Graph Neural Networks for Soft Semi-Supervised Learning on Hypergraphs

Graph-based semi-supervised learning (SSL) assigns labels to initially unlabelled vertices in a graph. Graph neural networks (GNNs), esp. graph convolutional networks (GCNs), inspired the current-state-of-the art models for graph-based SSL problems. GCNs inherently assume that the labels of interest are numerical or categorical variables. However, in many real-world applications such as co-authorship networks, recommendation networks, etc., vertex labels can be naturally represented by probability distributions or histograms. Moreover, real-world network datasets have complex relationships going beyond pairwise associations. These relationships can be modelled naturally and flexibly by hypergraphs. In this paper, we explore GNNs for graph-based SSL of histograms. Motivated by complex relationships (those going beyond pairwise) in real-world networks, we propose a novel method for directed hypergraphs. Our work builds upon existing works on graph-based SSL of histograms derived from the theory of optimal transportation. A key contribution of this paper is to establish generalisation error bounds for a one-layer GNN within the framework of algorithmic stability. We also demonstrate our proposed methods' effectiveness through detailed experimentation on real-world data. We have made the code available.