Higher-Order Topological Directionality and Directed Simplicial Neural Networks (2409.08389v3)

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.

• The paper introduces higher-order topological directionality using face maps to capture interactions beyond conventional pairwise relations.
• It proposes Dir-SNNs, a novel message-passing neural network that processes both lower and upper adjacencies in directed simplicial complexes.
• Empirical results show Dir-SNNs outperform traditional SNNs and Dir-GNNs in source localization and graph distinction tasks.

Higher-Order Topological Directionality and Directed Simplicial Neural Networks

"Directed Simplicial Neural Networks (Dir-SNNs)" by Manuel Lecha et al. introduces a novel approach in the field of Topological Deep Learning (TDL) by leveraging higher-order directionality in directed simplicial complexes. This paper establishes theoretical advancements and empirical validations that emphasize the expressivity and practical effectiveness of Dir-SNNs.

Key Contributions

The paper makes a three-fold contribution to the field of TDL:

1. Higher-Order Topological Directionality:
   • Introduces the notion of higher-order directionality in simplicial complexes based on face maps.
   • This approach captures relationships in directed simplicial complexes that go beyond the pairwise interactions in traditional graph structures.
2. Directed Simplicial Neural Networks (Dir-SNNs):
   • Presents Dir-SNNs as message-passing neural networks specifically designed to operate on directed simplicial complexes.
   • These networks utilize newly defined directed adjacencies among simplices, incorporating both lower and upper adjacencies.
3. Expressive Power:
   • Provides theoretical and empirical evidence that Dir-SNNs can distinguish isomorphic directed graphs more effectively than their directed graph counterparts.
   • Demonstrates that Dir-SNNs, through their higher-order topological inductive biases, surpass the expressive capabilities of current Directed Graph Neural Networks (Dir-GNNs).

Theoretical Foundation

At the heart of the Dir-SNN framework is the concept of higher-order directionality in simplicial complexes, formalized using face maps. A directed simplicial complex is defined as a pair $(V, \Sigma)$ where $\Sigma$ is a collection of ordered tuples of vertices that satisfy the inclusivity property. The authors introduce two main types of adjacency relations:

• Lower $(k, i, j)$-adjacency: Defined via shared lower-dimensional simplices.
• Upper $(k, i, j)$-adjacency: Defined via common higher-dimensional simplices.

This innovative method leverages the theory of directed simplicial paths, generalizing the concept of directed paths to higher-order simplices. It provides a consistent way to define neighborhoods through both down and up adjacencies. The paper formalizes this in a manner coherent with classical simplicial theory, ensuring that the new constructs retain mathematical rigor.

Directed Simplicial Neural Networks

The architecture of Dir-SNNs is grounded in message passing. The networks propagate information through simplices by aggregating messages from adjacent simplices defined via the new higher-order topological directionality constructs. The update rules in the Dir-SNN layers are designed to capture these adjacencies explicitly, thus enhancing the network’s ability to model complex directed interactions effectively.

Empirical Validation

To validate their theoretical contribution, the authors conducted experiments on synthetic and toy datasets. One key experiment involved a source localization task, confirming the enhanced performance of Dir-SNNs over traditional SNNs and Dir-GNNs.

• Source Localization:
  • Within directed flag complexes, Dir-SNNs demonstrated superior accuracy in a source localization task compared to both undirected SNNs and directed GNNs. This underscores the importance of leveraging directed higher-order topological information.

An additional experiment with a toy dataset confirmed the ability of Dir-SNNs to distinguish non-isomorphic graphs that Dir-GNNs fail to separate. This provided a clear illustration of the increased expressive power that Dir-SNNs offer due to their use of directed higher-order structures.

Theoretical Implications

The expressivity results of Dir-SNNs, based on the Directed Simplicial Weisfeiler-Leman (D-SWL) test, suggest a significant development in the theoretical understanding of neural networks operating on combinatorial topological spaces. The increased expressivity of Dir-SNNs holds promise for more accurate modeling and learning from data structures that encode complex, asymmetric relationships.

Future Directions

The paper opens several avenues for future research in TDL:

• Advanced Theoretical Characterization:
  • Further exploration of the theoretical properties of Dir-SNNs, particularly their expressivity under different settings and their relation to other topological deep learning models.
• Spectral Theory:
  • Development of a spectral theory specific to Dir-SNNs to provide insights into their filtering properties.
• Real-world Applications:
  • Comprehensive benchmarking of Dir-SNNs on diverse real-world datasets to validate the practical applicability of their theoretical advantages.

Conclusion

Manuel Lecha et al.'s work presents a substantial contribution to TDL by extending it to incorporate higher-order directionality within directed simplicial complexes. By demonstrating the superior expressivity and practical effectiveness of Dir-SNNs, this paper paves the way for future innovations in neural networks that fully leverage the richness of combinatorial topological spaces.