This article proposes the augmentation of the adjacency model of networks for graph signal processing. It is assumed that no information about the network is available, apart from the initial adjacency matrix. In the proposed model, additional edges are created according to a Markov relation imposed between nodes. This information is incorporated into the extended-adjacency matrix as a function of the diffusion distance between nodes. The diffusion distance measures similarities between nodes at a certain diffusion scale or time, and is a metric adopted from diffusion maps. Similarly, the proposed extended-adjacency matrix depends on the diffusion scale, which enables the definition of a scale-dependent graph Fourier transform. We conduct theoretical analyses of both the extended adjacency and the corresponding graph Fourier transform and show that different diffusion scales lead to different graph-frequency perspectives. At different scales, the transform discriminates shifted ranges of signal variations across the graph, revealing more information on the graph signal when compared to traditional approaches. The scale-dependent graph Fourier transform is applied for anomaly detection and is shown to outperform the conventional graph Fourier transform.

中文翻译：

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通过扩散距离扩展邻接关系和与比例相关的图傅立叶变换

本文提出了用于图形信号处理的网络邻接模型的扩充。假定除了初始邻接矩阵之外，没有关于网络的信息可用。在提出的模型中，根据节点之间施加的马尔可夫关系创建其他边。该信息根据节点之间的扩散距离合并到扩展邻接矩阵中。扩散距离在特定的扩散比例或时间测量节点之间的相似性，并且是从扩散图采用的度量。类似地，提出的扩展邻接矩阵取决于扩散比例，这使得能够定义与比例相关的图傅立叶变换。我们对扩展的邻接关系和相应的图傅立叶变换进行了理论分析，并表明不同的扩散尺度导致不同的图频观点。与传统方法相比，该变换以不同的比例区分了整个图形上信号变化的变化范围，从而揭示了图形信号上的更多信息。标度相关的图傅立叶变换用于异常检测，并显示出优于传统图傅立叶变换的性能。