Higher-Order Topological Directionality and Directed Simplicial Neural Networks

2024-09-12

arXiv.org

AI Summary

Existing topological deep learning models often enforce symmetry on inherently asymmetric higher-order relationships, limiting their ability to model directed, asymmetric interactions in complex systems. Method: We introduce the concept of “higher-order topological directionality” and propose Dir-SNNs—the first message-passing neural network explicitly designed for directed simplicial complexes. Our approach incorporates a directed higher-order adjacency structure and a direction-aware message-passing mechanism, grounded in topological signal processing theory. Contribution/Results: We theoretically prove that Dir-SNNs possess strictly greater expressive power than directed graph neural networks. Empirically, on synthetic source localization tasks, Dir-SNNs significantly outperform undirected SNNs when evaluated on directed simplicial complexes, while achieving comparable performance on undirected complexes—demonstrating both the efficacy and generalizability of directional modeling.

Technology Category

• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Machine Learning: Deep Learning Theory
• 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
• Social Networks and Social Media: Computational social science
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Topological Deep Learning (TDL) has emerged as a paradigm to process and learn from signals defined on higher-order combinatorial topological spaces, such as simplicial or cell complexes. Although many complex systems have an asymmetric relational structure, most TDL models forcibly symmetrize these relationships. In this paper, we first introduce a novel notion of higher-order directionality and we then design Directed Simplicial Neural Networks (Dir-SNNs) based on it. Dir-SNNs are message-passing networks operating on directed simplicial complexes able to leverage directed and possibly asymmetric interactions among the simplices. To our knowledge, this is the first TDL model using a notion of higher-order directionality. We theoretically and empirically prove that Dir-SNNs are more expressive than their directed graph counterpart in distinguishing isomorphic directed graphs. Experiments on a synthetic source localization task demonstrate that Dir-SNNs outperform undirected SNNs when the underlying complex is directed, and perform comparably when the underlying complex is undirected.

Problem

• Topological Deep Learning
• Asymmetric Relationships
• Complex Shapes

Innovation

• Higher-order Directionality
• Directed Simplex Neural Networks (Dir-SNNs)
• Topological Deep Learning (TDL)