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.

本研究解决了传统图移位算子在图表示学习中的局限性，提出了一种新的中心性图移位算子（CGSO），通过全球中心性指标对邻接矩阵进行归一化。研究表明，CGSO在图信号处理中的谱特性表现良好，并且在多种真实数据集上，采用CGSO的图卷积网络和图注意力网络变种展现出优异性能。

图神经网络的中心性图移位算子