Graph Shift Operators (GSOs), such as the adjacency and graph Laplacian matrices, play a fundamental role in graph theory and graph representation learning. Traditional GSOs are typically constructed by normalizing the adjacency matrix by the degree matrix, a local centrality metric. In this work, we instead propose and study Centrality GSOs (CGSOs), which normalize adjacency matrices by global centrality metrics such as the PageRank, $k$-core or count of fixed length walks. We study spectral properties of the CGSOs, allowing us to get an understanding of their action on graph signals. We confirm this understanding by defining and running the spectral clustering algorithm based on different CGSOs on several synthetic and real-world datasets. We furthermore outline how our CGSO can act as the message passing operator in any Graph Neural Network and in particular demonstrate strong performance of a variant of the Graph Convolutional Network and Graph Attention Network using our CGSOs on several real-world benchmark datasets.

翻译：图移位算子（GSOs），例如邻接矩阵和图拉普拉斯矩阵，在图论和图表示学习中扮演着基础性角色。传统的图移位算子通常通过度矩阵（一种局部中心性度量）对邻接矩阵进行归一化来构建。在本工作中，我们提出并研究了中心性图移位算子（CGSOs），它使用全局中心性度量（如PageRank、$k$-核或固定长度游走计数）对邻接矩阵进行归一化。我们研究了CGSOs的谱性质，从而能够理解它们对图信号的作用。我们基于不同的CGSOs定义并在多个合成与真实世界数据集上运行谱聚类算法，验证了这一理解。此外，我们概述了我们的CGSO如何作为任何图神经网络中的消息传递算子，并特别展示了使用我们的CGSOs的图卷积网络和图注意力网络变体在多个真实世界基准数据集上的优异性能。