This paper presents a novel study of the oversmoothing issue in diffusion-based Graph Neural Networks (GNNs). Diverging from extant approaches grounded in random walk analysis or particle systems, we approach this problem through operator semigroup theory. This theoretical framework allows us to rigorously prove that oversmoothing is intrinsically linked to the ergodicity of the diffusion operator. This finding further poses a general and mild ergodicity-breaking condition, encompassing the various specific solutions previously offered, thereby presenting a more universal and theoretically grounded approach to mitigating oversmoothing in diffusion-based GNNs. Additionally, we offer a probabilistic interpretation of our theory, forging a link with prior works and broadening the theoretical horizon. Our experimental results reveal that this ergodicity-breaking term effectively mitigates oversmoothing measured by Dirichlet energy, and simultaneously enhances performance in node classification tasks.

Graph contrastive learning (GCL) has recently emerged as an effective learning paradigm to alleviate the reliance on labelling information for graph representation learning. The core of GCL is to maximise the mutual information between the representation of a node and its contextual representation (i.e., the corresponding instance with similar semantic information) summarised from the contextual scope (e.g., the whole graph or 1-hop neighbourhood). This scheme distils valuable self-supervision signals for GCL training. However, existing GCL methods still suffer from limitations, such as the incapacity or inconvenience in choosing a suitable contextual scope for different datasets and building biased contrastiveness. To address aforementioned problems, we present a simple self-supervised learning method termed Unifying Graph Contrastive Learning with Flexible Contextual Scopes (UGCL for short). Our algorithm builds flexible contextual representations with tunable contextual scopes by controlling the power of an adjacency matrix. Additionally, our method ensures contrastiveness is built within connected components to reduce the bias of contextual representations. Based on representations from both local and contextual scopes, UGCL optimises a very simple contrastive loss function for graph representation learning. Essentially, the architecture of UGCL can be considered as a general framework to unify existing GCL methods. We have conducted intensive experiments and achieved new state-of-the-art performance in six out of eight benchmark datasets compared with self-supervised graph representation learning baselines. Our code has been open-sourced.