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

TL;DR

This work tackles the limitation of traditional GNNs in capturing higher-order interactions by introducing a higher-order Flower-Petals (FP) representation on simplicial complexes and a Higher-order Graph Convolutional Network (HiGCN) built upon FP Laplacians. By decomposing interactions into flower-core and petal bipartite graphs and applying learnable polynomial filters per FP domain, HiGCN learns multi-scale, order-specific patterns and enables quantification of higher-order interaction strengths through filter weights. The authors establish theoretical foundations via HWL/SHWL concepts, demonstrate expressiveness and equivariance, and validate results with state-of-the-art performance on node and graph classification, as well as simplicial data imputation. The framework offers a scalable, flexible approach to uncover and leverage higher-order structure in complex networks, with public-code availability for reproducibility.

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.

Figures (5)

• Figure 1: a shows several typical simplices and its collection forms SCs in b. Subfigures c and d visualize the higher-order incidence matrices $\mathcal{H}_p$ for $p=1$ and $2$, respectively.
• Figure 2: Visualization of the flower-petals model. Different HiGCN models employ different numbers of petals, with each petal containing simplices of identical order. a-d visualizes 1,2,4,3-HiGCN, respectively. The interaction between each petal and the flower core can be unwrapped as an individual bipartite graph. FP Laplacians are derived based on the random walk dynamics in the bipartite graphs, followed by various learnable convolution operations $g_p$ on each FP Laplacian basis.
• Figure 3: a, b, c and d visualize the stack of learned weights $|\gamma_{p,k}|$ under order $p=1,2,3,4 (P=4)$. e visualizes the stack of $|\gamma_{2,k}|$ for Texas under various relative densities $\rho_2$.
• Figure 4: Three non-isomorphic graphs that are indistinguishable by WL but distinguishable by HWL and SHWL with clique complex lifting. The incorporation of higher-order information in HWL and SHWL enables nodes to exhibit a more extensive diversity in color compared to the WL test, consequently yielding superior performance in both node-level and graph-level tasks.
• Figure 5: The modification process of the 1k null model. The left and right figures show the partial structures of the networks before and after modification, respectively.

Theorems & Definitions (22)

• Definition 3.1: Simplicial complexes, SCs
• Theorem 4.1
• Theorem 4.2
• Lemma A.1: Non-negativity of $\tilde{\mathcal{A}}_p$
• proof
• Lemma A.2: Non-negativity of $\mathcal{L}_p$
• proof
• Lemma A.3
• proof
• Theorem B.1