Higher-Order Topological Directionality and Directed Simplicial Neural Networks

TL;DR

The paper addresses the limitation of existing Topological Deep Learning methods in modeling asymmetric higher-order interactions by introducing higher-order directionality and Directed Simplicial Neural Networks (Dir-SNNs). Dir-SNNs operate on directed simplicial complexes using multiple directed adjacencies and boundary/coboundary operators to propagate messages along directed simplicial paths, extending both undirected TNNs and Dir-GNNs. The authors prove that Dir-SNNs have higher expressivity in distinguishing isomorphic structures and demonstrate superior performance on a synthetic edge-level source localization task when directionality is present, while remaining competitive on undirected data. This work enables principled modeling of complex asymmetric higher-order interactions and lays groundwork for theoretical analysis and real-data applications in topological deep learning.

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.

Figures (4)

• Figure 1: (Left)$2$-dimensional directed flag complexes homotopic to the $2$-sphere, thus indistinguishable by homology. (Right) However, examining paths along $(d_1, d_2)$ reveals circular flows: the first complex has a flow only in the upper hemisphere, while the second complex has flows in both the upper and lower hemispheres.
• Figure 2: Examples of edge adjacencies. $\sigma$ is the red edge in each subfigure. (a)/(b) $\mathcal{A}_{\downarrow,1}^{0,0}$/$\mathcal{A}_{\downarrow,1}^{1,1}$ connects $\sigma$ with the edges with whom it shares a target/source node; (c)/(d) $\mathcal{A}_{\downarrow,1}^{0,1}$/$\mathcal{A}_{\downarrow,1}^{1,0}$ connects $\sigma$ with the edges whom source/target node is the target/source node of $\sigma$.
• Figure 3: Examples of simplicial paths (in red) of 2-simplices (triangles, in blue). (a) The path along $(d_0,d_2)$, showing the simplices are equidirected; (b) the path along $(d_1,d_2)$, revealing a circular flow around a common source node.
• Figure 4: SNR vs classification accuracy of directed and undirected TNNs and GNNs on directed (left) and undirected (right) synthetic graphs.