Higher-Order Topological Directionality and Directed Simplicial Neural Networks
This work addresses the limitation of forcibly symmetrizing asymmetric relationships in complex systems for researchers in topological deep learning, representing an incremental advancement by extending existing models to incorporate directionality.
The paper tackles the problem of processing asymmetric relational structures in topological deep learning by introducing a novel notion of higher-order directionality and designing Directed Simplicial Neural Networks (Dir-SNNs), which outperform undirected SNNs on directed complexes and perform comparably on undirected ones in a synthetic source localization task.
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.