Convolving Directed Graph Edges via Hodge Laplacian for Brain Network Analysis

Abstract

A brain network, viewed as a graph wiring different regions of interest (ROIs) in the brain, has been widely used to investigate brain dysfunction with various graph neural networks (GNNs). However, existing GNNs are built upon graph convolution that transforms measurements on the nodes, where ROI-wise features are not always guaranteed for brain networks. Therefore, the majority of existing graph analysis methods that rely on node features are inapplicable for network analysis unless a proxy such as node degree is provided. Moreover, the complex neurological interactions across different brain regions cannot be directly expressed in a simple node-to-node (i.e., 0-simplex) representation. In this paper, we propose a novel method, Hodge-Graph Neural Network (Hodge-GNN), that allows the GNN to directly derive desirable representations of graph edges and capture complex edge-wise topological features spatially via the Hodge Laplacian. Specifically, representing a graph as a simplicial complex holds a significant advantage over conventional methods that extract higher-order connectivity of a graph through hierarchical convolution in the spatial domain or graph transformation. The superiority of our method is validated in the Alzheimer’s Disease Neuroimaging Initiative (ADNI) study, in comparison to benchmarking GNNs as well as state-of-the-art graph classification models.