Sheaves Reloaded: A Directional Awakening

TL;DR

This paper introduces the Directed Sheaf Neural Network (DSNN), a novel graph neural network that explicitly incorporates edge directionality through the Directed Cellular Sheaf and a new sheaf Laplacian, leading to improved performance on real-world tasks.

Contribution

It proposes the Directed Cellular Sheaf and Directed Sheaf Laplacian, enabling SNNs to model directional information explicitly, a feature absent in prior models.

Findings

DSNN outperforms baseline methods on nine benchmarks.

The new sheaf Laplacian effectively captures directional information.

Incorporating directionality improves graph learning performance.

Abstract

Sheaf Neural Networks (SNNs) represent a powerful generalization of Graph Neural Networks (GNNs) that significantly improve our ability to model complex relational data. While directionality has been shown to substantially boost performance in graph learning tasks and is key to many real-world applications, existing SNNs fall short in representing it. To address this limitation, we introduce the Directed Cellular Sheaf, a special type of cellular sheaf designed to explicitly account for edge orientation. Building on this structure, we define a new sheaf Laplacian, the Directed Sheaf Laplacian, which captures both the graph's topology and its directional information. This operator serves as the backbone of the Directed Sheaf Neural Network (DSNN), the first SNN model to embed a directional bias into its architecture. Extensive experiments on nine real-world benchmarks show that DSNN consistently outperforms baseline methods.