising approaches to mitigate this issue. PE schemes endow GNNs with location-aware features, while PH methods enhance GNNs with multiresolution topological features. However, a rigorous theoretical characterization of the relative merits and shortcomings of PE and PH has remained elusive. We bridge this gap by establishing that neither paradigm is more expressive than the other, providing novel constructions where one approach fails but the other succeeds. Our insights inform the design of a novel learnable method, PiPE (Persistence-informed Positional Encoding), which is provably more expressive than both PH and PE. PiPE demonstrates strong performance across a variety of tasks (e.g., molecule property prediction, graph classification, and out-of-distribution generalization), thereby advancing the frontiers of graph representation learning. Code is available at https://github.com/Aalto-QuML/PIPE.

Problem

• Enhancing GNNs with structural information via PE and PH
• Theoretical comparison of PE and PH expressiveness in GNNs
• Designing PiPE to outperform PE and PH in graph tasks

Innovation

• Combines Positional Encoding with Persistent Homology
• Introduces PiPE for enhanced expressiveness
• Improves graph representation learning performance