  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.
