Higher-order Graph Convolutional Network with Flower-Petals Laplacians on Simplicial Complexes

TL;DR

This paper introduces a higher-order graph neural network model based on Flower-Petals Laplacians within simplicial complexes, enabling better detection of complex, higher-order interactions in graphs, with theoretical and empirical validation.

Contribution

The paper proposes a novel HiGCN model utilizing FP Laplacians for higher-order interaction modeling, improving expressiveness and scalability over existing SC-based GNNs.

Findings

Achieves state-of-the-art performance on multiple graph tasks.

Provides a scalable, flexible framework for higher-order interaction analysis.

Demonstrates theoretical advantages in expressiveness of the proposed model.

Abstract

Despite the recent successes of vanilla Graph Neural Networks (GNNs) on various tasks, their foundation on pairwise networks inherently limits their capacity to discern latent higher-order interactions in complex systems. To bridge this capability gap, we propose a novel approach exploiting the rich mathematical theory of simplicial complexes (SCs) - a robust tool for modeling higher-order interactions. Current SC-based GNNs are burdened by high complexity and rigidity, and quantifying higher-order interaction strengths remains challenging. Innovatively, we present a higher-order Flower-Petals (FP) model, incorporating FP Laplacians into SCs. Further, we introduce a Higher-order Graph Convolutional Network (HiGCN) grounded in FP Laplacians, capable of discerning intrinsic features across varying topological scales. By employing learnable graph filters, a parameter group within each FP Laplacian domain, we can identify diverse patterns where the filters' weights serve as a quantifiable measure of higher-order interaction strengths. The theoretical underpinnings of HiGCN's advanced expressiveness are rigorously demonstrated. Additionally, our empirical investigations reveal that the proposed model accomplishes state-of-the-art performance on a range of graph tasks and provides a scalable and flexible solution to explore higher-order interactions in graphs. Codes and datasets are available at https://github.com/Yiminghh/HiGCN.