Higher-Order Topological Directionality and Directed Simplicial Neural Networks

TL;DR

This paper introduces Directed Simplicial Neural Networks (Dir-SNNs), a novel topological deep learning model that leverages higher-order directionality to better handle asymmetric relationships in complex systems, outperforming undirected models.

Contribution

The paper presents the first TDL model incorporating higher-order directionality, enabling Dir-SNNs to process directed simplicial complexes and capture asymmetric interactions.

Findings

Dir-SNNs are more expressive than directed graph models.

Dir-SNNs outperform undirected SNNs on directed complex tasks.

Dir-SNNs perform comparably to undirected models on undirected data.

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.