The emerging field of graph signal processing (GSP) allows one to transpose classical signal processing operations (e.g., filtering) to signals on graphs. The GSP framework is generally built upon the graph Laplacian, which plays a crucial role in studying graph properties and measuring graph signal smoothness. Here, instead, we propose the graph modularity matrix as the centerpiece of GSP to incorporate knowledge about graph community structure when processing signals on the graph but without the need for community detection. We study this approach in several generic settings, such as filtering, optimal sampling and reconstruction, surrogate data generation, and denoising. Feasibility is illustrated by a small-scale example and a transportation network data set as well as one application in human neuroimaging where community-aware GSP reveals relationships between behavior and brain features that are not shown by Laplacian-based GSP. This work demonstrates how concepts from network science can lead to new, meaningful operations on graph signals.

中文翻译：

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社区感知图信号处理：模块化定义了处理图信号的新方法

图信号处理 (GSP) 的新兴领域允许人们将经典的信号处理操作（例如，滤波）转换为图上的信号。GSP 框架通常建立在图拉普拉斯算子之上，它在研究图属性和测量图信号平滑度方面起着至关重要的作用。在这里，我们建议将图模块化矩阵作为 GSP 的核心，以便在处理图上的信号时结合有关图社区结构的知识，但不需要社区检测。我们在几种通用设置中研究这种方法，例如过滤、最佳采样和重建、替代数据生成和去噪。可行性通过一个小规模示例和交通网络数据集以及人类神经成像中的一个应用来说明，其中社区感知 GSP 揭示了行为与大脑特征之间的关系，而基于拉普拉斯的 GSP 未显示这些关系。这项工作展示了来自网络科学的概念如何导致对图形信号进行新的、有意义的操作。