In many real-world network datasets such as co-authorship, co-citation, email communication, etc., relationships are complex and go beyond pairwise. Hypergraphs provide a flexible and natural modeling tool to model such complex relationships.

Graph Convolutional Networks (GCNs) are known to suffer from performance degradation as the number of layers increases, which is usually attributed to over-smoothing. Despite the apparent consensus, we observe that there exists a discrepancy between the theoretical understanding of over-smoothing and the practical capabilities of GCNs. Specifically, we argue that over-smoothing does not necessarily happen in practice, a deeper model is provably expressive, can converge to global optimum with linear convergence rate, and achieve very high training accuracy as long as properly trained. Despite being capable of achieving high training accuracy, empirical results show that the deeper models generalize poorly on the testing stage and existing theoretical understanding of such behavior remains elusive. To achieve better understanding, we carefully analyze the generalization capability of GCNs, and show that the training strategies to achieve high training accuracy significantly deteriorate the generalization capability of GCNs. Motivated by these findings, we propose a decoupled structure for GCNs that detaches weight matrices from feature propagation to preserve the expressive power and ensure good generalization performance. We conduct empirical evaluations on various synthetic and real-world datasets to validate the correctness of our theory.

The relationships of many real-world networks are complex and go beyond pairwise associations. Hypergraphs provide a flexible and natural modeling tool to model such complex relationships. The authors propose HyperGCN, a novel way of training a GCN for semi-supervised learning on hypergraphs using tools from spectral theory of hypergraphs and introduce FastHyperGCN. They conduct some experiments on co-authorship and co-citation hypergraphs to demonstrate the effectiveness of HyperGCN, and provide theoretical analyses for the results. The paper proposes 1-HyperGCN and HyperGCN using the hypergraph Laplacian and the generalized hypergraph Laplacian with mediators.

Graph Convolutional Networks (GCNs) are known to suffer from performance degradation as the number of layers increases, which is usually attributed to over-smoothing. Despite the apparent consensus, we observe that there exists a discrepancy between the theoretical understanding of over-smoothing and the practical capabilities of GCNs. Specifically, we argue that over-smoothing does not necessarily happen in practice, a deeper model is provably expressive, can converge to global optimum with linear convergence rate, and achieve very high training accuracy as long as properly trained. Despite being capable of achieving high training accuracy, empirical results show that the deeper models generalize poorly on the testing stage and existing theoretical understanding of such behavior remains elusive. To achieve better understanding, we carefully analyze the generalization capability of GCNs, and show that the training strategies to achieve high training accuracy significantly deteriorate the generalization capability of GCNs. Motivated by these findings, we propose a decoupled structure for GCNs that detaches weight matrices from feature propagation to preserve the expressive power and ensure good generalization performance.

In many real-world network datasets such as co-authorship, co-citation, email communication, etc., relationships are complex and go beyond pairwise. Hypergraphs provide a flexible and natural modeling tool to model such complex relationships. A popular learning paradigm is hypergraph-based semi-supervised learning (SSL) where the goal is to assign labels to initially unlabeled vertices in a hypergraph. Motivated by the fact that a graph convolutional network (GCN) has been effective for graph-based SSL, we propose HyperGCN, a novel GCN for SSL on attributed hypergraphs. Additionally, we show how HyperGCN can be used as a learning-based approach for combinatorial optimisation on NP-hard hypergraph problems.

Graph convolutional networks (GCNs) have recently achieved great empirical success in learning graph-structured data. To address its scalability issue due to the recursive embedding of neighboring features, graph topology sampling has been proposed to reduce the memory and computational cost of training GCNs, and it has achieved comparable test performance to those without topology sampling in many empirical studies. To the best of our knowledge, this paper provides the first theoretical justification of graph topology sampling in training (up to) three-layer GCNs for semi-supervised node classification. We formally characterize some sufficient conditions on graph topology sampling such that GCN training leads to a diminishing generalization error. Moreover, our method tackles the nonconvex interaction of weights across layers, which is under-explored in the existing theoretical analyses of GCNs. This paper characterizes the impact of graph structures and topology sampling on the generalization performance and sample complexity explicitly, and the theoretical findings are also justified through numerical experiments.